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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| Abstract | 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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| AbstractList | 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. |
| Author | Mahdavi-Amiri, N. Salehi Sadaghiani, F. |
| Author_xml | – sequence: 1 givenname: N. surname: Mahdavi-Amiri fullname: Mahdavi-Amiri, N. email: nezamm@sharif.edu organization: Faculty of Mathematical Sciences, Sharif University of Technology – sequence: 2 givenname: F. surname: Salehi Sadaghiani fullname: Salehi Sadaghiani, F. organization: Faculty of Mathematical Sciences, Sharif University of Technology |
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| Cites_doi | 10.1023/B:ANOR.0000045279.46948.dd 10.1287/mnsc.30.11.1268 10.1007/BF01106606 10.1287/opre.37.1.118 10.1007/s00186-008-0216-y 10.1016/S0377-2217(77)81008-4 10.1287/opre.40.6.1148 10.1016/S0305-0548(03)00074-1 10.1137/1.9781611971453 10.1023/A:1022614029984 10.1023/A:1026465906067 10.1016/S0377-2217(97)00056-8 10.1016/0022-247X(84)90028-3 10.1115/1.1904639 10.1016/j.ejor.2010.09.024 10.1016/0377-2217(92)90228-2 10.1016/0377-2217(93)90194-R 10.1007/978-3-319-21091-9 10.1016/0377-2217(92)90284-G 10.1016/S0377-2217(96)90078-8 10.1016/S0898-1221(99)00062-0 10.1057/jors.1974.108 10.1007/978-3-642-48697-5 10.1137/060674831 10.1007/978-0-387-74503-9 10.1007/978-1-4614-7630-6 |
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| Keywords | 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 |
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| Snippet | 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... |
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| SubjectTerms | Business and Management Efficiency Industrial and Production Engineering Multiple objective analysis Nonlinear programming Operations research Operations Research/Decision Theory Optimization Research Paper Theorem proving |
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| Title | Strictly feasible solutions and strict complementarity in multiple objective linear optimization |
| URI | https://link.springer.com/article/10.1007/s10288-016-0338-7 https://www.proquest.com/docview/1934680874 |
| Volume | 15 |
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