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# Constrained Optimization Matlab

Main features. Equality constrained minimization • equality constrained minimization • eliminating equality constraints • Newton's method with equality constraints • infeasible start Newton method • implementation 11-1. that's mean that matlab don't help to solve an integer constrained optimization problem using genetic algorithme. This reference section describes the original programming interface (API) of NLopt, used in versions of NLopt prior to 2. bharani@aero. An improved (μ λ)-constrained differential evolution for constrained optimization. Are there any comparisons testing their robustness against each other? I know finding a global optimum is not easy. Gibson Department of Mathematics Applied Math and Computation Seminar October 21, 2011 Prof. Optimization is the science of making a best choice in the face of conflicting requirements. The constraints are: Sum of all the numbers in F (10+10+5) < sum of all the numbers in S (8+8+8+4). The secondary optimization goal was to minimize required maximum force, torque, work, and power while maintaining the original maximum club head velocity. Iam able to perform the optimization in MATLAB using the fmincon() function which seems to use an interior-point or a trust-region-reflective. FunctionTolerance and constraint violation is less than options. Professors Walter Murray and Michael Saunders lead the SOL research program on constrained optimization, in close cooperation with Professor Philip Gill at UC San Diego. Welcome to our tutorial about Solvers for Excel and Visual Basic -- the easiest way to solve optimization problems -- from Frontline Systems, developers of the Solver in Microsoft Excel. txt) or read online. Constraint Satisfaction • Global search algorithms – Genetic algorithms • What is a constraint satisfaction problem (CSP) • Applying search to CSP • Applying iterative improvement to CSP COMP-424, Lecture 5 - January 21, 2013 1 Recall from last time: Optimization problems. This step-by-step tutorial demonstrates fmincon solver on a nonlinear optimization problem with one equality and one inequality constraint. The BOBYQA algorithm for bound constrained optimization without derivatives by M. But, for a constrained minimizer of the -penalized least-squares optimization problem, this case does not occur. Matlab Optimization Problem. This is why the study of unconstrained optimization is useful as a start. Mykel Kochenderfer, professor of Aeronautics and Astronautics at Stanford University, through a grant from the National Science Foundation. If the model has two variables, the graphical method can be used to solve the model. Specifically Octave can solve problems in Linear Programming, Quadratic Programming, Nonlinear Programming, and Linear Least Squares Minimization. Moreover, the constraints that appear in these problems are typically nonlinear. Elena Gutierrez. This behavior can be explained by constrained optimization of cost of transport (metabolic cost/distance) where cost of transport is seen as the main component of an underlying objective function that is minimized within the limitations of. Calls with Gradients Supplied Matlab's HELP DESCRIPTION. We will cover the most fundamental concepts in the field of optimization including metaheuristics and swarm intelligence. • The difference: after a subspace solve: compute the new derivative and update TR. You can include nonlinear constraints by writing a function that computes both equality and inequality constraint values. It is MPC problem. Pso Clustering Matlab Code. This MATLAB function creates an N-by-1 array of empty optimization constraints. I'll start by showing you how to install Solver, how to organize a worksheet for use in Solver, and how to find a solution to an optimization problem. Optional numerical differentiation. This problem is a little different from the previous problems. but I am not sure how to use both function handles as constraints at once. Super useful! If you're seeing this message, it means we're having trouble. txt) or read online for free. Selected Bibliography [1] "The Convergence of Variable Metric Methods for Nonlinearly Constrained Optimization Calculations," Nonlinear Programming 3,. Please find below the ATOMS description of the toolbox. That way, the problem can be converted to a single-variable constrained optimization problem in t, which can be solved using fminbnd or fmincon or fseminf. Many economic models involve optimization over time. gz, 59K) for constrained global optimization problems together with test results for his branch and bound algorithm. Therefore I need to solve a constrained optimization problem, thus finding the vector that mimimises a certain multi-variable function, subject to constraints. For watching full course of Numerical Computations, visit this page. They have to agree with the. Constrained minimization is the problem of finding a vector x that is a local minimum to a scalar function f(x) subject to constraints on the allowable x:. Provides all the tools needed to begin solving optimization problems using MATLAB® The Second Edition of Applied Optimization with MATLAB® Programming enables readers to harness all the features of MATLAB® to solve optimization problems using a variety of linear and nonlinear design optimization techniques. Constraint optimization, or constraint programming (CP), is the name given to identifying feasible solutions out of a very large set of candidates, where the problem can be modeled in terms of arbitrary constraints. If you know of useful sources not listed here, please let us know. This work entails a comparative study of several popu-lar optimization methods applied to the solution of the fully constrained production optimization problem. Constrained minimization of multivariate scalar functions (minimize)¶ The minimize function also provides an interface to several constrained minimization algorithm. but I am not sure how to use both function handles as constraints at once. Abstrmt-The MATLAB. General-purpose Optimization Description. Bieniawski⁄ and Ilan M. CP problems arise in many scientific and engineering disciplines. minimize) with a large equality constraint matrix. Matlab for Optimization. This problem is a little different from the previous problems. Ibrahim Mtolera. Chance constraint modeling can be used to ensure a high performance level of the robots given uncertainties in the qualities of the different parts of the robots as well as in the construction of the robots. Constrained Nonlinear Optimization Algorithms Constrained Optimization Definition. Abramson, Mark A. under the following constraint sum of theta^2 + e = 1, where e is a constant. Lagrange multipliers and constrained optimization Math · Multivariable calculus · Applications of multivariable derivatives · Optimizing multivariable functions (articles) Maxima, minima, and saddle points. Power Generation Loading Optimization using a Multi-Objective Constraint-Handling Method via PSO Algorithm Lily D Li1, Xiaodong Li2, and Xinghuo Yu3 1School of Computing Sciences, Central Queensland University. Solve Constrained Nonlinear Optimization, Problem-Based. For large scale problems this approach quickly becomes too computationally expensive, and there is a huge field of mathematics devoted to more efficient methods. Very few real world problems involve only two variables. Abstrmt-The MATLAB. The algorithm is based on linear approximations to the objective function and each constraint. Watch Queue Queue. Bennett Office: Amos Eaton 329 Office Hours: Tuesday 10:15-11:30 AM, Thursday 10:15-11:30 AM Phone: 276-6899. Interior Methods for Constrained Optimization - Download as PDF File (. Optimization (nonlinear and quadratic) L-BFGS and CG Unconstrained optimization. Nonlinear Trajectory Optimization with Path Constraints Applied to Spacecraft Reconﬁguration Maneuvers by Ian Miguel Garc´ıa S. This borrowing constraint rules out Ponzi-schemes and if ebis a large enough (negative) number then this constraint is unlikely to be binding. The paper M. sequential quadratic programming algorithm nonconvex nonsmooth constrained optimization constraint function problem function matlab implementation line search algorithm open dense subset optimization problem type arise stationary point sequential quadratic programming gradient sampling sqp-gs method important application search direction. The approach in these methods is to transform the constrained optimization problem into an equivalent unconstrained problem and solved using one of the algorithms for unconstrained optimization problems. The holy grail of constrained optimization is the development of an efficient, scale invariant, and generic constraint-handling procedure in single- and multi-objective constrained optimization. It uses an interface very similar to the Matlab optimization Toolbox function fminunc, and can be called as a replacement for this function. Gibson (OSU) Gradient-based Methods for Optimization AMC 2011 1 / 40. Hello! I looking for some matlab function (Constrained Optimization) which one returns a vector. Follows an incomplete list of stuff missing in the optimization package to be matlab compatible. Optional numerical differentiation. You can view. 6 Solving Unconstrained and Constrained Optimization Problems This section describes how to define and solve unconstrained and constrained optimization problems. This is a rather more di-cult subject. Live tutors are available for 24x7 hours helping students in their Constrained Optimization Models related problems. End-of-Chapter Solutions and MATLAB Programs. This work entails a comparative study of several popu-lar optimization methods applied to the solution of the fully constrained production optimization problem. This example shows how to include nonlinear inequality constraints in a surrogate optimization by using a penalty function. We write g(x)+z = b, z ≥0. Robinson is an Assistant Professor in the Applied Mathematics and Statistics department at Johns Hopkins University. Is this a correct approach? Is there any efficient method to include this implicit constraints 0. 4 0 0 4 16256 14052. Solve constrained or unconstrained nonlinear problems with one or more objectives, in serial or parallel To set up a nonlinear optimization problem for solution, first decide between a problem-based approach and solver-based approach. Optimization of Tree Pipe Networks Layout and Size, Using Particle Swarm Optimization. MATLAB Central contributions by Xin-She Yang. You can include nonlinear constraints by writing a function that computes both equality and inequality constraint values. tion problem with bound constraints. Calls with Gradients Supplied Matlab's HELP DESCRIPTION. The method wraps a FORTRAN implementation of the algorithm. 4 seconds to fully settle down. We now proceed to the next level: solving two-variable problems in which there is a constraint on the actions of the optimizing agent. 4 0 0 4 16256 14052. the constraint set. Although stochastic optimization refers to any optimization method that employs randomness within some communities, we only consider settings where the objective function or constraints are random. The function what I try minimalize is in the attachment. TOMLAB, a commercial, Matlab based optimization environment for dense nonlinear programming. Method COBYLA uses the Constrained Optimization BY Linear Approximation (COBYLA) method , ,. Hello! I looking for some matlab function (Constrained Optimization) which one returns a vector. Sequential Quadratic Programming. The constraint is that the overall area of the poster must be 200 in 2 while we want to optimize the printed area (i. 2 refers to the corresponding Lemma. The paper M. Open Live Script. m Constrained boundary condition function simple_constraint. The obtained fastest response of the closed-loop system with the satisfied overshoot limit is shown in Figure 3, where the system takes more than 0. Constraint Satisfaction • Global search algorithms – Genetic algorithms • What is a constraint satisfaction problem (CSP) • Applying search to CSP • Applying iterative improvement to CSP COMP-424, Lecture 5 - January 21, 2013 1 Recall from last time: Optimization problems. Note the warning about the algorithm change. If the model has two variables, the graphical method can be used to solve the model. This (kinetic minimization) optimization was applied to subject 1 only. Development. There's s, the tons of steel that you're using, h the hours of labor, and then lambda, this Lagrange Multiplier we introduced that's basically a proportionality constant between the gradient vectors of the revenue function and the constraint function. The project is about Phase linear Optimization. com/patrickjmt !! Please consider supporting me. 25 Optimization. The values of the variables are constrained by upper and lower bounds. For constrained minimization of an objective function f(x) (for maximization use -f), Matlab provides the command fmincon. Shaw Research February 25, 2008 R. To evaluate a novel optimization algorithm’s property on different types of landscapes Compare different optimization algorithms Types of benchmarks Bound constrained problems (real, binary, discrete, mixed) Constrained problems Single / Multi-objective problems Static / Dynamic optimization problems Multimodal problems. Nonlinear Trajectory Optimization with Path Constraints Applied to Spacecraft Reconﬁguration Maneuvers by Ian Miguel Garc´ıa S. Global Optimization with MATLAB. optimization package for MATLAB. can I do it by global syntax?!. Pso Clustering Matlab Code. Wah, Yixin Chen and Tao Wang (September 1st 2008). Accept 5 answers given by other contributors. Search Search. 