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# Multi-objective optimization

Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many fields of science, including engineering, economics and logistics (see the section on applications for detailed examples) where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives.

For a nontrivial multi-objective optimization problem, there does not exist a single solution that simultaneously optimizes each objective. In that case, the objective functions are said to be conflicting, and there exists a (possibly infinite) number of Pareto optimal solutions. A solution is called nondominated, Pareto optimal, Pareto efficient or noninferior, if none of the objective functions can be improved in value without degrading some of the other objective values. Without additional subjective preference information, all Pareto optimal solutions are considered equally good (as vectors cannot be ordered completely). Researchers study multi-objective optimization problems from different viewpoints and, thus, there exist different solution philosophies and goals when setting and solving them. The goal may be to find a representative set of Pareto optimal solutions, and/or quantify the trade-offs in satisfying the different objectives, and/or finding a single solution that satisfies the subjective preferences of a human decision maker (DM).

## Introduction

A multi-objective optimization problem is an optimization problem that involves multiple objective functions.[1][2] In mathematical terms, a multi-objective optimization problem can be formulated as

{\displaystyle {\begin{aligned}\min &\left(f_{1}(x),f_{2}(x),\ldots ,f_{k}(x)\right)\\{\text{s.t. }}&x\in X,\end{aligned}}}

where the integer ${\displaystyle k\geq 2}$ is the number of objectives and the set ${\displaystyle X}$ is the feasible set of decision vectors. The feasible set is typically defined by some constraint functions. In addition, the vector-valued objective function is often defined as

${\displaystyle f:X\to \mathbb {R} ^{k},\ f(x)=(f_{1}(x),\ldots ,f_{k}(x))^{T}}$. If some objective function is to be maximized, it is equivalent to minimize its negative. The image of ${\displaystyle X}$ is denoted by ${\displaystyle Y\in \mathbb {R} ^{k}}$
Example of a Pareto frontier (in red), the set of Pareto optimal solutions (those that are not dominated by any other feasible solutions). The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence do lie on the frontier.

An element ${\displaystyle x^{*}\in X}$ is called a feasible solution or a feasible decision. A vector ${\displaystyle z^{*}:=f(x^{*})\in \mathbb {R} ^{k}}$ for a feasible solution ${\displaystyle x^{*}}$ is called an objective vector or an outcome. In multi-objective optimization, there does not typically exist a feasible solution that minimizes all objective functions simultaneously. Therefore, attention is paid to Pareto optimal solutions; that is, solutions that cannot be improved in any of the objectives without degrading at least one of the other objectives. In mathematical terms, a feasible solution ${\displaystyle x^{1}\in X}$ is said to (Pareto) dominate another solution ${\displaystyle x^{2}\in X}$, if

1. ${\displaystyle f_{i}(x^{1})\leq f_{i}(x^{2})}$ for all indices ${\displaystyle i\in \left\{{1,2,\dots ,k}\right\}}$ and
2. ${\displaystyle f_{j}(x^{1}) for at least one index ${\displaystyle j\in \left\{{1,2,\dots ,k}\right\}}$.

A solution ${\displaystyle x^{*}\in X}$ (and the corresponding outcome ${\displaystyle f(x^{*})}$) is called Pareto optimal, if there does not exist another solution that dominates it. The set of Pareto optimal outcomes is often called the Pareto front, Pareto frontier, or Pareto boundary.

The Pareto front of a multi-objective optimization problem is bounded by a so-called nadir objective vector ${\displaystyle z^{nad}}$ and an ideal objective vector ${\displaystyle z^{ideal}}$, if these are finite. The nadir objective vector is defined as

${\displaystyle z_{i}^{nad}=\sup _{x\in X{\text{ is Pareto optimal}}}f_{i}(x){\text{ for all }}i=1,\ldots ,k}$

and the ideal objective vector as

${\displaystyle z_{i}^{ideal}=\inf _{x\in X}{f_{i}(x)}{\text{ for all }}i=1,\ldots ,k.}$

In other words, the components of a nadir and an ideal objective vector define upper and lower bounds for the objective function values of Pareto optimal solutions, respectively. In practice, the nadir objective vector can only be approximated as, typically, the whole Pareto optimal set is unknown. In addition, a utopian objective vector ${\displaystyle z^{utopian}}$ with

${\displaystyle z_{i}^{utopian}=z_{i}^{ideal}-\epsilon {\text{ for all }}i=1,\ldots ,k,}$

where ${\displaystyle \epsilon >0}$ is a small constant, is often defined because of numerical reasons.

## Examples of multi-objective optimization applications

### Economics

In economics, many problems involve multiple objectives along with constraints on what combinations of those objectives are attainable. For example, consumer's demand for various goods is determined by the process of maximization of the utilities derived from those goods, subject to a constraint based on how much income is available to spend on those goods and on the prices of those goods. This constraint allows more of one good to be purchased only at the sacrifice of consuming less of another good; therefore, the various objectives (more consumption of each good is preferred) are in conflict with each other. A common method for analyzing such a problem is to use a graph of indifference curves, representing preferences, and a budget constraint, representing the trade-offs that the consumer is faced with.

Another example involves the production possibilities frontier, which specifies what combinations of various types of goods can be produced by a society with certain amounts of various resources. The frontier specifies the trade-offs that the society is faced with — if the society is fully utilizing its resources, more of one good can be produced only at the expense of producing less of another good. A society must then use some process to choose among the possibilities on the frontier.

Macroeconomic policy-making is a context requiring multi-objective optimization. Typically a central bank must choose a stance for monetary policy that balances competing objectives — low inflation, low unemployment, low balance of trade deficit, etc. To do this, the central bank uses a model of the economy that quantitatively describes the various causal linkages in the economy; it simulates the model repeatedly under various possible stances of monetary policy, in order to obtain a menu of possible predicted outcomes for the various variables of interest. Then in principle it can use an aggregate objective function to rate the alternative sets of predicted outcomes, although in practice central banks use a non-quantitative, judgement-based, process for ranking the alternatives and making the policy choice.

### Finance

In finance, a common problem is to choose a portfolio when there are two conflicting objectives — the desire to have the expected value of portfolio returns be as high as possible, and the desire to have risk, often measured by the standard deviation of portfolio returns, be as low as possible. This problem is often represented by a graph in which the efficient frontier shows the best combinations of risk and expected return that are available, and in which indifference curves show the investor's preferences for various risk-expected return combinations. The problem of optimizing a function of the expected value (first moment) and the standard deviation (square root of the second central moment) of portfolio return is called a two-moment decision model.

### Optimal control

In engineering and economics, many problems involve multiple objectives which are not describable as the-more-the-better or the-less-the-better; instead, there is an ideal target value for each objective, and the desire is to get as close as possible to the desired value of each objective. For example, energy systems typically have a trade-off between performance and cost[3][4] or one might want to adjust a rocket's fuel usage and orientation so that it arrives both at a specified place and at a specified time; or one might want to conduct open market operations so that both the inflation rate and the unemployment rate are as close as possible to their desired values.

Often such problems are subject to linear equality constraints that prevent all objectives from being simultaneously perfectly met, especially when the number of controllable variables is less than the number of objectives and when the presence of random shocks generates uncertainty. Commonly a multi-objective quadratic objective function is used, with the cost associated with an objective rising quadratically with the distance of the objective from its ideal value. Since these problems typically involve adjusting the controlled variables at various points in time and/or evaluating the objectives at various points in time, intertemporal optimization techniques are employed.[5][6]

### Optimal design

Product and process design can be largely improved using modern modeling, simulation and optimization techniques. The key question in optimal design is the measure of what is good or desirable about a design. Before looking for optimal designs it is important to identify characteristics which contribute the most to the overall value of the design. A good design typically involves multiple criteria/objectives such as capital cost/investment, operating cost, profit, quality and/or recovery of the product, efficiency, process safety, operation time etc. Therefore, in practical applications, the performance of process and product design is often measured with respect to multiple objectives. These objectives typically are conflicting, i.e. achieving the optimal value for one objective requires some compromise on one or more of other objectives.

