University:
The University Of Western Ontario
Course:
MATH 3020A | Introduction to Abstract Algebra
Academic year:

2024
Views:

257

Pages:

2
Author:

Kyuwon J.

Assignment 10 1. a) Provide a Slater point for the NLP. Also, provide a feasible point for the NLP that is not a Slater point. The constraints of the problem are: g1 (x1 , x2 ) = x1 + x2 − 2 ≤ 0, g2 (x1 , x2 ) = x1 + 2x2 − 3 ≤ 0. Solution: A Slater point satisfies gi (x) < 0 for all i. Slater Point: Choose x1 = 0.5, x2 = 0.5: g1 (0.5, 0.5) = −1.0 < 0, g2 (0.5, 0.5) = −2.0 < 0. Hence, (0.5, 0.5) is a Slater point. Non-Slater Point: A feasible but non-Slater point satisfies gi (x) ≤ 0 but not gi (x) < 0. Choose x1 = 1, x2 = 1: g1 (1, 1) = 0, g2 (1, 1) = 0. Thus, (1, 1) is feasible but not a Slater point. 1. b) Let x be a feasible solution of the NLP. Write down the optimality conditions for x. The objective function is: f (x1 , x2 ) = x21 − 14x1 + x22 − 6x2 − 7. The gradients are: 2x1 − 14 ∇f (x) = , 2x2 − 6 1 ∇g1 (x) = , 1 1 ∇g2 (x) = . 2 Solution: The KKT conditions are: • Stationarity: ∇f (x) + λ1 ∇g1 (x) + λ2 ∇g2 (x) = 0. • Complementary Slackness: λ1 g1 (x) = 0, λ2 g2 (x) = 0. • Dual Feasibility: λ1 , λ2 ≥ 0. • Primal Feasibility: g1 (x) ≤ 0, g2 (x) ≤ 0. 1. c) Using the results of (b), find an optimal solution of the NLP. Conclude that the solution you found is the unique optimal solution. Assume both constraints are active: x1 + x2 = 2, x1 + 2x2 = 3. Solving this system of equations: x1 = 1, x2 = 1. 1 Solution: Substitute x1 = 1, x2 = 1 into ∇f (x): −12 ∇f (1, 1) = . −4 From stationarity: 1 1 12 λ1 + λ2 = . 1 2 4 Solve: λ1 + λ2 = 12, λ1 + 2λ2 = 4. λ2 = −8. λ1 = 20, Thus, the optimal solution is x1 = 1, x2 = 1, and it is unique due to the convexity of f and the feasible region. 2