Multi-objective optimization based algorithms for solving mixed integer linear minimum multiplicative programs

Abstract

We present two new algorithms for a class of single-objective non-linear optimization problems, the so-called Mixed Integer Linear minimum Multiplicative Programs (MIL-mMPs). This class of optimization problems has a desirable characteristic: a MIL-mMP can be viewed as a special case of the problem of optimization over the efficient set in multi-objective optimization. The proposed algorithms exploit this characteristic and solve any MIL-mMP from the viewpoint of multi-objective optimization. A computational study on 960 instances demonstrates that the proposed algorithms outperform a generic-purpose solver, SCIP, by a factor of more than 10 on many instances. We numerically show that selecting the best algorithm among our proposed algorithms highly depends on the class of instances used. Specifically, since one of the proposed algorithms is a decision space search algorithm and the other one is a criterion space search algorithm, one can significantly outperform the other depending on the dimension of decision space and criterion space. Although it is possible to linearize some instances of MIL-mMPs, we show that a commercial solver, CPLEX, struggles to directly solve such linearized instances because linearization introduces additional constraints and binary decision variables.