Complementary Slackness Conditions | Lecture Note - Edubirdie

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

2024
Views:

444

Pages:

1
Author:

Kyuwon J.

Solution: Complementary Slackness Conditions The given primal problem (P) is: Maximize (1 2 1)x, subject to 1 1 0 1 x= , 2 1 0 1 x ≥ 0. The corresponding dual problem (D) is: Minimize y1 + y2 ,     1 2 1    subject to 0 1 y ≥ 2 , 1 0 1 y ≥ 0. The complementary slackness conditions state that for a pair of feasible solutions x for (P) and y for (D): 1. If xj > 0, then the j-th constraint in the  1 0 1 dual problem is satisfied with equality:     2 1 1 y  = 2 . 0 1 j j 2. If yi > 0, then the i-th constraint in the primal problem is satisfied with equality: 1 0 1 1 x = . 2 1 0 1 i i From these conditions, we derive the complementary slackness equations for the specific problem: 1. Primal Variables and Dual Constraints: For each xj , we have: x1 ((1)y1 + (2)y2 − 1) = 0, x2 ((0)y1 + (1)y2 − 2) = 0, x3 ((1)y1 + (0)y2 − 1) = 0. 2. Dual Variables and Primal Constraints: For each yi , we have: y1 ((1)x1 + (0)x2 + (1)x3 − 1) = 0, y2 ((2)x1 + (1)x2 + (0)x3 − 1) = 0. These conditions ensure that the primal and dual solutions align with their respective feasible sets while satisfying complementary slackness. Final Complementary Slackness Set: The complementary slackness conditions are: 1. x1 · (y1 + 2y2 − 1) = 0, x2 · (y2 − 2) = 0, x3 · (y1 − 1) = 0, 2. y1 · (x1 + x3 − 1) = 0, y2 · (2x1 + x2 − 1) = 0. 1

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