On Optimization Over the Efficient Set of a Multiple Objective Linear Programming Problem

• Published in: Journal of optimization theory and applications Vol. 172; no. 1; pp. 236 - 246
• Main Author: Sun, Erjiang
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2017; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Engineering; Equivalence; Integer programming; Linear programming; Mathematical analysis; Mathematical functions; Mathematical programming; Mathematics; Mathematics and Statistics; Multiple objective analysis; Operations Research/Decision Theory; Optimization; Programming; Solvers; Studies; Theory of Computation; Global optimization; Efficient set; Weakly efficient set; Mixed-integer programming; 65K; 90C; Multiple objective linear programming
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-016-1030-y

In this paper, we provide a mixed-integer programming approach for solving the problem of minimizing a real-valued function over the efficient set of a multiple objective linear program problem. Instead of solving the problem directly, we introduce a new problem of minimizing the objective function subject to some linear constraints with additional binary variables. We show under certain conditions that the two problems are equivalent. When the objective function of the original problem is a linear or convex function, the new problem is a linear or convex programming problem, respectively, with some binary variables. These problems can be solved as mixed-integer programs with current state-of-art mixed-integer programming solvers.