For example, in paper industry when designing a paper mill, one can seek to decrease the amount of capital invested in a paper mill and enhance the quality of paper simultaneously. If the design of a paper mill is defined by large storage volumes and paper quality is defined by quality parameters, then the problem of optimal design of a paper mill can include objectives such as: i) minimization of expected variation of those quality parameter from their nominal values, ii) minimization of expected time of breaks and iii) minimization of investment cost of storage volumes. Here, maximum volume of towers are design variables. This example of optimal design of a paper mill is a simplification of the model used in.[7] Multi-objective design optimization have also been implemented in engineering systems - e.g. design of nano-CMOS semiconductors,[8] design of solar-powered irrigation systems,[9] optimization of sand mould systems,[10][11] engine design,[12][13] optimal sensor deployment[14] and optimal controller design.[15][16]

### Process optimization

Multi-objective optimization has also been increasingly employed in chemical engineering. In Fiandaca and Fraga, (2009),[17] the multi-objective generic algorithm (MOGA) was used to optimize the design of the pressure swing adsorption process (cyclic separation process). The design problem considered in that work involves the bi-objective maximization of nitrogen recovery and nitrogen purity. The results obtained in that work provided a good approximation of the Pareto frontier with acceptable trade-offs between the objectives. A multi-objective chemical process problem for the thermal processing of food was solved by Sendín et al.,(2010).[18] In that work, two case studies (bi-objective and triple objective problems) with nonlinear dynamic models were tackled. A hybrid approach consisting of the weighted Tchebycheff and the Normal Boundary Intersection approaches were utilized. This novel hybrid approach was successful in constructing a Pareto optimal set for the thermal processing of foods. The multi-objective optimization of the combined carbon dioxide reforming and partial-oxidation of methane was carried out in the work of Ganesan et al.,(2013).[19] In that work, the objective functions were: methane conversion, carbon monoxide selectivity and the hydrogen to carbon monoxide ratio. The problem was tackled using the Normal Boundary Intersection (NBI) method in conjunction with two swarm-based techniques (Gravitational Search Algorithm (GSA) and Particle Swarm Optimization (PSO)). Similar multi-objective problems were encountered and solved for applications involving chemical extraction[20] and bioethanol production processes.[21] In Abakarov et al., (2013), [22] an alternative technique to solve multi-objective optimization problems arising in food engineering was proposed. The Aggregating Functions Approach, the Adaptive Random Search Algorithm, and the Penalty Functions Approach[23] were used to compute the initial set of the non-dominated or Pareto-optimal solutions. The well-known method AHP and Tabular Method were used simultaneously for choosing the best alternative among the computed subset of non-dominated solutions for osmotic dehydration processes.

The purpose of radio resource management is to satisfy the data rates that are requested by the users of a cellular network.[24] The main resources are time intervals, frequency blocks, and transmit powers. Each user has its own objective function that, for example, can represent some combination of the data rate, latency, and energy efficiency. These objectives are conflicting since the frequency resources are very scarce, thus there is a need for tight spatial frequency reuse which causes immense inter-user interference if not properly controlled. Multi-user MIMO techniques are nowadays used to reduce the interference by adaptive precoding. The network operator would like to both bring great coverage and high data rates, thus the operator would like to find a Pareto optimal solution that balance the total network data throughput and the user fairness in an appropriate subjective manner.

Radio resource management is often solved by scalarization; that is, selection of a network utility function that tries to balance throughput and user fairness. The choice of utility function has a large impact on the computational complexity of the resulting single-objective optimization problem.[24] For example, the common utility of weighted sum rate gives an NP-hard problem with a complexity that scales exponentially with the number of users, while the weighted max-min fairness utility results in a quasi-convex optimization problem with only a polynomial scaling with the number of users.[25]

### Electric power systems

Reconfiguration, by exchanging the functional links between the elements of the system, represents one of the most important measures which can improve the operational performance of a distribution system. The problem of optimization through the reconfiguration of a power distribution system, in terms of its definition, is a historical single objective problem with constraints. Since 1975, when Merlin and Back [26] introduced the idea of distribution system reconfiguration for active power loss reduction, until nowadays, a lot of researchers have proposed diverse methods and algorithms to solve the reconfiguration problem as a single objective problem. Some authors have proposed Pareto optimality based approaches (including active power losses and reliability indices as objectives). For this purpose, different artificial intelligence based methods have been used: microgenetic,[27] branch exchange,[28] particle swarm optimization [29] and non-dominated sorting genetic algorithm.[30]

## Solving a multi-objective optimization problem

As there usually exist multiple Pareto optimal solutions for multi-objective optimization problems, what it means to solve such a problem is not as straightforward as it is for a conventional single-objective optimization problem. Therefore, different researchers have defined the term "solving a multi-objective optimization problem" in various ways. This section summarizes some of them and the contexts in which they are used. Many methods convert the original problem with multiple objectives into a single-objective optimization problem. This is called a scalarized problem. If scalarization is done neatly, Pareto optimality of the solutions obtained can be guaranteed.

Solving a multi-objective optimization problem is sometimes understood as approximating or computing all or a representative set of Pareto optimal solutions.[31][32]

When decision making is emphasized, the objective of solving a multi-objective optimization problem is referred to supporting a decision maker in finding the most preferred Pareto optimal solution according to his/her subjective preferences.[1][33] The underlying assumption is that one solution to the problem must be identified to be implemented in practice. Here, a human decision maker (DM) plays an important role. The DM is expected to be an expert in the problem domain.

The most preferred results can be found using different philosophies. Multi-objective optimization methods can be divided into four classes.[2] In so-called no preference methods, no DM is expected to be available, but a neutral compromise solution is identified without preference information.[1] The other classes are so-called a priori, a posteriori and interactive methods and they all involve preference information from the DM in different ways.

In a priori methods, preference information is first asked from the DM and then a solution best satisfying these preferences is found. In a posteriori methods, a representative set of Pareto optimal solutions is first found and then the DM must choose one of them. In interactive methods, the decision maker is allowed to iteratively search for the most preferred solution. In each iteration of the interactive method, the DM is shown Pareto optimal solution(s) and describes how the solution(s) could be improved. The information given by the decision maker is then taken into account while generating new Pareto optimal solution(s) for the DM to study in the next iteration. In this way, the DM learns about the feasibility of his/her wishes and can concentrate on solutions that are interesting to him/her. The DM may stop the search whenever he/she wants to. More information and examples of different methods in the four classes are given in the following sections.

## Scalarizing multi-objective optimization problems

Scalarizing a multi-objective optimization problem is an a priori method, which means formulating a single-objective optimization problem such that optimal solutions to the single-objective optimization problem are Pareto optimal solutions to the multi-objective optimization problem.[2] In addition, it is often required that every Pareto optimal solution can be reached with some parameters of the scalarization.[2] With different parameters for the scalarization, different Pareto optimal solutions are produced. A general formulation for a scalarization of a multiobjective optimization is thus

${\displaystyle {\begin{array}{ll}\min &g(f_{1}(x),\ldots ,f_{k}(x),\theta )\\{\text{s.t }}x\in X_{\theta },\end{array}}}$

where ${\displaystyle \theta }$ is a vector parameter, the set ${\displaystyle X_{\theta }\subseteq X}$ is a set depending on the parameter ${\displaystyle \theta }$ and ${\displaystyle g:\mathbb {R} ^{k+1}\mapsto \mathbb {R} }$ is a function.

Very well-known examples are the so-called

• linear scalarization
${\displaystyle \min _{x\in X}\sum _{i=1}^{k}w_{i}f_{i}(x),}$
where the weights of the objectives ${\displaystyle w_{i}>0}$ are the parameters of the scalarization, and the
• ${\displaystyle \epsilon }$-constraint method (see, e.g.[1])
${\displaystyle {\begin{array}{ll}\min &f_{j}(x)\\{\text{s.t. }}&x\in X\\&f_{i}(x)\leq \epsilon _{j}{\text{ for }}i\in \{1,\ldots ,k\}\setminus \{j\},\end{array}}}$
where upper bounds ${\displaystyle \epsilon _{j}}$ are parameters as above and ${\displaystyle f_{j}}$ is the objective to be minimized.

Somewhat more advanced examples are the achievement scalarizing problems of Wierzbicki.[34] One example of the achievement scalarizing problems can be formulated as

${\displaystyle {\begin{array}{ll}\min &\max _{i=1,\ldots ,k}\left[{\frac {f_{i}(x)-{\bar {z}}_{i}}{z_{i}^{\text{nad}}-z_{i}^{\text{utopia}}}}\right]+\rho \sum _{i=1}^{k}{\frac {f_{i}(x)}{z_{i}^{nad}-z_{i}^{\text{utopian}}}}\\{\text{subject to }}&x\in S,\end{array}}}$

where the term ${\displaystyle \rho \sum _{i=1}^{k}{\frac {f_{i}(x)}{z_{i}^{nad}-z_{i}^{\text{utopia}}}}}$ is called the augmentation term, ${\displaystyle \rho >0}$ is a small constant, and ${\displaystyle z^{\text{nad}}}$ and ${\displaystyle z^{\text{utopian}}}$ are the nadir vector and a utopian vectors, respectively. In the above problem, the parameter is the so-called reference point ${\displaystyle {\bar {z}}}$ which represents objective function values preferred by the decision maker.

