MA576 - Midterm All answers must be motivated and clearly formulated. Explain each step in your solutior Your solutions should make very clear to the instructor that you understand all of the steps and the logic behind the steps. Correctly interpreting the statement of each proble is part of the test. Each problem is 25 points. Very importantHonor code applies fulgu must submit your own work only. This means you cannot consult any other person (except me) directly or indirectly abou anything related with the topics of this midterm during the time you alre solving it. particular, it is prohibited to post the following problems or ask anything on any website/forum/tutoring system or any other virtual means. Question 1 Let f(x) = log Xk Question 3 Consider the problem: i=1 for given, € R¹, B₁ € Randx € R¹. a) Give a formula for the gradient and the Hessian. b) Show it is convex. ex+B₂ Question 2 Let X CR¹ convex. Prove thatis an extreme pointXoff and only if the sets convex, where = CU {}. minimize x+y s.t. x+ 2y ≤9 (x-2) + (y- 4 ≤2 a) Describe the normal cone to the feasible set at the point (1 b) Use the necessary optimality conditions to solve the problem. Are these condition sufficient? Question 4 Compute the subdifferentifilxof= max(1x,0) forx € R.

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