2) versus the solving the constrained problem (2. m – l1 constrained optimization. Hi, I am facing a complex optimization problem. Solving Optimization Problems using the Matlab Optimization Toolbox - a Tutorial TU-Ilmenau, Fakultät für Mathematik und Naturwissenschaften Dr. Global Optimization with MATLAB. MATLAB Optimization Techniques. Essentials of Convex Optimization Max Welling Department of Computer Science University of Toronto 10 King’s College Road Toronto, M5S 3G5 Canada welling@cs. Global Optimization. m Constrained boundary condition function simple_constraint. GOV Journal Article: Modeling an Augmented Lagrangian for Improved Blackbox Constrained Optimization. Specifically Octave can solve problems in Linear Programming, Quadratic Programming, Nonlinear Programming, and Linear Least Squares Minimization. MOSQP is a multiobjective optimization solver for bound, linear, and nonlinear constrained problems (for which the derivatives of the objective functions and constraints exist). This example shows how to solve a constrained nonlinear problem using an Optimization Toolbox™ solver. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. Solve multiobjective optimization problems in serial or parallel Solve problems that have multiple objectives by the goal attainment method. Constrained Optimization. Welcome to Solving Optimization and Scheduling Problems in Excel. The Design Optimization Toolkit (DOTk) is a stand-alone C++ software package intended to solve complex design optimization problems. All books are in clear copy here, and all files are secure so don't worry about it. TOMLAB Base Module efficiently integrates five Fortran solvers: Tlsqr, LSEI, Tnnls, QLD and Tfzero. In this video tutorial, “Optimization” has been reviewed and implemented using MATLAB. His research focuses on large-scale optimization methods with applications in machine learning, optimal control, trajectory optimization, financial mathematics, and computational mathematics. We have to set it through the usual optimoptions function: opts = optimoptions(@fmincon,'Algorithm','sqp') MATLAB assumes the following form for a constrained problem:. This technique enables solvers that do not normally accept nonlinear constraints to attempt to solve a nonlinearly constrained problem. Skickas inom 3-6 vardagar. Constrained nonlinear optimization Version: 1-11-2012 Material (for details see ) Chapter 12 in [FKS] (pp. Quadratic programming (QP) is the process of solving a special type of mathematical optimization problem—specifically, a (linearly constrained) quadratic optimization problem, that is, the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables. You can set or change the values of these parameters using the optimset function. An Exact ε-constraint Method for Bi-objective Combinatorial Optimization Problems – Application to the Traveling Salesman Problem with Profits Jean-François Bérubé1,*, Michel Gendreau1 , Jean-Yves Potvin1 1 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT),. The obtained fastest response of the closed-loop system with the satisfied overshoot limit is shown in Figure 3, where the system takes more than 0. MatLab Optimization Tool Box) Where m are the number of inequality constraints and q the number of equality constraints Denoting the optimization variables X, as a n-dimensional vector, where the n variables are its componets, and the objective function F(X) we search for : Regional constraints Behavior constraints. matlab constrained-optimization MATLAB Updated Feb 1, 2019. OPTI Toolbox Wiki If you are interested in optimization, use MATLAB and like free stuff, OPTI could be for you. txt (source code zip) Adaptive Diferential Evolution Algorithms, Handbook of Optimization, 2012 Source code in Matlab or C download readmeHO. Toolbox solvers include surrogate, pattern search, genetic algorithm, particle swarm, simulated annealing, multistart, and global search. If you know of useful sources not listed here, please let us know. This borrowing constraint rules out Ponzi-schemes and if ebis a large enough (negative) number then this constraint is unlikely to be binding. KNITRO is preferred only if second derivatives can be supplied. Find more Mathematics widgets in Wolfram|Alpha. In this work, we introduce volume optimization problems where enables us to obtain a convex formulation for different challenging problems in the area of system and control. The "Lagrange multipliers" technique is a way to solve constrained optimization problems. The function we seek to maximize is an unbounded plane, while the constraint is a unit circle. 3 Representation of constraints We may wish to impose a constraint of the form g(x) ≤b. Discover what MATLAB. Provide your first answer ever to someone else's question. The toolbox includes routines for many types of optimization including: •Unconstrained nonlinear minimization •Constrained nonlinear minimization, including goal attainment problems,. Hello! I looking for some matlab function (Constrained Optimization) which one returns a vector. Artificial Bee Colony (ABC) Algorithm Artificial Bee Colony (ABC) is one of the most recently defined algorithms by Dervis Karaboga in 2005, motivated by the intelligent behavior of honey bees. Optimization routines in the IMSL Libraries include unconstrained, linearly and nonlinearly constrained minimizations, and linear programming algorithms. Stochastic ranking for constrained evolutionary optimization (Matlab code by Runarsson and Yao) GLOBAL, Derivative-Free Boender-Timmer-Rinnoy Kan Algorithm in Fortran 77 and C (by Tibor Csendes) Fortran 95 version (translated by Alan Miller) StoGO, stochastic bound constrained global optimization using gradients (in C++). imposing some kind of borrowing constraint. Because other solvers like LINGO or CEPLEX find the global optimal solution with all constraints (equality and inequality) satisfaction. Since , we aren't really changing the original function, provided that the constraint is met!. ConstraintTolerance. PDCO: MATLAB software implementing a primal-dual interior method for sparse linear programming, least squares, or convex optimization subject to linear constraints $$Ax=b$$, $$l \le x \le u$$. Search MATLAB Documentation Release Notes for Global Optimization Toolbox, On this page, R2015a Global Optimization Toolbox Release Notes. A NumPy-based solution would be ideal, but any language will do. The developed results are illustrated with several examples, including linear and nonlinear ones. There's s, the tons of steel that you're using, h the hours of labor, and then lambda, this Lagrange Multiplier we introduced that's basically a proportionality constant between the gradient vectors of the revenue function and the constraint function. Basic properties of solutions and algorithms. When our constraints also have inequalities, we need to extend the method to the Karush-Kuhn-Tucker (KKT) conditions. You can include nonlinear constraints by writing a function that computes both equality and inequality constraint values. • Firms make production decisions to maximize their profits subject to. A multi-objective resource-constrained optimization of time-cost trade-off problems in scheduling project: Article 8, Volume 8, Issue 4, Autumn 2015, Page 653-685 PDF (457. Development. gz, 59K) for constrained global optimization problems together with test results for his branch and bound algorithm. Solve constrained or unconstrained nonlinear problems with one or more objectives, in serial or parallel To set up a nonlinear optimization problem for solution, first decide between a problem-based approach and solver-based approach. A multi-objective resource-constrained optimization of time-cost trade-off … 655. Constrained Optimization Engineering design optimization problems are very rarely unconstrained. There is a lot of output, that mostly tells us the function worked as expected. This is likely to be slow, but it seems like the only option unless you want to recode that app's objective function within MATLAB. Today, we use the builtin function fmincon in Matlab to solve the same problem. global-optimization black-box-optimization constrained-optimization surrogate-based-optimization optimization solver matlab open-source MATLAB Updated Jun 24, 2019 yonghoonlee / MO-ASMO-II. Constrained Multiobjective Optimization Problem. Basic Calls (without any special options) Example1 Example 2 B. Constrained minimization is the problem of finding a vector x that is a local minimum to a scalar function f(x) subject to constraints on the allowable x:. Bennett Office: Amos Eaton 329 Office Hours: Tuesday 10:15-11:30 AM, Thursday 10:15-11:30 AM Phone: 276-6899. Optimization (nonlinear and quadratic) L-BFGS and CG Unconstrained optimization. Introduction to non-linear optimization Ross A. —In this paper, we propose a distributed algorithm to solve the Security Constrained Optimal Power Flow (SC-OPF) Problem. Heinkenschloss: Numerical Solution of Implicitly Constrained Optimization Problems discusses the application of optimization algorithms for the solution of (3). A constraint is a hard limit placed on the value of a variable, which prevents us. Best Max Stall Generation Func-count f(x) Constraint Generations 1 2670 13578. Awarded to Sukanya J Nair on 03 Aug 2017. Solve constrained or unconstrained nonlinear problems with one or more objectives, in serial or parallel To set up a nonlinear optimization problem for solution, first decide between a problem-based approach and solver-based approach. The given objective function is subject to nonlinear. bharani@aero. Wed, Feb 12th: Constrained optimization, SQP method & Matlab Study this before the lecture! Questions to be considered -What is the basic idea of gradient projection? -What is the basic idea of active set methods? -What is the basic idea of Sequential Quadratic Programming (SQP)? spring 2014 TIES483 Nonlinear optimization. Process optimization is the discipline of adjusting a process so as to optimize (make the best or most effective use of) some specified set of parameters without violating some constraint. [SOLVED] Multivariate Linear Regression With Coefficient Constraint I'm attempting a multivariate linear regression (mvlr) by method of least squares. Wolpertz NASA Ames Research Center, Moﬁett Field, CA 94035. MATLAB Central contributions by Xin-She Yang. Absolute value functions themselves are very difficult to perform standard optimization procedures on. Hacky unconstrained vs. 0 (NonLinear Programming LIBrary); a set of Matlab solvers, test computational utilities for unconstrained and constrained optimization, quadratic programming, unconstrained and constrained nonlinear least. Optimization (e. Solving and Optimizing in Matlab 1 SOLVING AND OPTIMIZING IN MATLAB Note that you can copy code from the pdf and paste into a Matlab editor window to try out the code, or look for the code in Matlab/SuppExamples. pyOpt is an open-source software distributed under the tems of the GNU Lesser General Public License. LINDO Solver with an API for large scale optimization of linear, integer, quadratic, conic and general nonlinear programs with stochastic programming extensions. The MATLAB Optimization Toolbox