in the first writing assignment, you explored the “energy landscape” for the imprisoned micro-robot Xerxa in two and three dimensions. Using graphical methods (mostly), you estimated the position of the critical points of the energy function, and you also estimated the energy at those critical points. Since the first writing assignment, you have developed more sophisticated tools for analyzing minima and critical points (both symbolic and numeric). Further, you have learned about the frameworks of global optimization and constrained optimization. Your goal for this project is not only to apply these new methods and ideas to accurately and precisely solve an optimization problem, but to write the problem itself! By this, I mean the following: start with Xerxa’s energy function from the first writing assignment and impose a non-trivial constraint on Xerxa’s situation. You may also extend the problem in other ways, but you must have a constraint as part of your extension. You then need to translate your constrained problem into a mathematical model, solve the optimization problem you defined and, finally, explain the practical implications of your solution for Xerxa. You should write up your problem, your solution, and your conclusions in a 3–5page paper. Do not forget to include an explanation as to why the minimum & maximum you find are global extrema.
To ensure that we are on the same page, here is a sample (and reasonably simple) constrained problem that builds on the first writing project.
Xerxa’s partners in crime have been spotted near the prison. To ensure that Xerxa remains securely imprisoned, the warden has attached Xerxa to an inflexible rod of length 1mm whose other end is anchored to the point(1,1/2). Of course, the springs from the first problem are still in place. At what position does Xerxa reach maximal and minimal energies?
To model this problem mathematically, we note that the constraint forces Xerxa to live on a circle of radius 1 centered at (1,1/2). Thus, we are trying to minimize and maximize the function E(x, y) from before, subject to the constraint that Xerxa’s position (x, y) must satisfy (x−1)^2+ (y−1/2)^2 = 1.You should strive to make your story & model sufficiently interesting and challenging. It should involve constraints that are at least as complex as those described here, possibly more so. After you submit a proposed problem, I can help you scale it back (if it is too complicated), or to ramp it up, if it is not challenging enough.
the first wrinting project is going in the jpg format as additional files
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