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44Please use the original answer and I'll give it an upvote, thanks! Please make sure the answer is correct!
44Please use the original answer and I'll give it an upvote,
thanks! Please make sure the answer is correct!
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
thanks! Please make sure the answer is correct!
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
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