includes solvers for unconstrained and constrained nonlinear optimization, least-squares optimization, as well as algorithms for linear and quadratic programming. Tutorial for the Optimization Toolbox™ Tutorial example showing how to solve nonlinear problems and pass extra parameters. I am not sure if I really need optimization toolbox or not. Gradient for nonlinear constraint functions defined by the user. The constraint is that the overall area of the poster must be 200 in 2 while we want to optimize the printed area (i. “Every problem is an optimization problem. Because other solvers like LINGO or CEPLEX find the global optimal solution with all constraints (equality and inequality) satisfaction. Both the constraint and the function we are going to optimize are areas. Solving the bound constrained subproblem • It is an iterative bound constrained optimization algorithm with trust-region: • Each step solves a bound constrained QP (not necessarily PD), same as in your homework 4. We have to set it through the usual optimoptions function: opts = optimoptions(@fmincon,'Algorithm','sqp') MATLAB assumes the following form for a constrained problem:. Documentation. Example problems include analyzing design tradeoffs, selecting optimal product or process designs, or any other application where you need an optimal solution with tradeoffs between two or more conflicting objectives. Uncertainty, Design, and Optimization Department of Civil and Environmental Engineering Duke University Henri P. Box and linearly constrained optimization Linearly equality/inequality (and box) constrained optimization. More than 36 million people use GitHub to discover, fork, and contribute to over 100 million projects. Optimization is the science of making a best choice in the face of conflicting requirements. Gradient-based Methods for Optimization. You can order your own copy of Matlab for \$99 or you can use Matlab on the Courant Sparcstation network (or dial in from home). Given a transformation between input and output values, described by a mathematical function f, optimization deals with generating and selecting a best solution from some set of available alternatives, by systematically choosing input values from within an allowed set, computing the output of the function, and recording the best output values found during the process. In this tutorial, I use Matlab's fmincon—alternatives include SNOPT and IPOPT. 021e-06 0 3 8394 14034. Nonlinear Constrained Optimization: Methods and Software Sven Leyfferyand Ashutosh Mahajan z March 17, 2010 Abstract We survey the foundations of nonlinearly constrained optimization methods, emphasiz-ing general methods and highlighting their key components, namely, the local model and global convergence mechanism. From ( 2 ), the - regularization minimum-variance portfolio model also has the following equivalent multivariate regression form: The Lagrangian corresponding to the optimization problem stated in ( 34 ) is When , we have and. Global Optimization Toolbox provides methods that search for global solutions to problems that contain multiple maxima or minima. I understand that fmincon is what I should use by playing with the argument @mycon b. We have to set it through the usual optimoptions function: opts = optimoptions(@fmincon,'Algorithm','sqp') MATLAB assumes the following form for a constrained problem:. Quasi-Newton methods. Constrained Solver-Based Applications. 4: Lagrange Multipliers and Constrained Optimization A constrained optimization problem is a problem of the form maximize (or minimize) the function F(x,y) subject to the condition g(x,y) = 0. ca with a subject heading \A3 solutions". Gibson (OSU) Gradient-based Methods for Optimization AMC 2011 1 / 40. We consider a network of devices, each with its own dynamic constraints and objective, subject to reliability constraints across. All of the toolbox functions are MATLAB M-files, made up of MATLAB statements that implement specialized optimization algorithms. Suppose we seek to minimize the function $$f(x,y)=x+y$$ subject to the constraint that $$x^2 + y^2 = 1$$. 2 Posynomial 492 8. 7 0 0 5 18856 13573. Iam able to perform the optimization in MATLAB using the fmincon() function which seems to use an interior-point or a trust-region-reflective. For example, you can express the constraints that each row of a matrix variable x sums to one in this single expression. M-files accompanying the webinar titled "New Approaches to Constrained Optimization in MATLAB" held on November 05, 2005 These files provide what you need to run the two demos: 1) The first demo is a demonstration of bound constrained optimization of non-smooth objective function using GA and also using a Hybrid function (using GA with another. • Firms make production decisions to maximize their profits subject to. Numerical Operations: U of M-Dearborn ECE Department Introduction to MATLAB and its Toolboxes Optimization 2 Optimization Topics n Unconstrained Optimization Unconstrained Optimization n Constrained Optimization Constrained Optimization n Constrained Box Example Constrained Box Example. 0 are the algorithms for general and separable nonlinear least squares parameter estimation. When set to the default, false, fmincon estimates gradients of the nonlinear constraints by finite differences. † Joint Chance Constraints ƒ T (!) xed, h … P, with P quasi-concave) K1(ﬁ) is convex Use Lagrangian Approach a Reduced Gradient NLP approach † Otherwise Very Hard. Iterative Linear Algebra for Constrained Optimization Hilary Dollar, Keble College, University of Oxford A thesis submitted for the degree of Doctor of Philosophy Michaelmas 2005 Abstract Each step of an interior point method for nonlinear optimization requires the solution of a symmetric indeﬁnite linear system known. the constraint set. 07 KB, 22 pages and we collected some download links, you can download this pdf book for free. CUTEr is a versatile testing environment for optimization and linear algebra solvers. constrained numerical Learn more about constrained optimization MATLAB. Basically, I'm solving a matrix of the following form for $$\beta_p,$$[tex]. It is based on Stephen Boyd’s book, chapter 5 (available online). Box and linearly constrained optimization Linearly equality/inequality (and box) constrained optimization. • Constrained optimization and convex optimization Several computation projects which require students to use Matlab to implement specific numerical optimization techniques and to solve interesting application problems will be. bharani@aero. So I use fmincon and it works. matlab constrained-optimization MATLAB Updated Feb 1, 2019. Constrained Problem-Based Applications. the area of the poster with the margins taken out). Wolpertz NASA Ames Research Center, Moﬁett Field, CA 94035. In particular, the methodology known as sequential quadratic programming(SQP) has had one of the longest and richest histories [21, 36]. Author links open overlay panel H. Optimization with absolute values is a special case of linear programming in which a problem made nonlinear due to the presence of absolute values is solved using linear programming methods. txt (source code zip) Adaptive Diferential Evolution Algorithms, Handbook of Optimization, 2012 Source code in Matlab or C download readmeHO. L-BFGS-B is a collection of Fortran 77 routines for solving nonlinear optimization problems with bound constraints on the variables. 1 Lagrangians and all that. The given objective function is a simple function. The constraint function you posted is linear, and I believe you wouldn't use it as a nonlinear constraint, but as a linear one. This example shows how to solve a constrained nonlinear problem using an Optimization Toolbox™ solver. Optimization Problem that can be solve in MATLAB (iii(Optimization Too lb )lbox) • Constrained and Unconstrained continues and discrete - Linear - Quadratic - BinaryBinary Integer - Nonlinear - M lti bj tiMu ltio bjec tive PblPro blems 4. 80 (released December 2006) Author: Hartmut Pohlheim The Genetic and Evolutionary Algorithm Toolbox (GEATbx) implements a wide range of genetic and evolutionary algorithms to solve large and complex real-world problems. Super useful! If you're seeing this message, it means we're having trouble. Evaporation Rate based Water Cycle Algorithm (ER-WCA) Code for Constrained Optimization Problems (Direct Method for handling constraints) (Written in MATLAB) Water Cycle Algorithm for solving Unconstrained combinatorial optimization problems (Discrete Version 1 of WCA) (Source Code) (Written in MATLAB). That way, the problem can be converted to a single-variable constrained optimization problem in t, which can be solved using fminbnd or fmincon or fseminf. minFunc is a Matlab function for unConstrained optimization of differentiable real-valued multivariate functions using line-search methods. The MATLAB Optimization Toolbox includes solvers for unconstrained and constrained nonlinear optimization, least-squares optimization, as well as algorithms for linear and quadratic programming. Chapter 10: Constrained Optimization via Calculus Introduction You have learned how to solve one-variable and two-variable unconstrained optimization problems. Back to Continuous Optimization. m – l1 constrained optimization. Welcome! This site aims at helping you identify ready to use solutions for your optimization problem, or at least to find some way to build such a solution using work done by others. So I use fmincon and it works. To evaluate a novel optimization algorithm’s property on different types of landscapes Compare different optimization algorithms Types of benchmarks Bound constrained problems (real, binary, discrete, mixed) Constrained problems Single / Multi-objective problems Static / Dynamic optimization problems Multimodal problems. Because other solvers like LINGO or CEPLEX find the global optimal solution with all constraints (equality and inequality) satisfaction. Iterative Linear Algebra for Constrained Optimization Hilary Dollar, Keble College, University of Oxford A thesis submitted for the degree of Doctor of Philosophy Michaelmas 2005 Abstract Each step of an interior point method for nonlinear optimization requires the solution of a symmetric indeﬁnite linear system known. This article in JEQ. " (2002) https://hdl. ConstraintTolerance. Construct the Lagrange multiplier augmented function. But, for a constrained minimizer of the -penalized least-squares optimization problem, this case does not occur. Provides all the tools needed to begin solving optimization problems using MATLAB® The Second Edition of Applied Optimization with MATLAB® Programming enables readers to harness all the features of MATLAB® to solve optimization problems using a variety of linear and nonlinear design optimization techniques. ON THE IMPLEMENTATION OF AN ALGORITHM FOR LARGE-SCALE EQUALITY CONSTRAINED OPTIMIZATION∗ MARUCHA LALEE †, JORGE NOCEDAL , AND TODD PLANTENGA‡ SIAM J. TOMLAB is a general purpose, open and integrated Matlab development environment for research and teaching in optimization on Unix and PC systems. Constrained Optimization Engineering design optimization problems are very rarely unconstrained. This paper addresses the issue of obt. Course Schedule. The core of trajectory optimization is constrained function optimization. KNITRO is preferred only if second derivatives can be supplied. " (2002) https://hdl. Discover what MATLAB. It is a multi-objective version of PSO which incorporates the Pareto Envelope and grid making technique, similar to Pareto Envelope-based Selection Algorithm to handle the multi-objective optimization problems. com offer finest service of Operations Management assignment help and Operations Management homework help. Example of nonlinear programming with constraints using the Optimization app. It can be seen that with various population size the optimal value of the func- tion reached the acceptable level. Development. I would like to perform How can I implement the method of Lagrange multipliers to find constrained extrema? without specifying the objective function. Yining Geng MATLAB preparatory class September 2015, Barcelona GSE Day 7 Constrained Optimization Categories of. Numerical optimization involves fundamental research on mathematical methods for linear and nonlinear programming, as well as techniques for. Constrained Nonlinear Optimization Algorithms Constrained Optimization Definition. Aeronautics and Astronautics, Massachusetts Institute of.


