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. 

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

  • Dont think so, and changing the deadline would mean nothing

  • 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.

  • Do you need a code or just the algorithm?

  • code is preferable

  • 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.

  • thats fine just answer every part of the problem

  • 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.

  • I suppose thats good enough

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

The answer is accepted.