For example, portfolio optimization is often conducted in terms of mean-variance analysis. In this context, the efficient set is a subset of the portfolios parametrized by the portfolio mean return ${\displaystyle \mu _{P}}$ in the problem of choosing portfolio shares so as to minimize the portfolio's variance of return ${\displaystyle \sigma _{P}}$ subject to a given value of ${\displaystyle \mu _{P}}$; see Mutual fund separation theorem for details. Alternatively, the efficient set can be specified by choosing the portfolio shares so as to maximize the function ${\displaystyle \mu _{P}-b\sigma _{P}}$; the set of efficient portfolios consists of the solutions as b ranges from zero to infinity.

## No-preference methods

Multi-objective optimization methods that do not require any preference information to be explicitly articulated by a decision maker can be classified as no-preference methods.[2] A well-known example is the method of global criterion,[35] in which a scalarized problem of the form

{\displaystyle {\begin{aligned}\min &\|f(x)-z^{ideal}\|\\{\text{s.t. }}&x\in X\end{aligned}}}

is solved. In the above problem, ${\displaystyle \|\cdot \|}$ can be any ${\displaystyle L_{p}}$ norm, with common choices including ${\displaystyle L_{1}}$, ${\displaystyle L_{2}}$ and ${\displaystyle L_{\infty }}$.[1] The method of global criterion is sensitive to the scaling of the objective functions, and thus, it is recommended that the objectives are normalized into a uniform, dimensionless scale.[1][33]

## A priori methods

A priori methods require that sufficient preference information is expressed before the solution process.[2] Well-known examples of a priori methods include the utility function method, lexicographic method, and goal programming.

In the utility function method, it is assumed that the decision maker's utility function is available. A mapping ${\displaystyle u\colon Y\rightarrow \mathbb {R} }$ is a utility function if for all ${\displaystyle \mathbf {y} ^{1},\mathbf {y} ^{2}\in Y}$ if it holds that ${\displaystyle u(\mathbf {y} ^{1})>u(\mathbf {y} ^{2})}$ if the decision maker prefers ${\displaystyle \mathbf {y} ^{1}}$ to ${\displaystyle \mathbf {y} ^{2}}$, and ${\displaystyle u(\mathbf {y} ^{1})=u(\mathbf {y} ^{2})}$ if the decision maker is indifferent between ${\displaystyle \mathbf {y} ^{1}}$ and ${\displaystyle \mathbf {y} ^{2}}$. The utility function specifies an ordering of the decision vectors (recall that vectors can be ordered in many different ways). Once ${\displaystyle u}$ is obtained, it suffices to solve

${\displaystyle \max \;u(\mathbf {f} (\mathbf {x} )){\text{ subject to }}\mathbf {x} \in X,}$

but in practice it is very difficult to construct a utility function that would accurately represent the decision maker's preferences[1] - particularly since the Pareto front is unknown before the optimization begins.

The lexicographic method assumes that the objectives can be ranked in the order of importance. We can assume, without loss of generality, that the objective functions are in the order of importance so that ${\displaystyle f_{1}}$ is the most important and ${\displaystyle f_{k}}$ the least important to the decision maker. The lexicographic method consists of solving a sequence of single-objective optimization problems of the form

{\displaystyle {\begin{aligned}\min &f_{l}(\mathbf {x} )\\{\text{s.t. }}&f_{j}(\mathbf {x} )\leq \mathbf {y} _{j}^{*},\;j=1,\dotsc ,l-1,\\&\mathbf {x} \in X,\end{aligned}}}

where ${\displaystyle \mathbf {y} _{j}^{*}}$ is the optimal value of the above problem with ${\displaystyle l=j}$. Thus, ${\displaystyle \mathbf {y} _{1}^{*}:=\min\{f_{1}(\mathbf {x} )\mid \mathbf {x} \in X\}}$ and each new problem of the form in the above problem in the sequence adds one new constraint as ${\displaystyle l}$ goes from ${\displaystyle 1}$ to ${\displaystyle k}$.

## A posteriori methods

A posteriori methods aim at producing all the Pareto optimal solutions or a representative subset of the Pareto optimal solutions. Most a posteriori methods fall into either one of the following two classes: mathematical programming-based a posteriori methods, where an algorithm is repeated and each run of the algorithm produces one Pareto optimal solution, and evolutionary algorithms where one run of the algorithm produces a set of Pareto optimal solutions.

Well-known examples of mathematical programming-based a posteriori methods are the Normal Boundary Intersection (NBI),[36] Modified Normal Boundary Intersection (NBIm) [37] Normal Constraint (NC),[38][39] Successive Pareto Optimization (SPO)[40] and Directed Search Domain (DSD)[41] methods that solve the multi-objective optimization problem by constructing several scalarizations. The solution to each scalarization yields a Pareto optimal solution, whether locally or globally. The scalarizations of the NBI, NBIm, NC and DSD methods are constructed with the target of obtaining evenly distributed Pareto points that give a good evenly distributed approximation of the real set of Pareto points.

Evolutionary algorithms are popular approaches to generating Pareto optimal solutions to a multi-objective optimization problem. Currently, most evolutionary multi-objective optimization (EMO) algorithms apply Pareto-based ranking schemes. Evolutionary algorithms such as the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) [42] and Strength Pareto Evolutionary Algorithm 2 (SPEA-2)[43] have become standard approaches, although some schemes based on particle swarm optimization and simulated annealing[44] are significant. The main advantage of evolutionary algorithms, when applied to solve multi-objective optimization problems, is the fact that they typically generate sets of solutions, allowing computation of an approximation of the entire Pareto front. The main disadvantage of evolutionary algorithms is their lower speed and the Pareto optimality of the solutions cannot be guaranteed. It is only known that none of the generated solutions dominates the others.

In 2015, a new paradigm for multi-objective optimization based on novelty was presented using evolutionary algorithms.[45] This new paradigm searches for novel solutions in objective space (i.e., novelty search[46] on objective space) in addition to the search for non-dominated solutions. Novelty search is like stepping stones guiding the search to previously unexplored places. It is specially useful in overcoming bias and plateaus as well as guiding the search in many-objective optimization problems. Subpopulation Algorithm based on Novelty[45] is the first algorithm based on this paradigm.

Commonly known a posteriori methods are listed below:

## Interactive methods

In interactive methods, the solution process is iterative and the decision maker continuously interacts with the method when searching for the most preferred solution (see e.g.[1][51]). In other words, the decision maker is expected to express preferences at each iteration in order to get Pareto optimal solutions that are of interest to him/her and learn what kind of solutions are attainable. The following steps are commonly present in interactive methods:[51]

1. initialize (e.g. calculate ideal and approximated nadir objective vectors and show them to the decision maker)
2. generate a Pareto optimal starting point (by using e.g. some no-preference method or solution given by the decision maker)
3. ask for preference information from the decision maker (e.g. aspiration levels or number of new solutions to be generated)
4. generate new Pareto optimal solution(s) according to the preferences and show it/them and possibly some other information about the problem to the decision maker
5. if several solutions were generated, ask the decision maker to select the best solution so far
6. stop, if the decision maker wants to; otherwise, go to step 3).

Above, aspiration levels refer to desirable objective function values forming a reference point. Instead of mathematical convergence that is often used as a stopping criterion in mathematical optimization methods, a psychological convergence is emphasized in interactive methods. Generally speaking, a method is terminated when the decision maker is confident that (s)he has found the most preferred solution available.