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Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. Further, the ideals $(x_{i_1},...,x_{i_j})$ with $i_1,...,i_j\in {1,...,n}$, are prime.

Can we make similar statements about any other kind of prime ideals in a regular local ring $R$. R$? Specifically, do any other prime ideals satisfy the condition: if the ideal is minimally generated by a certain set of generators, then every subset of the generators defines a prime ideal. ? One example in light of the first paragraph, are the prime ideals generated by a subset of the regular sequence that generates the maximal ideal. Also, when does a regular sequence define a prime ideal in a regular local ring$R$R$, and when does a maximal regular sequence define a maximal ideal.?

2 added 42 characters in body

Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. Further, the ideals $(x_{i_1},...,x_{i_j})$ with $i_1,...,i_j\in {1,...,n}$, are prime.

Can we make similar statements about any other kind of prime ideals in a regular local ring $R$. Specifically, do any other prime ideals satisfy the condition: if the ideal is minimally generated by a certain set of generators, then every subset of the generators defines a prime ideal. One example in light of the first paragraph, are the prime ideals generated by a subset of the regular sequence that generates the maximal ideal.

Also, when does a regular sequence define a prime ideal in a regular local ring $R$ and when does a maximal regular sequence define a maximal ideal.

1

# prime ideals in regular local rings

Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. Further, the ideals $(x_{i_1},...,x_{i_j})$ with $i_1,...,i_j\in {1,...,n}$, are prime.

Can we make similar statements about any other kind of prime ideals in $R$. Specifically, do any other prime ideals satisfy the condition: if the ideal is minimally generated by a certain set of generators, then every subset of the generators defines a prime ideal. One example in light of the first paragraph, are the prime ideals generated by a subset of the regular sequence that generates the maximal ideal.

Also, when does a regular sequence define a prime ideal in $R$ and when does a maximal regular sequence define a maximal ideal.

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# Revisiting Systems of Linear Equations

Almost every reader would have seen systems of linear equations from their high school days. Whether they liked it or not is a separate story. But, in all likelihood, they would have solved these equations by gradually removing variables one by one by substitution. In this way, three equations with three variables(or unknowns) gets transformed to two equations in two variables and one further step of reduction gives us an equation with only one variable which is readily solvable. Then the final solution is obtained by back substituting the obtained value of the variable into remaining equations. This method, in mathematical jargon, is called Gaussian elimination and back substitution.

It turns out (surprisingly) that linear systems form the basis of many interesting engineering applications. Ultimately the problem boils down to solution (or approximate solution) of a system of linear equations. So a thorough understanding of linear systems is essential to appreciate the applications. In this post we will outline all possible cases of finding solutions to linear systems and briefly outline two most important applications.

We will use matrix notation to represent the equations succinctly. It also gives us better insight into their solution. Using matrix notation the system can be represented as $\textbf{Ax}=\textbf{b}$ Where $$\textbf{A}$$ is the matrix of coefficients of size $$(m\times n)$$, $$\textbf{x}$$ is a vector of variables of size $$(n\times 1)$$, and $$\textbf{b}$$ is a vector of size $$(m\times 1)$$ representing constant right hand sides. Note that $$\textbf{b}$$ can be a vector of all zeros, i.e., $$\textbf{b} = \textbf{0}$$ or it can be any arbitrary vector with some nonzero values, i.e.,$$\textbf{b}\neq \textbf{0}$$. The solution(s) of linear systems depend to a large extent on what the right hand side is as we will see shortly.

Apart from notation, we need two other concepts from matrix theory. One is of rank and other is the range space (or column space) of a matrix. Rank $$(Rank(\textbf{A}))$$ of a matrix (say, $$\textbf{A}$$) is defined as number of independent rows or columns of a matrix. It is a well known result in matrix theory that row rank (number of independent rows) is equal to column rank (number of independent columns) and $$Rank(\textbf{A})\leq min(m,n)$$.

Range space $$(\mathcal{R}(A))$$(in short, Range) of a matrix is the vector space of all possible linear combinations of columns of the matrix. As we take all possible linear combination of columns, it is also known as column space. Readers who are slightly more familiar with linear algebra may know that Range is the span of columns of $$\textbf{A}$$. Zero vector $$(\textbf{0})$$ is always in the range of $$\textbf{A}$$ because if we take linear combination of columns of $$\textbf{A}$$ with all coefficients as 0’s, we get zero vector. Hence $$\textbf{b}=0 \in \mathcal{R}(\textbf{A})$$ is always true.

Let’s now discuss different cases separately and their solutions. We will assume that our system of equations has real entries.

