Answer Happy

Problem 4. (25%) Consider the following convex optimization min -21 -3.02 1,22 ER s.t. -X1 +2:02 < y1 (P) 2x1 + x2 < y2 (a) (10%) Let y1, yz be fixed non-negative numbers. Write down the Karush-Kuhn-Tucker (KKT) optimality conditions for the above problem. (b) (15%) Consider the following bi-level optimization problem: min Yi + y2 + +22 91,92,11,12 ER arg min -21 - 3.02 21,02ER s.t. (x1, x2) € s.t. -X1 + 2x2 < y1 2x1 + x2 < y2 20 > yi > 0, 20 > 42 > 0. In other words, the above problem minimizes the objective function yı + y2 + x1 + x2, subject to the constraint that 21, 22 is an optimal solution to the 'inner' optimization (P). The latter is parameterized by y1, 42, which is a variable in the 'outer' optimization. Using the KKT conditions derived in (a), write down a mixed integer program that is equivalent to (P). You may assume that for any y1, y2 satisfying 20 > yi > 0,20 > y2 > 0, the KKT points for (P) are bounded. Therefore, the functions defined on them will also be bounded by a large number M. Note that the formulated mixed integer program should consist of only convex constraints except for the integer requirement for a few variables.