Different interactive methods involve different types of preference information. For example, three types can be identified: methods based on 1) trade-off information, 2) reference points and 3) classification of objective functions.[51] On the other hand, a fourth type of generating a small sample of solutions is included in[52] and.[53] An example of interactive method utilizing trade-off information is the Zionts-Wallenius method,[54] where the decision maker is shown several objective trade-offs at each iteration, and (s)he is expected to say whether (s)he likes, dislikes or is indifferent with respect to each trade-off. In reference point based methods (see e.g.[55][56]), the decision maker is expected at each iteration to specify a reference point consisting of desired values for each objective and a corresponding Pareto optimal solution(s) is then computed and shown to him/her for analysis. In classification based interactive methods, the decision maker is assumed to give preferences in the form of classifying objectives at the current Pareto optimal solution into different classes indicating how the values of the objectives should be changed to get a more preferred solution. Then, the classification information given is taken into account when new (more preferred) Pareto optimal solution(s) are computed. In the satisficing trade-off method (STOM)[57] three classes are used: objectives whose values 1) should be improved, 2) can be relaxed, and 3) are acceptable as such. In the NIMBUS method,[58][59] two additional classes are also used: objectives whose values 4) should be improved until a given bound and 5) can be relaxed until a given bound.

## Hybrid methods

Different hybrid methods exist, but here we consider hybridizing MCDM (multi-criteria decision making) and EMO (evolutionary multi-objective optimization). A hybrid algorithm in the context of multi-objective optimization is a combination of algorithms/approaches from these two fields (see e.g.[51]). Hybrid algorithms of EMO and MCDM are mainly used to overcome shortcomings by utilizing strengths. Several types of hybrid algorithms have been proposed in the literature, e.g. incorporating MCDM approaches into EMO algorithms as a local search operator and to lead a DM to the most preferred solution(s) etc. A local search operator is mainly used to enhance the rate of convergence of EMO algorithms.

The roots for hybrid multi-objective optimization can be traced to the first Dagstuhl seminar organized in November 2004 (see, here). Here some of the best minds in EMO (Professor Kalyanmoy Deb, Professor Jürgen Branke etc.) and MCDM (Professor Kaisa Miettinen, Professor Ralph E. Steuer etc.) realized the potential in combining ideas and approaches of MCDM and EMO fields to prepare hybrids of them. Subsequently many more Dagstuhl seminars have been arranged to foster collaboration. Recently, hybrid multi-objective optimization has become an important theme in several international conferences in the area of EMO and MCDM (see e.g.[60] and.[61])

## Visualization of the Pareto front

Visualization of the Pareto front is one of the a posteriori preference techniques of multi-objective optimization. The a posteriori preference techniques (see, for example,[1]) provide an important class of multi-objective optimization techniques. Usually the a posteriori preference techniques include four steps: (1) computer approximates the Pareto front, i.e. the Pareto optimal set in the objective space; (2) the decision maker studies the Pareto front approximation; (3) the decision maker identifies the preferred point at the Pareto front; (4) computer provides the Pareto optimal decision, which output coincides with the objective point identified by the decision maker. From the point of view of the decision maker, the second step of the a posteriori preference techniques is the most complicated one. There are two main approaches to informing the decision maker. First, a number of points of the Pareto front can be provided in the form of a list (interesting discussion and references are given in[62]) or using Heatmaps.[63]

### Visualization in bi-objective problems: tradeoff curve

In the case of bi-objective problems, informing the decision maker concerning the Pareto front is usually carried out by its visualization: the Pareto front, often named the tradeoff curve in this case, can be drawn at the objective plane. The tradeoff curve gives full information on objective values and on objective tradeoffs, which inform how improving one objective is related to deteriorating the second one while moving along the tradeoff curve. The decision maker takes this information into account while specifying the preferred Pareto optimal objective point. The idea to approximate and visualize the Pareto front was introduced for linear bi-objective decision problems by S.Gass and T.Saaty.[64] This idea was developed and applied in environmental problems by J.L. Cohon.[65] A review of methods for approximating the Pareto front for various decision problems with a small number of objectives (mainly, two) is provided in.[66]

### Visualization in high-order multi-objective optimization problems

There are two generic ideas how to visualize the Pareto front in high-order multi-objective decision problems (problems with more than two objectives). One of them, which is applicable in the case of a relatively small number of objective points that represent the Pareto front, is based on using the visualization techniques developed in statistics (various diagrams, etc. – see the corresponding subsection below). The second idea proposes the display of bi-objective cross-sections (slices) of the Pareto front. It was introduced by W.S. Meisel in 1973[67] who argued that such slices inform the decision maker on objective tradeoffs. The figures that display a series of bi-objective slices of the Pareto front for three-objective problems are known as the decision maps. They give a clear picture of tradeoffs between three criteria. Disadvantages of such an approach are related to two following facts. First, the computational procedures for constructing the bi-objective slices of the Pareto front are not stable since the Pareto front is usually not stable. Secondly, it is applicable in the case of only three objectives. In the 1980s, the idea W.S. Meisel of implemented in a different form – in the form of the Interactive Decision Maps (IDM) technique.[68]

## References

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text:
v Visualizing Classification Results: Confusion Star and Confusion Gear · Mirko Mazzoleni

# Visualizing Classification Results: Confusion Star and Confusion Gear

### Abstract

Recent developments in machine learning applications are deeply concerned with the poor interpretability of most of these techniques. To gain some insights in the process of designing data-based models it is common to graphically represent the algorithm’s results, either in their final or intermediate stage. Specially challenging is the task of plotting multiclass classification results as they involve categorical variables (classes) rather than numeric results. Using the well-known MNIST dataset and a simple neural network as an example, this paper reviews the existing techniques to visualize classification results, from those centered on a particular instance or set of instances, to those representing an overall performance metric. As classification results are commonly summarized in the form of a confusion matrix, special attention is paid to its graphical representation. From this analysis, a new visualization tool is derived, which is presented in two forms: confusion star and confusion gear. The confusion star is centered on the classification errors, while the confusion gear focuses on the classification hits. The proposed visualization tools are also evaluated when facing: (i) balanced and imbalanced classifiers issues; (ii) the problem of representing errors with different orders of magnitude. By using shapes instead of colors to represent the value of each matrix cell, the new tools significantly improve the readability of the confusion matrices. Furthermore, we show how the area enclosed by the confusion stars and gears are directly related to standard classification metrics. The new graphic tools can be also usefully employed to visualize the performances of a sequence of classifiers. [Paper, Code]

#### Reference

A. Luque, M. Mazzoleni, A. Carrasco and A. Ferramosca, "Visualizing Classification Results: Confusion Star and Confusion Gear," in IEEE Access, doi: 10.1109/ACCESS.2021.3137630 , 2021.

#### Bibtex

@ARTICLE{9658486,
author={Luque, Amalia and Mazzoleni, Mirko and Carrasco, Alejandro and Ferramosca, Antonio},
journal={IEEE Access},
title={Visualizing Classification Results: Confusion Star and Confusion Gear},
year={2021},
volume={},
number={},
pages={1-1},
doi={10.1109/ACCESS.2021.3137630}
}

format: plain


text:
# Proving that matrix in equation is invertible

The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible.

I have tried to solve it like a quadratic, but that doesn't work. Any help is appreciated!

$A^2-4A-7I=0$ $\Longrightarrow$ $A^2-4A=7I$ $\Longrightarrow$ $A\cdot \frac17(A-4I)=I$. So $A$ is invertible. Its inverse is $\frac17(A-4I)$.


text:
Generic simple coverings of the affine plane
Compositio Mathematica, Tome 93 (1994) no. 3, pp. 243-253.
@article{CM_1994__93_3_243_0,
author = {Blass, Joseph and Blass, Piotr and Lang, Jeffrey},
title = {Generic simple coverings of the affine plane},
journal = {Compositio Mathematica},
pages = {243--253},
volume = {93},
number = {3},
year = {1994},
zbl = {0839.14005},
mrnumber = {1300763},
language = {en},
url = {www.numdam.org/item/CM_1994__93_3_243_0/}
}
Blass, Joseph; Blass, Piotr; Lang, Jeffrey. Generic simple coverings of the affine plane. Compositio Mathematica, Tome 93 (1994) no. 3, pp. 243-253. http://www.numdam.org/item/CM_1994__93_3_243_0/

1 Blass, P., Picard groups of Zariski surfaces I, Comp. Math., 54 (1985), 3-86. | Numdam | MR 782383 | Zbl 0624.14021

2 Blass, P. and Lang, J., Zariski Surfaces and Differential Equations in Characteristic p > 0, Marcel Decker, Monographs in Pure and Applied Math., 106, 1987. | MR 879599 | Zbl 0614.14011