• Case - I: $$(m = n)$$

• $$Rank(\textbf{A}) = m$$

• $$\textbf{b} \in \mathcal{R}(\textbf{A})$$ : Unique solution (for any $$\textbf{b}$$). For example,$$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 8 \\ 3 & 5 & 7 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 3 \\ 5 \\ 7 \end{bmatrix}$$$ This system has unique solution.
• $$\textbf{b} \not\in \mathcal{R}(\textbf{A})$$ : Impossible (This case will never happen because $$Rank(\textbf{A})=m$$)
• $$Rank(\textbf{A}) < m$$

• $$\textbf{b} \in \mathcal{R}(\textbf{A})$$ : Infinitely many solutions. For example,$$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 5 & 7 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \\ 8 \end{bmatrix}$$$ This system has infinitely many solutions.
• $$\textbf{b} \not\in \mathcal{R}(\textbf{A})$$ : No solution. For example,$$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 5 & 7 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 1 \\ 5 \\ 7 \end{bmatrix}$$$ This system has no solution.
• Case - II: $$(m > n)$$

• $$Rank(\textbf{A}) = n$$

• $$\textbf{b} \in \mathcal{R}(\textbf{A})$$ : Unique solution. For example,$$$\begin{bmatrix} 1 & 2 \\ 2 & 7 \\ 3 & 8 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 3 \\ 9 \\ 11 \end{bmatrix}$$$ This system has unique solution.
• $$\textbf{b} \not\in \mathcal{R}(\textbf{A})$$ : No solution. For example,$$$\begin{bmatrix} 1 & 2 \\ 2 & 7 \\ 3 & 8 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 3 \\ 9 \\ 11 \end{bmatrix}$$$ This system has no solution. But this case is immensely useful from application point of view. Sometimes it is not desirable to obtain the exact solution. Rather an approximate solution suffices for all practical purposes. Finding an approximate solution to an overdetermined system leads to the famous Least Squares problem.
• $$Rank(\textbf{A}) < n$$

• $$\textbf{b} \in \mathcal{R}(\textbf{A})$$ : Infinitely many solutions. For example,$$$\begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \\ 9 \end{bmatrix}$$$ It has infinitely many solutions.
• $$\textbf{b} \not\in \mathcal{R}(\textbf{A})$$ : No solution. For example,$$$\begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 3 \\ 6 \\ 8 \end{bmatrix}$$$ This system has no solution.
• Case - III: $$(m < n)$$

• $$Rank(\textbf{A}) = m$$ :

• $$\textbf{b} \in \mathcal{R}(\textbf{A})$$ : Infinitely many solutions. For example, $$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$$$ This system has infinitely many solutions. This case is also used in many applications. As there are infinitely many solutions, a natural choice is to choose the best solution. The qualifier ‘best’ determines what application we have in our mind. If we seek minimum $$(l_2)$$ norm, we get the so called minimum energy solution, a concept used in signal processing. Yet another concern is to seek for the sparsest solution (a solution with only a few nonzero entries and all other entries being zero). This idea is used in Compressed Sensing, an active research area with many interesting applications.
• $$\textbf{b} \not\in \mathcal{R}(\textbf{A})$$ : Impossible. This case will never happen since $$Rank(\textbf{A})=m$$.
• $$Rank(\textbf{A}) < m$$

• $$\textbf{b} \in \mathcal{R}(\textbf{A})$$ : Infinitely many solutions. For example, $$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 4 \\ 8 \end{bmatrix}$$$ This system has infinitely many solutions.
• $$\textbf{b} \not\in \mathcal{R}(\textbf{A})$$ : No solution. For example, $$$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 1 \\ 5 \end{bmatrix}$$$ This system has no solution.

Hope this post gives a clear overview of linear systems of equations. Interested reader may explore further applications. Comments and clarifications are welcome.

##### Biswajit Sahoo
###### PhD Student

My research interests include machine learning, deep learning, signal processing and data-driven machinery condition monitoring.


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ttfquery Package¶

ttfquery Package¶

Fonttools package for querying and sorting system fonts

_scriptregistry Module¶

Provides primitive font registry lookup/location for scripts

The singleton “registry” object (an instance of ttfquery.ttffiles.Registry is the only member of note). This registry with be created in a file in the user’s $HOME directory .font.cache if the$HOME environment variable is defined. Otherwise will be created in the ttfquery source code directory (which is obviously not a very good solution).

describe Module¶

Extract meta-data from a font-file to describe the font

ttfquery.describe.family(font)[source]

Get the family (and sub-family) for a font

ttfquery.describe.familyNames(familyID, subFamilyID=0)[source]

Convert family integers to human-readable names

ttfquery.describe.guessEncoding(font, given=None)[source]

Attempt to guess/retrieve an encoding from the font itself

Basically this will try to get the given encoding (unless it is None).

If given is a single integer or a single-item tuple, we will attempt scan looking for any table matching given as the platform ID and returning the first sub table.

If given is a two-value tuple, we will require explicit matching, and raise errors if the encoding cannot be retrieved.

if given is None, we will return the first encoding in the font.

XXX This needs some work, particularly for non-win32
platforms, where there is no preference embodied for the native encoding.
ttfquery.describe.modifiers(font)[source]

Get weight and italic modifiers for a font

weight is taken from the OS/2 usWeightClass field italic is taken from either OS/2 fsSelection or head macStyle, if either indicates italics we report italics

ttfquery.describe.openFont(filename)[source]

Get a new font object

ttfquery.describe.shortName(font)[source]

Get the short name from the font’s names table

ttfquery.describe.weightName(number)[source]

Convert integer number to a human-readable weight-name

ttfquery.describe.weightNumber(name)[source]

Convert a string-name to a weight number compatible with this module

findsystem Module¶

Find system fonts (only works on Linux and Win32 at the moment)

ttfquery.findsystem.findFonts(paths=None)[source]

Find fonts in paths, or the system paths if not given

XXX Doesn’t current support OS-X system paths

ttfquery.findsystem.linuxFontDirectories()[source]

Get system font directories on Linux/Unix

Uses /usr/sbin/chkfontpath to get the list of system-font directories, note that many of these will not be truetype font directories.

If /usr/sbin/chkfontpath isn’t available, uses returns a set of common Linux/Unix paths

ttfquery.findsystem.win32FontDirectory()[source]

Get User-specific font directory on Win32

ttfquery.findsystem.win32InstalledFonts(fontDirectory=None)[source]

Get list of explicitly installed font names

glyph Module¶

Representation of a single glyph including contour extraction

class ttfquery.glyph.Glyph(glyphName)[source]

Bases: object

Object encapsulating metadata regarding a particular glyph

calculateContours(font)[source]

Given a character, determine contours to draw

returns a list of contours, with each contour being a list of ((x,y),flag) elements. There may be only a single contour.

compile(font, steps=3)[source]

Compile the glyph to a set of poly-line outlines

ttfquery.glyph.decomposeOutline(contour, steps=3)[source]

Decompose a single TrueType contour to a line-loop

In essence, this is the “interpretation” of the font as geometric primitives. I only support line and quadratic (conic) segments, which should support most TrueType fonts as far as I know.

The process consists of first scanning for any multi- off-curve control-point runs. For each pair of such control points, we insert a new on-curve control point.

Once we have the “expanded” control point array we scan through looking for each segment which includes an off-curve control point. These should only be bracketed by on-curve control points. For each found segment, we call our integrateQuadratic method to produce a set of points interpolating between the end points as affected by the middle control point.

All other control points merely generate a single line-segment between the endpoints.

Get points on curve for quadratic w/ end points A and C

Basis Equations are taken from here:
http://www.truetype.demon.co.uk/ttoutln.htm

This is a very crude approach to the integration, everything is coded directly in Python, with no attempts to speed up the process.

XXX Should eventually provide adaptive steps so that
the angle between the elements can determine how many steps are used.

glyphquery Module¶

Glyph-specific queries on font-files

ttfquery.glyphquery.charDescent(font)[source]

Determine the general descent for the font (for scaling)

ttfquery.glyphquery.charHeight(font)[source]

Determine the general character height for the font (for scaling)

ttfquery.glyphquery.explicitGlyph(font, char, encoding=None)[source]

Return glyphName or None if there is not explicit glyph for char

ttfquery.glyphquery.glyphName(font, char, encoding=None, warnOnFailure=1)[source]

Retrieve the glyph name for the given character

XXX
Not sure what the effect of the Unicode mapping will be given the use of ord...
ttfquery.glyphquery.hasGlyph(font, char, encoding=None)[source]

Check to see if font appears to have explicit glyph for char

ttfquery.glyphquery.lineHeight(font)[source]

Get the base-line to base-line height for the font

XXX
There is some fudging going on here as I workaround what appears to be a problem with the specification for sTypoDescender, which states that it should normally be a negative value, but winds up being positive in at least one font that defines points below the zero axis.
XXX The entire OS/2 table doesn’t appear in a few
fonts (symbol fonts in particular), such as Corel’s BeeHive and BlackLight 686.
ttfquery.glyphquery.width(font, glyphName)[source]

Retrieve the width of the giving character for given font

The horizontal metrics table provides both the width and the left side bearing, we should really be using the left side bearing to adjust the character, but that’s a later project.

guessdescription Module¶

Heuristic attempting to extract descriptions from font names

This should only be used as a backup in case there is no proper available querying mechanism (i.e. fonttools), and even then, we really would rather avoid using this extremely fragile mechanism.

Basically what happens is that a name compression function is run over the font name, as seen in the Windows Registry, producing a base name and a set of modifiers, with the modifiers being the weight (as a string) and a flag indicating whether the font appears to be italicised.

XXX This is English-specific and entirely dependent on the
naming conventions commonly seen when naming fonts, so there will be a considerable number of cases where the resulting name and flags will be incorrect.

