Answer Happy

Problem 4. (25%) Consider the following convex optimization -21-322 min 21,72ER s.t. (P) 2x1 + x2 < y2 (a) (10%) Let y₁, 92 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 192,1₂ER Y₁+Y2 +1 +₂ arg min -₁- 3x2 11,72ER s.t. (P¹) −11+ 2x2 ≤ yı 2x1 + x2 ≤Y2 20≥ ₁ ≥0, 20≥ 2 ≥ 0. In other words, the above problem minimizes the objective function y₁ + y2 + £1 + £2, subject to the constraint that I1, I2 is an optimal solution to the 'inner' optimization (P). The latter is parameterized by 91, 92, which is a variable in the 'outer' optimization. Using the KKT conditions derived in (a), write down a mixed integer program (or a linear program, see the remark below) that is equivalent to (P). You may assume that for any 31, 32 satisfying 20 ≥ ₁ ≥ 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 Homework 6 3 the formulated mixed integer program should consist of only convex constraints except for the integer requirement for a few variables. Remark: An alternative approach is to solve the KKT conditions in (a) and express the solution in terms of y1,32. This will result in a linear program that is equivalent to (P). s.t. (11, 12) € 1+2x2 < y1