3 Fossum, R., The Divisor Class Group of a Krull Domain, Springer-Verlag, New York, 1973. | MR 382254 | Zbl 0256.13001

4 Grant, A. and Lang, J., Applications of the fundamental group and purely inseparable descent to the study of curves on Zariski surfaces, J. Alg., 132:2 (1990), 340-360. | MR 1061484 | Zbl 0738.14021

5 Lang, J., The divisor class group of the surface zpm = G(x, y) over fields of characteristic p > 0, J. Alg., 84 (1983). | MR 723398 | Zbl 0528.14017

6 Lang, J., Generic Zariski surfaces, Compositio Mathematica, 73 (1990), 345-361. | Numdam | MR 1044434 | Zbl 0738.14022

7 Samuel, P., Lectures on unique factorization domains, Tata Lectures Notes, 1964. | MR 214579 | Zbl 0184.06601

8 Shafarevich, I.R., Basic Algebraic Geometry, Grundlehren Math. Wiss., No. 213, Springer-Verlag, Heidelberg, 1974. | MR 366917 | Zbl 0284.14001

9 Steenbrink, J., On the Picard Groups of Certain Smooth Surfaces in Weighted Projective Spaces, LNM, 961, Springer-Verlag, 1982, pp. 302-313. | MR 708341 | Zbl 0507.14025


text:
## Is human sentence parsing serial or parallel? : Evidence from event-related brain potentials

2003
Hopf, Jens-Max
Meng, Michael
##### Project
Sprachverstehen und variable Wortstellung: Syntaktische und außersyntaktische Faktoren bei der Verarbeitung deutscher Sätze
Journal article
##### Published in
Cognitive Brain Research ; 15 (2003). - pp. 165-177
##### Abstract
In this ERP study we investigate the processes that occur in syntactically ambiguous German sentences at the point of disambiguation. Whereas most psycholinguistic theories agree on the view that processing difficulties arise when parsing preferences are disconfirmed (so-called garden-path effects), important differences exist with respect to theoretical assumptions about the parser s recovery from a misparse. A key distinction can be made between parsers that compute all alternative syntactic structures in parallel (parallel parsers) and parsers that compute only a single preferred analysis (serial parsers). To distinguish empirically between parallel and serial parsing models, we compare ERP responses to garden-path sentences with ERP responses to truly ungrammatical sentences. Garden-path sentences contain a temporary and ultimately curable ungrammaticality, whereas truly ungrammatical sentences remain so permanently a difference which gives rise to different predictions in the two classes of parsing architectures. At the disambiguating word, ERPs in both sentence types show negative shifts of similar onset latency, amplitude, and scalp distribution in an initial time window between 300 and 500 ms. In a following time window (500 700 ms), the negative shift to garden-path sentences disappears at right central parietal sites, while it continues in permanently ungrammatical sentences. These data are taken as evidence for a strictly serial parser. The absence of a difference in the early time window indicates that temporary and permanent ungrammaticalities trigger the same kind of parsing responses. Later differences can be related to successful reanalysis in garden-path but not in ungrammatical sentences.
##### Subject (DDC)
400 Philology, Linguistics
##### Keywords
ERP,Sentence-processing,Syntactic reanalysis,N400,LAN
##### Cite This
ISO 690HOPF, Jens-Max, Markus BADER, Michael MENG, Josef BAYER, 2003. Is human sentence parsing serial or parallel? : Evidence from event-related brain potentials. In: Cognitive Brain Research. 15, pp. 165-177. Available under: doi: 10.1016/S0926-6410(02)00149-0
BibTex
@article{Hopf2003human-3691,
year={2003},
doi={10.1016/S0926-6410(02)00149-0},
title={Is human sentence parsing serial or parallel? : Evidence from event-related brain potentials},
volume={15},
journal={Cognitive Brain Research},
pages={165--177},
author={Hopf, Jens-Max and Bader, Markus and Meng, Michael and Bayer, Josef}
}

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text:
# Calculating $f(0), f'(0), f''(0)$ for a function $f$ satisfying $\lim_{x \to 0} (1 + x + f(x)/x)^{1/x} = e^3$.

I'm trying to do the following problem and could use some help(from Apostol, Calculus, Volume I, 7.11 Ex. 33 p. 291):

A funtion $f$ has a continuous third derivative everywhere and satisfies the relation

$$\lim_{x \to 0} \left(1 + x + \dfrac{f(x)}{x}\right)^{1/x} = e^3.$$

Compute $f(0), f'(0), f''(0),$ and $\lim_{x \to 0} \left(1 + \frac{f(x)}{x}\right)^{1/x}$.

[Hint: If $\lim_{x \to 0} g(x) = A$, then $g(x) = A + o(1)$ as $x \to 0$.]

The book gives the following as answers: $f(0) = 0, f'(0) = 0, f''(0) = 4$ and the limit $= e^2$.

I cannot seem to make any forward progress on this. It is the last Exercise in a section of exercises on taking limits by using polynomial expansions of functions. (This set of Exercises is immediately before the section on L'Hopital's rule... I don't know if that is applicable here, but a solution without it is appreciated since Apostol intends this to be done without it.)

My initial attempts involve writing:

\begin{align*} & \lim_{x \to 0} \left(1 + x + \dfrac{f(x)}{x}\right)^{1/x} &= e^3.\\ \implies & \lim_{x \to 0} \left(e^{(1/x)\log(1 + x + f(x)/x)}\right) &= e^3.\\ \implies & \lim_{x \to 0} \left(e^{-\frac{\log x}{x} + \frac{1}{x} \log (x + x^2 + f(x))}\right) &= e^3. \end{align*}

I wanted to do this to attempt to get to a point that I could write a polynomial expansion of $\log$ at $0$, but I can't seem to make any progress on that front. Maybe there is a better way to simplify things?

Thanks for any help. Full solutions or hints are equally welcome.

• @PeterT.off It was some error in your typing. – AD. Apr 10 '12 at 20:56
• What is going on with this question. I can't see any formulas only here, but not in other questions! – Norbert Apr 10 '12 at 21:04
• @PeterT.off: I might well be that because of your comment, all the math on this site is broken. Could you please try to delete your comment? – Fabian Apr 10 '12 at 21:25
• @Fabian Really? How is that even possible? – Pedro Tamaroff Apr 10 '12 at 21:28
• @PeterT.off: don't know how this is possible. I just know that now the site is fine again (if you remember the content of your comment then maybe you should start a thread on meta) – Fabian Apr 10 '12 at 21:29

The hypothesis is that $\left(1+x+x^{-1}f(x)\right)^{1/x}=\mathrm e^{3+o(1)}$, that is, $1+x+x^{-1}f(x)=\mathrm e^{3x+o(x)}$. Since $\mathrm e^{3x+o(x)}=1+3x+o(x)$, one gets $x^{-1}f(x)=2x+o(x)$. Thus $f(x)=2x^2+o(x^2)$.
Assume that $f$ is twice differentiable at $0$. Then $f(x)=f(0)+f'(0)x+\frac12f''(0)x^2+o(x^2)$, by Taylor's theorem. By the uniqueness of the coefficients of the expansion of $f$ at $0$, one gets $f(0)=f'(0)=0$ and $f''(0)=4$.
Note that the fact that $f$ is twice differentiable at $0$ is a hypothesis, and certainly not a consequence of the expansion of $f$ at $0$. (On the other hand, the hypothesis that $f$ has a continuous third derivative everywhere is stronger than needed.)
Finally, $x^{-1}f(x)=o(1)$ hence $\log(1+x^{-1}f(x))=x^{-1}f(x)+o(x^{-1}f(x))=2x+o(x)$ and $x^{-1}\log(1+x^{-1}f(x))=2+o(1)$.
Taking exponentials of both sides yields $\left(1+x^{-1}f(x)\right)^{1/x}=\mathrm e^{2+o(1)}$, which is equivalent to the statement that $\left(1+x^{-1}f(x)\right)^{1/x}\to\mathrm e^{2}$ when $x\to0$.


text:
# Rotation and shear mapping of Linear algebra

I created simple examples of the rotation and shear mapping of linear algebra for understanding.

My examples this repository.

This visualized example was very helpful for understanding eigenvectors and eigenvalues.

# Rotation

The rotation uses below rotation matrix.
$A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

A rotated vector is represented like this;

$rotatedVector = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} v$

E.g.

$\begin{bmatrix} \cos90 & -\sin90 \\ \sin90 & \cos90 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$

## Example

This is my example code.

Origin image here.

Rotate it 90 degrees.

# Shear mapping

The shear mapping uses below matrix.
$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$

A sheer mapping vector is represented like this;

$shearedVector = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} v$

E.g.