Add a font with name and file to this module’s registry

ttfquery.guessdescription.get(name, style=None)[source]

Get a font by name and optional style

Style defaults to (“normal”, 0)

ttfquery.guessdescription.interpretModifiers(name)[source]

Heuristic attempt to get weight and italic data from font-name

return base_font_name, (weight, italic)

Query for meta-data for a given font

ttffamily Module¶

Query for font-members of a particular family

ttfquery.ttffamily.main()[source]

ttffiles Module¶

Registry of available TrueType font files

XXX Currently two copies of exactly the same font
will likely confuse the registry because the specificFonts set will only have one of the metrics sets. Nothing breaks at the moment because of this, but it’s not ideal.
class ttfquery.ttffiles.Registry[source]

Bases: object

Object providing centralized registration of TTF files

Attributes:
families – mapping from TrueType font families
to sub-families and then to general fonts (as a set of font-names).
fonts – mapping from general fonts to modifiers to
specific font instances
specificFonts – mapping from specific font names
to the entire “metrics” set for the particular font.
files – mapping from (absolute) filenames to
specific font names
shortFiles – mapping from font filename basenames
to font-file-lists
DIRTY – flag indicating whether the registry has
had a new font registered (i.e. whether it should be saved out to disk).
DIRTY = 0
clear()[source]

Clear out the all tables and mark unchanged

dirty(dirty=1)[source]

Mark the registry as changed/unchanged

familyMembers(major, minor=None)[source]

Get all (general) fonts for a given family

filename = ''
fontFile(specificName)[source]

Return the absolute path-name for a given specific font

fontForms(fontName)[source]

Retrieve the set of font-forms (weight,italics) available in a font

fontMembers(fontName, weight=None, italics=None)[source]

Get specific font names for given generic font name

weight – if specified, only members with the given weight italics – if specified, only members where the flag matches

returns list of specific font names

file – a file open in binary read mode or a filename clearFirst – if true, clear tables first, and reset DIRTY to 0 after finished

matchName(name, single=0)[source]

Try to find a general font based on a name

filename – fully specified path to the font file force – if false, and the metadata is already available for this file, do not access the font file to retrieve, just return the existing metadata.

return value:
tuple of:
filename – fully specified absolute path modifiers – (weightInteger, italicsFlag) specificName – specific name of the particular font stored in the given file, the name of the “modified” font fontName – name of the general font which the modifiers are specialising specifier – family specifier, two-tuple of high-level and sub-level font classifications based on font characteristics as encoded in the font file.
register(filename, modifiers=None, specificName=None, fontName=None, familySpecifier=None, force=0)[source]

Do the actual registration of a filename & metadata

See metadata function for description of the various arguments. If modifiers == None then the metadata function will be used to scan for the metadata.

have the meta-data for the file loaded.
save(file=None, force=0)[source]

Attempt to save the font metadata to a pickled file

file – a file open in binary write mode or a filename force – if not true and DIRTY false, then don’t actually save anything

returns number of records saved

scan(paths=None, printErrors=0, force=0)[source]

Scan the given paths registering each found font

Construct registry from saved file

Assembly creates a Registry object and calls the load method with the file as argument.

ttfquery.ttffiles.main()[source]

ttfgroups Module¶

Demo script to print ordered set of system fonts

ttfquery.ttfgroups.buildTable(filenames=None, failureCallback=None)[source]

Build table mapping {family:(font:{modifiers:(name,file)})}

filenames – if provided, list of filenames to scan, otherwise the full set of system fonts provided by findsystem will be used. failureCallback – if provided, a function taking three arguments, the failing filename, an error-type code, and the error object. If processing should stop, raise an error. codes:

0 – couldn’t open the font file 1 – couldn’t find modifiers in the font file 2 – couldn’t find font-name in the font file 3 – couldn’t find the generic family specifier for the font
ttfquery.ttfgroups.interactiveCallback(file, code, err)[source]

Simple error callback for interactive use

ttfquery.ttfgroups.main()[source]


text:
# Interpreting confidence interval of regression coefficient.

In a Simple Linear Regression analysis, independent variable is weekly income and dependent variable is weekly consumption expenditure.

Here $95$% confidence interval of regression coefficient, $\beta_1$ is $(.4268,.5914)$.

So i have interpreted as :

"The data provides much evidence to conclude that the true slope of the regression line lies between $.4268$ and $.5914$ at $\alpha=5$% level of significance."

But it is not understandable to those who don't know statistics.

How can i interpret it more generally?

Our model is $Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$. This means that $\beta_1$ is the amount that we expect $Y$ to increase by when $X$ increases by 1 [or decrease if $\beta_1 < 0$].
In this problem $X$ is weekly income and $Y$ is weekly consumption expenditure so $\hat{\beta_1}$ is our estimate of the amount that weekly consumption expenditure increases for every $\$1$increase in weekly income. We are 95% confidence that it is between 0.43 and 0.59 [where by "95%" confidence we mean that if we were to collect new data generated from the same distribution then in 19 out of every 20 experiments we'd get$\hat{\beta_1}\$ in this interval].


text:
I’ve been fiddling with z3py, figuring out some functionality and realizing some interesting things you could do with it. I think I’m at a point where it is nice to checkpoint myself with a blog post.

I’m a little surprised z3py doesn’t overload the & and | operators and some kind of implies operator for BoolRef. You can insert them later using this.

from z3 import *
# useful non default operator definitions for z3 bools
BoolRef.__and__ = lambda self, rhs: And(self,rhs)
BoolRef.__or__ = lambda self, rhs: Or(self,rhs)
BoolRef.__xor__ = lambda self, rhs: Xor(self,rhs)
BoolRef.__invert__ = lambda self: Not(self)
BoolRef.__rshift__ = lambda self, rhs: Implies(self,rhs)

#### Functional Programming

Python is not the best functional programming environment imo. And by functional programming I implicitly mean roughly ML-like FP a la Haskell or OCaml. I don’t venture much into lisp land.

The lack of good algebraic datatypes (the class syntax is so ungainly) and a type system hurts. The lack of pattern matching hurts. The lambda keyword is so long it makes me sad.

But you have full access to z3 from the python bindings. Z3 does have algebraic data types, and a type system. It has built in substitution mechanisms and evaluation. And it has insane search procedures and the ability to prove things. Pretty damn cool!

Unfortunately the type system is rather simplistic, being basically simply typed rather than polymorphic or something else. But using python a a schema/macro system for z3 seems a plausible way forward.

To build templated types, you can have constructor functions in python for the appropriate types.

def Tuple(a,b):
Type = Datatype('Tuple(f{(a.name(),b.name())})')
Type.declare('pair', ('fst', a), ('snd', b))
Type = Type.create()
return Type
def Either(a,b):
Type = Datatype('Either(f{(a.name(),b.name())})')
Type.declare('left', ('getLeft', a))
Type.declare('right', ('getRight', b))
Type = Type.create()
return Type
def Maybe(a):
Type = Datatype('Maybe(f{(a.name())})')
Type.declare('Just', ('fromJust', a))
Type.declare("Nothing")
Type = Type.create()
return Type
def List(a):
Type = Datatype('List(f{(a.name())})')
Type.declare('Cons', ('car', a), ('cdr', Type))
Type.declare("Nil")
Type = Type.create()
return Type
'''
Note in regards to List. Z3 has a built in type Seq that I think it has built in smarts about. You might be better off using that rather than a custom List. YMMV
'''

You can access the constructors from the returned types. Check this out. You get detector functions is_Nothing and is_Just , the extractor function fromJust and constructor functions Nothing and Just. I do a lot of dir exploration with z3py. It’s hard to know what’s available sometimes

# dir(Maybe(IntSort())) returns
[
'Just',
'Nothing',
... underscore junk ... ,
'accessor',
'as_ast',
'ast',
'cast',
'constructor',
'ctx',
'ctx_ref',
'eq',
'fromJust',
'get_id',
'hash',
'is_Just',
'is_Nothing',
'kind',
'name',
'num_constructors',
'recognizer',
'sexpr',
'subsort',
'translate',
'use_pp']

It’s possible to build a general purpose match combinator on these types since you can introspect the number of constructors of the ADT using num_constructors, constructor, recognizer, and accessor. There might be a match inside z3py somewhere? I think it’s part of the SMTLIB standard now.

def match(x, **kwargs):
t = x.sort()
nc = t.num_constructors()
acc = kwargs["_"] # default argument
for c in range(nc):
con = t.constructor(c)
rec = t.recognizer(c)
nfields = con.arity()
if nfields == 0:
res = kwargs[con.name()]
else:
res = kwargs[con.name()](  *[t.accessor(c,a)(x) for a in range(nfields)] )
acc = If(rec(x), res, acc)
return acc

Example usage:

match(Const("x", Maybe(IntSort())), Just=lambda y : y + 1, Nothing = IntVal(3), _=IntVal(10))
# returns If(is(Nothing, x), 3, If(is(Just, x), fromJust(x) + 1, 10))

Z3 has a substitution mechanism built in. This is useful for instantiating ForAll and for evaluating Lambda. The substitute_vars function is what you want like so substitute_vars(f.body(), x, y, z)

It is possible to reflect the syntax in a fairly straightforward way back into python via a lambdify function, mimicking the equivalent very useful function from sympy. Lambdify is basically an interp function. Here is a start for such a function. I by no means have implemented interpretation of the entirety of z3. Also I feel like this implementation is very clunky. Some kind of CPS?

def lift1(f,x):
return lambda *args: f(x(*args))

def lift2(op,l,r):
return lambda *args: op(l(*args), r(*args))

