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Problem 4. (25%) Consider the following convex optimization min 21,22 ER -x13x2 s.t. x₁ + 2x≤ yı (P) 2x² + x2 < y2 ✓(10%) (10%) Let y₁, y2 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 Y₁ + y2 + x1 + x2 91,92,21,22 ER 1 - 3x2 s.t. x₁ + 2x² ≤ yı 2x² + x2 < y2 20 y10, 20 ≥ 2 ≥ 0. In other words, the above problem minimizes the objective function y₁ + y2 + x₁ + x2, subject to the constraint that x₁, x2 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 that is equivalent to (P). You may assume that for any y1, y2 satisfying 20 y₁ ≥ 0,20 ≥ 2 ≥ 0, the KKT points for (P) are bounded. Therefore, the functions defined on them will also be bounded by a large number M. s.t. (x1, x₂) € arg min 21,22 ER
Problem 4. (25%) Consider the following convex optimization min 21,22 ER -x13x2 s.t. x₁ + 2x≤ yı (P) 2x² + x2 < y2 ✓(10%) (10%) Let y₁, y2 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 Y₁ + y2 + x1 + x2 91,92,21,22 ER 1 - 3x2 s.t. x₁ + 2x² ≤ yı 2x² + x2 < y2 20 y10, 20 ≥ 2 ≥ 0. In other words, the above problem minimizes the objective function y₁ + y2 + x₁ + x2, subject to the constraint that x₁, x2 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 that is equivalent to (P). You may assume that for any y1, y2 satisfying 20 y₁ ≥ 0,20 ≥ 2 ≥ 0, the KKT points for (P) are bounded. Therefore, the functions defined on them will also be bounded by a large number M. s.t. (x1, x₂) € arg min 21,22 ER
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