$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

The x-axis of the vecter is increased, but the y-axis is not changed. Any other vectors are the same behaviour, only the x-axis values are changed.

# Example

This is my example code.

Sheard image.


text:
• ### The pion vector form factor from Lattice QCD at the physical point(1710.10401)

Oct. 28, 2017 hep-ph, hep-lat
We present an investigation of the electromagnetic pion form factor, $F_\pi(Q^2)$, at small values of the four-momentum transfer $Q^2$ ($\lesssim 0.25$ GeV$^2$), based on the gauge configurations generated by European Twisted Mass Collaboration with $N_f = 2$ twisted-mass quarks at maximal twist including a clover term. Momentum is injected using non-periodic boundary conditions and the calculations are carried out at a fixed lattice spacing ($a \simeq 0.09$ fm) and with pion masses equal to its physical value, 240 MeV and 340 MeV. Our data are successfully analyzed using Chiral Perturbation Theory at next-to-leading order in the light-quark mass. For each pion mass two different lattice volumes are used to take care of finite size effects. Our final result for the squared charge radius is $\langle r^2 \rangle_\pi = 0.443~(29)$ fm$^2$, where the error includes several sources of systematic errors except the uncertainty related to discretization effects. The corresponding value of the SU(2) chiral low-energy constant $\overline{\ell}_6$ is equal to $\overline{\ell}_6 = 16.2 ~ (1.0)$.
• ### Isospin-0 $\pi\pi$ s-wave scattering length from twisted mass lattice QCD(1612.02061)

Oct. 3, 2017 hep-lat
We present results for the isospin-0 $\pi\pi$ s-wave scattering length calculated with Osterwalder-Seiler valence quarks on Wilson twisted mass gauge configurations. We use three $N_f = 2$ ensembles with unitary (valence) pion mass at its physical value (250$\sim$MeV), at 240$\sim$MeV (320$\sim$MeV) and at 330$\sim$MeV (400$\sim$MeV), respectively. By using the stochastic Laplacian Heaviside quark smearing method, all quark propagation diagrams contributing to the isospin-0 $\pi\pi$ correlation function are computed with sufficient precision. The chiral extrapolation is performed to obtain the scattering length at the physical pion mass. Our result $M_\pi a^\mathrm{I=0}_0 = 0.198(9)(6)$ agrees reasonably well with various experimental measurements and theoretical predictions. Since we only use one lattice spacing, certain systematics uncertainties, especially those arising from unitary breaking, are not controlled in our result.
• ### First Physics Results at the Physical Pion Mass from $N_f = 2$ Wilson Twisted Mass Fermions at Maximal Twist(1507.05068)

June 26, 2017 hep-lat
We present physics results from simulations of QCD using $N_f = 2$ dynamical Wilson twisted mass fermions at the physical value of the pion mass. These simulations were enabled by the addition of the clover term to the twisted mass quark action. We show evidence that compared to previous simulations without this term, the pion mass splitting due to isospin breaking is almost completely eliminated. Using this new action, we compute the masses and decay constants of pseudoscalar mesons involving the dynamical up and down as well as valence strange and charm quarks at one value of the lattice spacing, $a \approx 0.09$ fm. Further, we determine renormalized quark masses as well as their scale-independent ratios, in excellent agreement with other lattice determinations in the continuum limit. In the baryon sector, we show that the nucleon mass is compatible with its physical value and that the masses of the $\Delta$ baryons do not show any sign of isospin breaking. Finally, we compute the electron, muon and tau lepton anomalous magnetic moments and show the results to be consistent with extrapolations of older ETMC data to the continuum and physical pion mass limits. We mostly find remarkably good agreement with phenomenology, even though we cannot take the continuum and thermodynamic limits.
• ### Isospin-0 $\pi\pi$ scattering from twisted mass lattice QCD(1701.08961)

Jan. 31, 2017 hep-lat
We present results for the isospin-0 $\pi\pi$ s-wave scattering length calculated in twisted mass lattice QCD. We use three $N_f = 2$ ensembles with unitary pion mass at its physical value, 240~MeV and 330~MeV respectively. We also use a large set of $N_f = 2 + 1 +1$ ensembles with unitary pion masses varying in the range of 230~MeV - 510~MeV at three different values of the lattice spacing. A mixed action approach with the Osterwalder-Seiler action in the valence sector is adopted to circumvent the complications arising from isospin symmetry breaking of the twisted mass quark action. Due to the relatively large lattice artefacts in the $N_f = 2 + 1 +1$ ensembles, we do not present the scattering lengths for these ensembles. Instead, taking the advantage of the many different pion masses of these ensembles, we qualitatively discuss the pion mass dependence of the scattering properties of this channel based on the results from the $N_f = 2 + 1 +1$ ensembles. The scattering length is computed for the $N_f = 2$ ensembles and the chiral extrapolation is performed. At the physical pion mass, our result $M_\pi a^\mathrm{I=0}_0 = 0.198(9)(6)$ agrees reasonably well with various experimental measurements and theoretical predictions.
• ### Review of lattice results concerning low-energy particle physics(1607.00299)

July 1, 2016 hep-ph, hep-lat
We review lattice results related to pion, kaon, D- and B-meson physics with the aim of making them easily accessible to the particle physics community. More specifically, we report on the determination of the light-quark masses, the form factor f+(0), arising in the semileptonic K -> pi transition at zero momentum transfer, as well as the decay constant ratio fK/fpi and its consequences for the CKM matrix elements Vus and Vud. Furthermore, we describe the results obtained on the lattice for some of the low-energy constants of SU(2)LxSU(2)R and SU(3)LxSU(3)R Chiral Perturbation Theory. We review the determination of the BK parameter of neutral kaon mixing as well as the additional four B parameters that arise in theories of physics beyond the Standard Model. The latter quantities are an addition compared to the previous review. For the heavy-quark sector, we provide results for mc and mb (also new compared to the previous review), as well as those for D- and B-meson decay constants, form factors, and mixing parameters. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. Finally, we review the status of lattice determinations of the strong coupling constant alpha_s.
• ### Mass of the b-quark and B-decay constants from Nf=2+1+1 twisted-mass Lattice QCD(1603.04306)

June 10, 2016 hep-ph, hep-lat
We present precise lattice computations for the b-quark mass, the quark mass ratios mb/mc and mb/ms as well as the leptonic B-decay constants. We employ gauge configurations with four dynamical quark flavors, up/down, strange and charm, at three values of the lattice spacing (a ~ 0.06 - 0.09 fm) and for pion masses as low as 210 MeV. Interpolation in the heavy quark mass to the bottom quark point is performed using ratios of physical quantities computed at nearby quark masses exploiting the fact that these ratios are exactly known in the static quark mass limit. Our results are also extrapolated to the physical pion mass and to the continuum limit and read: mb(MSbar, mb) = 4.26(10) GeV, mb/mc = 4.42(8), mb/ms = 51.4(1.4), fBs = 229(5) MeV, fB = 193(6) MeV, fBs/fB = 1.184(25) and (fBs/fB)/(fK/fpi) = 0.997(17).
• ### Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass(1507.04936)

Jan. 23, 2016 hep-ph, nucl-ex, nucl-th, hep-lat
We present results on the nucleon scalar, axial and tensor charges as well as on the momentum fraction, and the helicity and transversity moments. The pion momentum fraction is also presented. The computation of these key observables is carried out using lattice QCD simulations at a physical value of the pion mass. The evaluation is based on gauge configurations generated with two degenerate sea quarks of twisted mass fermions with a clover term. We investigate excited states contributions with the nucleon quantum numbers by analyzing three sink-source time separations. We find that, for the scalar charge, excited states contribute significantly and to a less degree to the nucleon momentum fraction and helicity moment. Our analysis yields a value for the nucleon axial charge agrees with the experimental value and we predict a value of 1.027(62) in the $\overline{\text{MS}}$ scheme at 2 GeV for the isovector nucleon tensor charge directly at the physical point. The pion momentum fraction is found to be $\langle x\rangle_{u-d}^{\pi^\pm}=0.214(15)(^{+12}_{-9})$ in the $\overline{\rm MS}$ at 2 GeV.
• ### \Delta S=2 and \Delta C=2 bag parameters in the SM and beyond from Nf=2+1+1 twisted-mass LQCD(1505.06639)

