A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

Vahid Mahmoodian (), Iman Dayarian (), Payman Ghasemi Saghand (), Yu Zhang () and Hadi Charkhgard ()
Additional contact information
Vahid Mahmoodian: Department of Industrial and Management Systems Engineering, University of South Florida, Tampa, Florida 33620
Iman Dayarian: Culverhouse College of Business, The University of Alabama, Tuscaloosa, Alabama 35487
Payman Ghasemi Saghand: Department of Industrial and Management Systems Engineering, University of South Florida, Tampa, Florida 33620
Yu Zhang: Department of Civil and Environmental Engineering, University of South Florida, Tampa, Florida 33620
Hadi Charkhgard: Department of Industrial and Management Systems Engineering, University of South Florida, Tampa, Florida 33620

INFORMS Journal on Computing, 2022, vol. 34, issue 3, 1453-1470

Abstract: This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

Keywords: multiobjective optimization; maximum multiplicative programming; optimization over the efficient set; Nash social welfare; mixed-integer programming (search for similar items in EconPapers)
Date: 2022
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