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.
(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.
Rhystic53
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Erdos
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For the gradient descent method see: https://en.wikipedia.org/wiki/Gradient_descent
The answer is accepted.
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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