Aug. 7, 2015 hep-ph, hep-lat
We present unquenched lattice QCD results for the matrix elements of four-fermion operators relevant to the description of the neutral K and D mixing in the Standard Model and its extensions. We have employed simulations with Nf = 2 + 1 + 1 dynamical sea quarks at three values of the lattice spacings in the interval 0.06 - 0.09 fm and pseudoscalar meson masses in the range 210 - 450 MeV. Our results are extrapolated to the continuum limit and to the physical pion mass. Renormalization constants have been determined non-perturbatively in the RI-MOM scheme. In particular, for the Kaon bag-parameter, which is relevant for the \overline{K}^0-K^0 mixing in the Standard Model, we obtain B_K^{RGI} = 0.717(24).
• ### Leptonic decay constants fK, fD and fDs with Nf = 2+1+1 twisted-mass lattice QCD(1411.7908)

Feb. 25, 2015 hep-ph, hep-lat
We present a lattice QCD calculation of the pseudoscalar decay constants fK, fD and fDs performed using the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2 + 1 + 1 dynamical quarks, which include in the sea, besides two light mass degenerate quarks, also the strange and charm quarks with masses close to their values in the real world. The simulations are based on a unitary setup for the two light mass-degenerate quarks and on a mixed action approach for the strange and charm quarks. We use data simulated at three different values of the lattice spacing in the range 0.06 - 0.09 fm and at pion masses in the range 210 - 450 MeV. Our main results are: fK+ / fpi+ = 1.184 (16), fK+ = 154.4 (2.0) MeV, which incorporate the leading strong isospin breaking correction due to the up- and down-quark mass difference, and fK = 155.0 (1.9) MeV, fD = 207.4 (3.8) MeV, fDs = 247.2 (4.1) MeV, fDs / fD = 1.192 (22) and (fDs / fD) / (fK / fpi) = 1.003 (14) obtained in the isospin symmetric limit of QCD. Combined with the experimental measurements of the leptonic decay rates of kaon, pion, D- and Ds-mesons our results lead to the following determination of the CKM matrix elements: |Vus| = 0.2269 (29), |Vcd| = 0.2221 (67) and |Vcs| = 1.014 (24). Using the latest value of |Vud| from superallowed nuclear beta decays the unitarity of the first row of the CKM matrix is fulfilled at the permille level.
• ### Progress in Simulations with Twisted Mass Fermions at the Physical Point(1411.6842)

Nov. 25, 2014 hep-lat
In this contribution, results from $N_f=2$ lattice QCD simulations at one lattice spacing using twisted mass fermions with a clover term at the physical pion mass are presented. The mass splitting between charged and neutral pions (including the disconnected contribution) is shown to be around $20(20)~\mathrm{MeV}$. Further, a first measurement using the clover twisted mass action of the average momentum fraction of the pion is given. Finally, an analysis of pseudoscalar meson masses and decay constants is presented involving linear interpolations in strange and charm quark masses. Matching to meson mass ratios allows the calculation of quark mass ratios: $\mu_s/\mu_l=27.63(13)$, $\mu_c/\mu_l=339.6(2.2)$ and $\mu_c/\mu_s=12.29(10)$. From this mass matching the quantities $f_K=153.9(7.5)~\mathrm{MeV}$, $f_D=219(11)~\mathrm{MeV}$, $f_{D_s}=255(12)~\mathrm{MeV}$ and $M_{D_s}=1894(93)~\mathrm{MeV}$ are determined without the application of finite volume or discretization artefact corrections and with errors dominated by a preliminary estimate of the lattice spacing.
• ### Lattice QCD Study of $B$-meson Decay Constants from ETMC(1411.5566)

Nov. 20, 2014 hep-ph, hep-lat
We discuss a lattice QCD computation of the $B$-meson decay constants by the ETM collaboration where suitable ratios allow to reach the bottom quark sector by combining simulations around the charm-quark mass with an exactly known static limit. The different steps involved in this ratio method are discussed together with an account of the assessment of various systematic effects. A comparison of results from simulations with two and four flavour dynamical quarks is presented.
• ### Heavy flavour precision physics from Nf = 2 + 1 + 1 lattice simulations(1411.0484)

Nov. 3, 2014 hep-ph, hep-lat
We present precision lattice calculations of the pseudoscalar decay constants of the charmed sector as well as determinations of the bottom quark mass and its ratio to the charm quark mass. We employ Nf=2+1+1 dynamical quark gauge configurations generated by the European Twisted Mass Collaboration, using data at three values of the lattice spacing and pion masses as low as 210 MeV. Strange and charm sea quark masses are close to their physical values.
• ### Up, down, strange and charm quark masses with Nf = 2+1+1 twisted mass lattice QCD(1403.4504)

Aug. 21, 2014 hep-lat
We present a lattice QCD calculation of the up, down, strange and charm quark masses performed using the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2 + 1 + 1 dynamical quarks, which include in the sea, besides two light mass degenerate quarks, also the strange and charm quarks with masses close to their physical values. The simulations are based on a unitary setup for the two light quarks and on a mixed action approach for the strange and charm quarks. The analysis uses data at three values of the lattice spacing and pion masses in the range 210 - 450 MeV, allowing for accurate continuum limit and controlled chiral extrapolation. The quark mass renormalization is carried out non-perturbatively using the RI-MOM method. The results for the quark masses converted to the bar{MS} scheme are: mud(2 GeV) = 3.70(17) MeV, ms(2 GeV) = 99.6(4.3) MeV and mc(mc) = 1.348(46) GeV. We obtain also the quark mass ratios ms/mud = 26.66(32) and mc/ms = 11.62(16). By studying the mass splitting between the neutral and charged kaons and using available lattice results for the electromagnetic contributions, we evaluate mu/md = 0.470(56), leading to mu = 2.36(24) MeV and md = 5.03(26) MeV.
• ### D-Dbar Mixing in the Standard Model and Beyond from Nf=2 Twisted Mass QCD(1403.7302)

June 13, 2014 hep-ph, hep-lat
We present the first unquenched lattice QCD results for the bag parameters controlling the short distance contribution to D meson oscillations in the Standard Model and beyond. We have used the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2 dynamical quarks, at four lattice spacings and light meson masses in the range 280-500 MeV. Renormalization is carried out non-perturbatively with the RI-MOM method. The bag-parameter results have been used to constrain New Physics effects in D-Dbar mixing, to put a lower bound to the generic New Physics scale and to constrain off-diagonal squark mass terms for TeV-scale Supersymmetry.
• ### B-physics from Nf=2 tmQCD: the Standard Model and beyond(1308.1851)

March 5, 2014 hep-ph, hep-lat
We present a lattice QCD computation of the b-quark mass, the B and B_s decay constants, the B-mixing bag parameters for the full four-fermion operator basis as well as determinations for \xi and f_{Bq}\sqrt{B_i^{(q)}} extrapolated to the continuum limit and to the physical pion mass. We used N_f = 2 twisted mass Wilson fermions at four values of the lattice spacing with pion masses ranging from 280 to 500 MeV. Extrapolation in the heavy quark mass from the charm to the bottom quark region has been carried out on ratios of physical quantities computed at nearby quark masses, exploiting the fact that they have an exactly known infinite mass limit. Our results are m_b(m_b, \overline{\rm{MS}})=4.29(12) GeV, f_{Bs}=228(8) MeV, f_{B}=189(8) MeV and f_{Bs}/f_B=1.206(24). Moreover with our results for the bag-parameters we find \xi=1.225(31), B_1^{(s)}/B_1^{(d)}=1.01(2), f_{Bd}\sqrt{\hat{B}_{1}^{(d)}} = 216(10) MeV and f_{Bs}\sqrt{\hat{B}_{1}^{(s)}} = 262(10) MeV. We also computed the bag parameters for the complete basis of the four-fermion operators which are required in beyond the SM theories. By using these results for the bag parameters we are able to provide a refined Unitarity Triangle analysis in the presence of New Physics, improving the bounds coming from B_{(s)}-\bar B_{(s)} mixing.
• ### A first look at maximally twisted mass lattice QCD calculations at the physical point(1311.4522)

Nov. 18, 2013 hep-lat
In this contribution, a first look at simulations using maximally twisted mass Wilson fermions at the physical point is presented. A lattice action including clover and twisted mass terms is presented and the Monte Carlo histories of one run with two mass-degenerate flavours at a single lattice spacing are shown. Measurements from the light and heavy-light pseudoscalar sectors are compared to previous $N_f = 2$ results and their phenomenological values. Finally, the strategy for extending simulations to $N_f = 2 + 1 + 1$ is outlined.
• ### Pseudoscalar decay constants $f_K/f_\pi$, $f_D$ and $f_{D_s}$ with $N_f = 2 + 1 + 1$ ETMC configurations(1311.3080)