# interp is useful for transferring expressions into numpy, sympy
# but also for program extraction

from functools import reduce
import operator as op
def interp(a, *args):
if is_true(a):
return lambda *args: True
elif is_false(a):
return lambda *args: False
elif is_int_value(a):
return lambda *args: a.as_long()
elif is_rational_value(a):
n = a.numerator_as_long()
d = a.denominator_as_long()
return lambda *args: n / d
#elif is_algebraic_value(a):
#    pass
elif is_const(a): # is free variable
loc = [ind for ind, b in enumerate(args) if eq(a,b)]
if len(loc) == 0:
return a
else:
ind = loc[0]
return lambda *args2: args2[ind]
b = [interp(c, *args) for c in a.children()]
if is_and(a):
return lambda *args: reduce(op.and_, [f(*args) for f in b])
elif is_or(a):
return lambda *args: reduce(op.or_, [f(*args) for f in b])
elif is_app_of(a, Z3_OP_XOR):
return lambda *args: reduce(op.xor, [f(*args) for f in b])
return lambda *args: reduce(op.add, [f(*args) for f in b])
elif is_mul(a):
return lambda *args: reduce(op.mul, [f(*args) for f in b])
elif len(b) == 1:
n = b[0]
if is_not(a):
return lift1(op.invert , n)# lambda *args: ~n
elif len(b) == 2:
l,r = b
if is_eq(a):
return lift2(op.eq, l,r) #lambda *args: l == r
elif is_distinct(a): # This can be multi_argument
return lift2(op.ne, l,r) # lambda *args: l != r
elif is_sub(a):
return lift2(op.sub, l,r) # lambda *args: l - r
elif is_app_of(a, Z3_OP_POWER):
return lift2(op.pow, l,r) #lambda *args: l ** r
elif is_div(a):
return  lift2(op.truediv, l,r)# lambda *args: l / r
elif is_idiv(a):
return lift2(op.floordiv, l,r) # lambda *args: l // r
elif is_mod(a):
return lift2(op.mod, l,r) # lambda *args: l % r
elif is_le(a):
return lift2(op.le, l,r) # lambda *args: l <= r
elif is_lt(a):
return lift2(op.lt, l,r) # lambda *args: l < r
elif is_ge(a):
return lift2(op.ge, l,r) #lambda *args: l > r
elif is_gt(a):
return lift2(op.gt, l,r) # lambda *args: l >= r
elif is_implies(a):
return lambda *args: (~ l(*args) ) & r(*args)
print("unrecognized constructor: ", type(a))
assert(False)

#example usage
a = Bool('a')
interp(Xor(a & a | a, a), a)(True)
x, y = Reals('x y')
interp(x + y + y + y * x, x ,y)(3,2)

There is the ability to define recursive functions in z3. It is also plausible to define them via. In this way you can get a reversible functional programming language, maybe some subset of mercury / curry’s power.

fac = RecFunction('fac', IntSort(), IntSort())
n = Int('n')
RecAddDefinition(fac, n, If(n == 0, 1, n*fac(n-1)))

s = Solver()
s.check()
s.model()
#  returns [n = 3, fac = [0 → 1, else → fac(-1 + ν0)·ν0]]

### Interactive Theorem Proving

Z3 is awesome at thoerem proving. But somethings it just doesn’t handle right and needs human guidance.

Through searching, there are a couple interesting python interactive theorem prover projects. Cody pointed me to a project he worked on a while back, Boole https://github.com/avigad/boole . It has a dependently typed lambda calculus in it with the purpose of gluing together many systems, I think. He implemented a lot of stuff from scratch. I think I want to try to get less and do less. There is also holpy https://arxiv.org/abs/1905.05970 which appears to be being actively developed. It’s roughly a translation of hol to python I think. It’s available from a strange chinese github on the author’s website if you go looking for it.

This suggests an interesting approach. Most interactive theorem provers start unautomated and add it later. Instead we can iteratively build an interface to de-automate z3.

Altogether, this approach is more HOL flavored than Coq/Agda flavored. z3 terms are our logic and python is our manipulation metal language. Ideally, one would want to verify that every.

Python is so unprincipled that I can’t imagine that you could ever build a system up to the trustworthiness of the other theorem provers. But this is freeing in many ways. Since that is off the table, we can just do the best we can.

Using the z3 syntax tree and the z3 proof automation and z3 substitution mechanisms gives us a HUGE step up from implementing them from scratch. Ideally, we’d want to write as little python as possible, and especially as little python as possible that has to be trusted to be implemented correctly.

One big concern is accidental mutation of the proof under our feet by python. Perhaps using hashes and checking them might be a way to at least detect this. I need to have a good think about how to factor out a trusted core from all possible tactics.

I think it helps a little that z3 often will be able to verify the equivalence of small steps in proofs even if it can’t do the entire proof itself.

I think induction principles will need to be injected by hand. Z3 doesn’t really have that built in. There are definitely situations that after you introduce the induction, z3 can slam all the cases no problem. For example, check this out.

Another thing that might be nice is integration/translation to sympy. Sympy has a ton of useful functionality, at the very least differentiation.

Translation and integration with cvxpy for sum of squares proofs would also be quite neat. I already did something with this using sympy. I’m not super sure how you extract exact proofs from the floating point solutions SCS returns? I think there is a thing. I’ve heard the LLL algorithm can be used for this somehow (finding likely algebraic number matches to floating point numbers)?

So here are some sketched out ideas for tactics.

class Proof():
def __init__(self, goal, name=None): # Taken a name for the theorem?
self.goals = [([],goal)]
self.proven = False
self.name = name
#def intros(self): #intro_all
#    self.goals.append( (ctx, goal.intros())  )
#    return self
def equiv(self, goal2):
ctx, goal1 = self.goals.pop()
if prove2(Implies(And(*ctx), goal1 == goal2)):
g = goal2
else:
g = goal1
self.goals.append( (ctx, g))
return self
def __eq__(self,rhs):
return self.equiv(rhs)
#def assert(): #put new goal in stack with current context. Put into context of 1 below top
#def assume(): #just put crap in the context.
def intro_all(self): #name = hint maybe later
ctx, goal = self.goals.pop()
assert(goal.is_forall())
vs = [FreshConst(goal.var_sort(i) , prefix=goal.var_name(i)) for i in range(goal.num_vars())]
g = instantiate(goal,*vs)
self.goals.append( (ctx + vs, g)) # wait. I should keep propositions and variables seperate
return self
def intro_imp(self): #intro_impl
ctx, goal = self.goals.pop()
if is_implies(goal):
a, b = goal.children()
ctx.append(a)
self.goals.append((ctx,b))
else:
self.goals.append((ctx,goal))
return self
def split(self): #z3 tactic split-clauses?
ctx, goal = self.goals.pop()
if is_and(goal):
for c in goal.children():
self.goals.append((ctx,c))
else:
self.goals.append((ctx,goal))
return self
def z3_tactic(self,t):
t = Tactic(t)
ctx, goal = self.goals.pop()
#g = t(Implies(And(*ctx), goal)).as_expr()
g = t(goal).as_expr()
self.goals.append(([],g))
return self
def simpl(self):
return self.z3_tactic("simplify")
def congruence(self):
#maybe search for equalities. And put them in the goal
return self.z3_tactic("solve-eqs")
def smt(self):
ctx, goal = self.goals.pop()
s = Solver()
#s.set(**keywords)
claim = Implies(And(*ctx), goal)
r = s.check()
if r  == sat:
print("Countermodel : " + str(s.model()))
assert(r == unsat)
return self
def destruct(self):
ctx, goal = self.goals.pop()
if is_bool(goal):
ctx1 = ctx.copy()
ctx2 = ctx.copy()
ctx1.append(goal == True)
ctx2.append(goal == False)
self.goals.append((ctx2, BoolVal(False) ))
self.goals.append((ctx1, BoolVal(True) ))
else:
self.goals.append((ctx, goal))
return self
def forget(self,n):
ctx, goal = self.goals.pop()
ctx.pop(n)
self.goals.append((ctx, goal))
return self
def qed(self):
if len(self.goals) == 0:
self.proven = True
# add self to global proof context if self.name is not None
def get_ctx(self,n):
return self.goals[-1][0][n]
def __str__(self):
if len(self.goals) >= 1:
ctx, goal = self.goals[-1]
return "".join([f"[{i}] {str(c)} : {str(c.sort())}\n" for i, c in enumerate(ctx)]) + "----------------\n" + f"{str(goal)} : {str(goal.sort())}"
else:
return "No Goals Left"
def __repr__(self):
return str(self)

x = Real("x")
Proof(x**2 - 1 == 0).equiv((x+1)*(x-1) == 0).equiv((x == 1) | (x == -1))

a, b = Bools('a b')
p = Proof((a & b) > b)
p.intro_imp().destruct()
.smt() \
.smt() \
.qed()

Another question is how to implement an apply tactic gracefully. Fully deconstructing syntax trees and unifying ourselves is not utilizing z3 well. If you have a good idea how to get unification out of z3, I’d be interested to hear from you here. https://stackoverflow.com/questions/59398955/getting-z3-instantiations-of-quantified-variables/59400838#59400838

Here’s an idea though. In the cold light of day, I am still not sure this reasoning makes much sense. Suppose we’re trying to apply forall x. a(x) -> b(x) to a c(y). If forall x. b(x) -> c(y) we’re good and by assumption that is obvious for some reason, like the syntactic instantiation of b gives c. We can ask z3 to prove that and it will hopefully easy. If we can prove forall x. a(x) in the current context, that would be sufficient, but not true typically. It is an overly difficult request. We really only need to prove a(x) for values pertinent to the proof of c(y). Here’s a suspicious strategem. Any a -> b can be weakened to (q -> a) -> (q -> b). In particular we can choose to weaken forall x. a(x) -> b(x) to forall x. ((c(y) -> b(x)) -> a(x)) -> ((c(y) -> b(x)) -> b(x)). Then we can replace the goal with forall x. ((c(y) -> b(x)) -> a(x)) after we prove that (forall x. (c(y) -> b(x)) -> b(x)) -> c(y). Maybe c(y) -> b(x) is sufficient to restrict the values of x? Not sure.

