Technical Note—Complementary Slackness Theorem in Multiple Objective Linear Programming
TL;DR: A complementary slackness theorem in multiple objective linear programming (MOLP) is proved and it is shown that the necessary and sufficient condition for existence of a pair of primal and dual efficient solutions with equal objective values is existence if a variable in the primal problem is positive.
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Abstract: A complementary slackness theorem in multiple objective linear programming (MOLP) is proved. MOLP problems may have many efficient solutions, and the objective values of the solutions may not be equal to those of efficient solutions of their dual problems. We show that the necessary and sufficient condition for existence of a pair of primal and dual efficient solutions with equal objective values is as follows if a variable in the primal problem is positive, then the corresponding relation in the dual is equality, and if a relation in the dual problem is not equality, then the corresponding variable in the primal must be zero.
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Citations
Sensitivity Analysis for Convex Multiobjective Programming in Abstract Spaces
TL;DR: In this article, it was shown that for a linear or convex multiobjective program, a dual program can be obtained which gives the primal sensitivity without any special hypothesis about the way of choosing the optimal solution in the efficient set.
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Weak duality theorem and complementary slackness theorem for linear matrix programming problems
Ferenc Gyetvàn,Yong Shi +1 more
TL;DR: A weak duality theorem and a complementary slackness theorem are presented, which were omitted from the duality theory of the Gale-Kuhn-Tucker matrix optimization problem, and it is shown that the results are closely related to the dualities of multiple-criteria and multiple-constraint level (MC^2) programming and theDuality of multiobjective linear programming respectively.
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