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I will be really grateful if someone can give me som advice. I have the following problem:

Given the program:

\begin{align*} & \mathrm{opt.}:\quad &&(x_{1} - 4)^2 + (x_{2} -3)^2\\ & \mathrm{subject \ to}: &&x^2_{1} +x^2_{2} \leq 25\\ & &&x_{1} + x_{2} \leq 5,\\ & &&x_{1} + x_{2} \geq -5, \end{align*}

a) Find all the possible maximuns and minimus with the help of the Kuhn-Tucker conditions.\ b) Classify the points found in the previous item.

I am not sure how to tackle this problem. Should I define the minimization problem and then the maximization problem and solve them separately?

My other question is, once I define the minimization (or maximization) problem, Should I find the solutions for the eight different cases that are possible?

Thank you again for your advice.

Jorge
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  • The task is straight forward. You have $f(x) = (x_1-4)^2+(x_2-3)^2$ and $g_1(x) = x_1^2+x_2^2-25$, $g_2(x) = x_1+x_2-5$, and $g_3(x) = -x_1-x_2-5$. Now write down the KKT conditions. – amsmath Oct 08 '19 at 23:13
  • Start with setting $\mu_1=0$. – amsmath Oct 08 '19 at 23:39
  • Thanks for your answer. But then again, should I set the conditions for the maximization problem, minimization problem, or both? Secondly, is there any particular reason for setting $\mu_{1} =0$ first? why not strating with $\mu_{1}=\mu_{2} = \mu_{3} = 0$? – Jorge Oct 09 '19 at 16:04
  • Sure, you can start with setting them all to zero. The KKT conditions yield points that are candidates for minima. So, find the minimum first. Then minimize $-f(x)$ to find the maximum. – amsmath Oct 09 '19 at 16:09

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