Another rough sketch of induction on Nat. Not right yet.

def inductionNat(self):
assert(self.num_vars() == 1 and self.var_sort(0) == IntSort() and self.is_forall())
n = FreshInt()
return instantiate(self, IntVal(0)) & ForAll([n],instantiate(self, n) & (n > 0) > instantiate(self, n+1))

We could also make a simple induction for ADTs based on the similar introspection we used for match above. It’s ugly but I think it works.

def induction(self):
assert(is_quantifier(self) and self.is_forall() and self.num_vars() == 1) #we can eventually relax vars = 1
t = self.var_sort(0)
nc = t.num_constructors()
th = []
for i in range(nc):
con = t.constructor(i)
nfields = con.arity()
if nfields == 0:
th += [substitute_vars(self.body(), con())]
else:
hyp = []
args = []
for d in range(nfields):
td = con.domain(d)
x = FreshConst(td)
print(x)
if td == t:
hyp += [substitute_vars(self.body(), x)]
args += [x]
th += [ForAll(args, Implies(And(*hyp), substitute_vars(self.body(), con(*args))))]
print(th)
return And(*th)

I haven’t really though much about tacticals yet.

# describe_tactics() gives a list of all z3 tactics
ackermannize_bv : A tactic for performing full Ackermannization on bv instances.
subpaving : tactic for testing subpaving module.
horn : apply tactic for horn clauses.
horn-simplify : simplify horn clauses.
nlsat : (try to) solve goal using a nonlinear arithmetic solver.
qfnra-nlsat : builtin strategy for solving QF_NRA problems using only nlsat.
nlqsat : apply a NL-QSAT solver.
qe-light : apply light-weight quantifier elimination.
qe-sat : check satisfiability of quantified formulas using quantifier elimination.
qe : apply quantifier elimination.
qsat : apply a QSAT solver.
qe2 : apply a QSAT based quantifier elimination.
qe_rec : apply a QSAT based quantifier elimination recursively.
psat : (try to) solve goal using a parallel SAT solver.
sat : (try to) solve goal using a SAT solver.
sat-preprocess : Apply SAT solver preprocessing procedures (bounded resolution, Boolean constant propagation, 2-SAT, subsumption, subsumption resolution).
ctx-solver-simplify : apply solver-based contextual simplification rules.
smt : apply a SAT based SMT solver.
psmt : builtin strategy for SMT tactic in parallel.
unit-subsume-simplify : unit subsumption simplification.
aig : simplify Boolean structure using AIGs.
add-bounds : add bounds to unbounded variables (under approximation).
card2bv : convert pseudo-boolean constraints to bit-vectors.
degree-shift : try to reduce degree of polynomials (remark: :mul2power simplification is automatically applied).
diff-neq : specialized solver for integer arithmetic problems that contain only atoms of the form (<= k x) (<= x k) and (not (= (- x y) k)), where x and y are constants and k is a numeral, and all constants are bounded.
eq2bv : convert integer variables used as finite domain elements to bit-vectors.
factor : polynomial factorization.
fix-dl-var : if goal is in the difference logic fragment, then fix the variable with the most number of occurrences at 0.
fm : eliminate variables using fourier-motzkin elimination.
lia2card : introduce cardinality constraints from 0-1 integer.
lia2pb : convert bounded integer variables into a sequence of 0-1 variables.
nla2bv : convert a nonlinear arithmetic problem into a bit-vector problem, in most cases the resultant goal is an under approximation and is useul for finding models.
normalize-bounds : replace a variable x with lower bound k <= x with x' = x - k.
pb2bv : convert pseudo-boolean constraints to bit-vectors.
propagate-ineqs : propagate ineqs/bounds, remove subsumed inequalities.
purify-arith : eliminate unnecessary operators: -, /, div, mod, rem, is-int, to-int, ^, root-objects.
recover-01 : recover 0-1 variables hidden as Boolean variables.
bit-blast : reduce bit-vector expressions into SAT.
bv1-blast : reduce bit-vector expressions into bit-vectors of size 1 (notes: only equality, extract and concat are supported).
bv_bound_chk : attempts to detect inconsistencies of bounds on bv expressions.
propagate-bv-bounds : propagate bit-vector bounds by simplifying implied or contradictory bounds.
propagate-bv-bounds-new : propagate bit-vector bounds by simplifying implied or contradictory bounds.
reduce-bv-size : try to reduce bit-vector sizes using inequalities.
bvarray2uf : Rewrite bit-vector arrays into bit-vector (uninterpreted) functions.
dt2bv : eliminate finite domain data-types. Replace by bit-vectors.
elim-small-bv : eliminate small, quantified bit-vectors by expansion.
max-bv-sharing : use heuristics to maximize the sharing of bit-vector expressions such as adders and multipliers.
blast-term-ite : blast term if-then-else by hoisting them.
cofactor-term-ite : eliminate term if-the-else using cofactors.
collect-statistics : Collects various statistics.
ctx-simplify : apply contextual simplification rules.
der : destructive equality resolution.
distribute-forall : distribute forall over conjunctions.
dom-simplify : apply dominator simplification rules.
elim-term-ite : eliminate term if-then-else by adding fresh auxiliary declarations.
elim-uncnstr : eliminate application containing unconstrained variables.
injectivity : Identifies and applies injectivity axioms.
snf : put goal in skolem normal form.
nnf : put goal in negation normal form.
occf : put goal in one constraint per clause normal form (notes: fails if proof generation is enabled; only clauses are considered).
pb-preprocess : pre-process pseudo-Boolean constraints a la Davis Putnam.
propagate-values : propagate constants.
reduce-args : reduce the number of arguments of function applications, when for all occurrences of a function f the i-th is a value.
reduce-invertible : reduce invertible variable occurrences.
simplify : apply simplification rules.
elim-and : convert (and a b) into (not (or (not a) (not b))).
solve-eqs : eliminate variables by solving equations.
special-relations : detect and replace by special relations.
split-clause : split a clause in many subgoals.
symmetry-reduce : apply symmetry reduction.
tseitin-cnf : convert goal into CNF using tseitin-like encoding (note: quantifiers are ignored).
tseitin-cnf-core : convert goal into CNF using tseitin-like encoding (note: quantifiers are ignored). This tactic does not apply required simplifications to the input goal like the tseitin-cnf tactic.
qffd : builtin strategy for solving QF_FD problems.
pqffd : builtin strategy for solving QF_FD problems in parallel.
smtfd : builtin strategy for solving SMT problems by reduction to FD.
fpa2bv : convert floating point numbers to bit-vectors.
qffp : (try to) solve goal using the tactic for QF_FP.
qffpbv : (try to) solve goal using the tactic for QF_FPBV (floats+bit-vectors).
qffplra : (try to) solve goal using the tactic for QF_FPLRA.
default : default strategy used when no logic is specified.
sine-filter : eliminate premises using Sine Qua Non
qfbv-sls : (try to) solve using stochastic local search for QF_BV.
nra : builtin strategy for solving NRA problems.
qfaufbv : builtin strategy for solving QF_AUFBV problems.
qfauflia : builtin strategy for solving QF_AUFLIA problems.
qfbv : builtin strategy for solving QF_BV problems.
qfidl : builtin strategy for solving QF_IDL problems.
qflia : builtin strategy for solving QF_LIA problems.
qflra : builtin strategy for solving QF_LRA problems.
qfnia : builtin strategy for solving QF_NIA problems.
qfnra : builtin strategy for solving QF_NRA problems.
qfuf : builtin strategy for solving QF_UF problems.
qfufbv : builtin strategy for solving QF_UFBV problems.
qfufbv_ackr : A tactic for solving QF_UFBV based on Ackermannization.
ufnia : builtin strategy for solving UFNIA problems.
uflra : builtin strategy for solving UFLRA problems.
auflia : builtin strategy for solving AUFLIA problems.
auflira : builtin strategy for solving AUFLIRA problems.
aufnira : builtin strategy for solving AUFNIRA problems.
lra : builtin strategy for solving LRA problems.
lia : builtin strategy for solving LIA problems.
lira : builtin strategy for solving LIRA problems.
skip : do nothing tactic.
fail : always fail tactic.
fail-if-undecided : fail if goal is undecided.
macro-finder : Identifies and applies macros.
quasi-macros : Identifies and applies quasi-macros.
ufbv-rewriter : Applies UFBV-specific rewriting rules, mainly demodulation.
bv : builtin strategy for solving BV problems (with quantifiers).
ufbv : builtin strategy for solving UFBV problems (with quantifiers).