Nov. 13, 2013 hep-lat
We present a lattice QCD calculation of the pseudoscalar decay constants $f_K$, $f_D$ and $f_{D_s}$ performed by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$ dynamical fermions. We simulated at three different values of the lattice spacing, the smallest being approximately $0.06fm$, and with pion masses as small as $210$MeV. Our main results are: $f_{K^+}/f_{\pi^+}=1.183(17)$, $f_{K^+}=154.4(2.1)$MeV, $f_{D_s}=242.1(8.3)$MeV, $f_D=201.9(8.0)$MeV, $f_{D_s}/f_D=1.199(25)$ and $(f_{D_s}/f_D) / (f_K/f_\pi) = 1.005(15)$.
• ### A determination of the average up-down, strange and charm quark masses from $N_f=2+1+1$(1311.2793)

Nov. 13, 2013 hep-lat
We present a lattice QCD determination of the average up-down, strange and charm quark masses based on simulations performed by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$ dynamical fermions. We simulated at three different values of the lattice spacing, the smallest being approximately $0.06fm$, and with pion masses as small as $210 \text{MeV}$. Our results are: $m_{ud}(2\text{GeV})=3.70(17)\text{MeV}$, $m_s(2\text{GeV})=99.2(3.9)\text{MeV}$, $m_c(m_c)=1.350(49)\text{GeV}$, $m_s/m_{ud}=26.64(30)$ and $m_c/m_s=11.65(12)$.
• ### A $N_f = 2 + 1 + 1$ "twisted" determination of the $b$-quark mass, $f_{B}$ and $f_{B_s}$(1311.2837)

Nov. 12, 2013 hep-lat
We present a lattice QCD determination of the $b$-quark mass and of the $f_{B_s}$ and $f_B$ decay constants performed with $N_f = 2 + 1 + 1$ twisted mass Wilson fermions. We have used simulations at three values of the lattice spacing generated by ETMC with pion masses ranging from 210 to 440 MeV. To obtain physical quantities we performed a combined chiral and continuum limit and an extrapolation in the heavy quark mass from the charm to the $b$-quark region using suitable ratios calculated at nearby heavy-quark masses having an exactly known static limit. Our results are: $m_b(m_b) = 4.29 (13)$ GeV, $f_B = 196 (9)$ MeV, $f_{B_s} = 235 (9)$ MeV, $f_{B_s} / f_B = 1.201 (25)$, $(f_{B_s}/f_B)/(f_K/f_\pi) = 1.007 (16)$ and $(f_{B_s}/f_B)/(f_{D_s}/f_D) = 1.008 (13)$.
• ### B-physics computations from Nf=2 tmQCD(1310.1851)

Oct. 7, 2013 hep-lat
We present an accurate lattice QCD computation of the b-quark mass, the B and Bs decay constants, the B-mixing bag-parameters for the full four-fermion operator basis, as well as estimates for \xi and f_{Bq}\sqrt{B_q} extrapolated to the continuum limit and the physical pion mass. We have used Nf = 2 dynamical quark gauge configurations at four values of the lattice spacing generated by ETMC. Extrapolation in the heavy quark mass from the charm to the bottom quark region has been carried out using ratios of physical quantities computed at nearby quark masses, having an exactly known infinite mass limit.
• ### Kaon Mixing Beyond the SM from Nf=2 tmQCD and model independent constraints from the UTA(1207.1287)

July 26, 2013 hep-ph, hep-lat
We present the first unquenched, continuum limit, lattice QCD results for the matrix elements of the operators describing neutral kaon oscillations in extensions of the Standard Model. Owing to the accuracy of our calculation on \Delta S=2 weak Hamiltonian matrix elements, we are able to provide a refined Unitarity Triangle analysis improving the bounds coming from model independent constraints on New Physics. In our non-perturbative computation we use a combination of Nf=2 maximally twisted sea quarks and Osterwalder-Seiler valence quarks in order to achieve both O(a)-improvement and continuum-like renormalization properties for the relevant four-fermion operators. The calculation of the renormalization constants has been performed non-perturbatively in the RI-MOM scheme. Based on simulations at four values of the lattice spacing and a number of quark masses we have extrapolated/interpolated our results to the continuum limit and physical light/strange quark masses.
• ### B-physics from lattice QCD...with a twist(1212.0301)

Dec. 3, 2012 hep-ph, hep-ex, hep-lat
We present a precise lattice QCD determination of the b-quark mass, of the B and Bs decay constants and first results for the B-meson bag parameters. For our computation we employ the so-called ratio method and our results benefit from the use of improved interpolating operators for the B-mesons. QCD calculations are performed with Nf = 2 dynamical light-quarks at four values of the lattice spacing and the results are extrapolated to the continuum limit. The preliminary results are mb(mb) = 4.35(12) GeV for the MSbar b-quark mass, fBs = 234(6) MeV and fB = 197(10) MeV for the B-meson decay constants, BBs(mb) = 0.90(5) and BB(mb) = 0.87(5) for the B-meson bag parameters.
• ### Neutral meson oscillations in the Standard Model and beyond from Nf=2 Twisted Mass Lattice QCD(1211.0565)

Nov. 2, 2012 hep-ph, hep-lat
We present the ETMC results for the bag parameters describing the neutral kaon mixing in the Standard Model and beyond and preliminary results for the bag parameters controlling the short distance contributions in the D^0-\bar{D}^0 oscillations. We also present preliminary results for the B_{Bd}, B_{Bs}, B_{Bs}/B_{Bd} and \xi -parameter controlling B^0_-\bar{B}^0 oscillations in the Standard Model employing the so-called ratio method. Using Nf=2 maximally twisted sea quarks and Osterwalder-Seiler valence quarks we achieve both O(a)-improvement and continuum like renormalization pattern. Simulations are performed at three-values of the lattice spacing and several values of quark masses in the light, strange, charm region and above charm up to ~2.5m_c. Our results are extrapolated to the continuum limit and extrapolated/interpolated to the physical quark masses.
• ### B-physics from the ratio method with Wilson twisted mass fermions(1211.0568)

Nov. 2, 2012 hep-lat
We present a precise lattice QCD determination of the b-quark mass, of the B and Bs decay constants and first preliminary results for the B-mesons bag parameter. Simulations are performed with Nf = 2 Wilson twisted mass fermions at four values of the lattice spacing and the results are extrapolated to the continuum limit. Our calculation benefits from the use of improved interpolating operators for the B-mesons and employs the so-called ratio method. The latter allows a controlled interpolation at the b-quark mass between the relativistic data around and above the charm quark mass and the exactly known static limit.
• ### Isospin breaking effects due to the up-down mass difference in Lattice QCD(1110.6294)

April 13, 2012 hep-ph, hep-lat
We present a new method to evaluate with high precision isospin breaking effects due to the small mass difference between the up and down quarks using lattice QCD. Our proposal is applicable in principle to any hadronic observable which can be computed on the lattice. It is based on the expansion of the path-integral in powers of the small parameter md-mu. In this paper, we apply this method to compute the leading isospin breaking effects for several physical quantities of interest: the kaon meson masses, the kaon decay constant, the form factors of semileptonic Kl3 decays and the neutron-proton mass splitting.


text:
# American Institute of Mathematical Sciences

## A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition

 1 Nanhu College, Jiaxing University, Jiaxing, 314001, China 2 Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China, Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China 3 College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

* Corresponding author: P. Zhu

Received  September 2018 Revised  September 2019 Published  November 2020

Fund Project: P. Zhu is supported by Zhejiang Provincial Natural Science Foundation of China (Grant No.LY19A010008)

A reliable and efficient a posteriori error estimator is presented for a weak Galerkin finite element method without stabilizer for the second order elliptic equation with mixed boundary conditions. The upper bound of the estimator is proved by Helmholtz decomposition technique and lower bound is hold naturally. The performance of the estimator is illustrated by numerical experiments.

Citation: Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020340
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##### References:
Effectivity index for Example 1
Example 1. Final adaptive refinement mesh and WG solution
Effectivity index for Example 2
Example 2. Final adaptive refinement mesh and WG solution
Effectivity index for Example 3
Example 3. Final adaptive refinement mesh and WG solution
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