Optimization problem

Consider the problem: min f(x) s.t. h(x) ? 0 where f (x) = $(x_{1} -1)^2 +2(x_{2}-2)^2$  and h(x) = $[1-(x_{1})^2 -(x_{2})^2,x_{1}+x_{2}]^T$ .

(a) Plot the contour of f(x) and the feasible set on one single figure, i.e., overlay the feasible set on the contour plot of f(x);

(b) Find a solution to the problem using the natural logarithmic barrier function, i.e., the barrier function is  -$log(h_{1}(x))-log(h_{2}(x))$ . Use initialization vector [0.5 0.5]$^{T} $  and the initial penalty parameter equal to 1 and reduce it by $\frac{1}{2} $ in each iteration. Use a stopping threshold of 0.002;

(c) In a 2-D figure, plot the trajectory (i.e., the values connected by lines with arrows) of the computed solution vector as the number of iteration progresses. 

Optimization Calculus Multivariable Calculus
Rhystic53 Rhystic53
Matchmaticians Optimization problem File #1 File #1 (jpg)
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  • Daniel90 Daniel90

    This sounds like a small optimization project. Your offered bounty is too low. I would also suggest to extend your deadline.

  • Rhystic53 Rhystic53

    Dont think so, and changing the deadline would mean nothing

  • Nirenberg Nirenberg

    Is this a question for numerical analysis? It is not possible to compute the solution analytically, you should use methods like gradient descent algorithm to find the minimizer.

  • Nirenberg Nirenberg

    Do you need a code or just the algorithm?

  • Rhystic53 Rhystic53

    code is preferable

  • Nirenberg Nirenberg

    That's too much work! I can write a phsodocode, if you want to implement the code yourself, and graph solutions with a computer on your own.

  • Rhystic53 Rhystic53

    thats fine just answer every part of the problem

  • Nirenberg Nirenberg

    Part c) can not be answered without a code and computing the solutions. So can only answer parts a) and b) and provide a phsodocode.

  • Rhystic53 Rhystic53

    I suppose thats good enough

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Nirenberg Nirenberg
  • Nirenberg Nirenberg

    For the gradient descent method see: https://en.wikipedia.org/wiki/Gradient_descent

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