Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships...

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Bibliographic Details
Published in:4OR Vol. 15; no. 3; pp. 303 - 326
Main Authors: Mahdavi-Amiri, N., Salehi Sadaghiani, F.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2017
Springer Nature B.V
Subjects:
Business and Management
Efficiency
Industrial and Production Engineering
Multiple objective analysis
Nonlinear programming
Operations research
Operations Research/Decision Theory
Optimization
Research Paper
Theorem proving
90C46 (Optimality conditions, duality)
90C29 (Multi-objective and goal programming)
Strictly complementary conditions
90C05 (Linear programming)
Constraint qualification
Strictly feasible points
Multiple objective programming
Farkas’ lemma
Duality
Primal-dual weakly efficient solutions
ISSN:1619-4500, 1614-2411
Online Access:Get full text
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Summary:Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.
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ISSN:1619-4500
1614-2411
DOI:10.1007/s10288-016-0338-7