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

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.

An Efficient Polynomial Time Algorithm for a Class of Generalized Linear Multiplicative Programs with Positive Exponents

This paper explains a region-division-linearization algorithm for solving a class of generalized linear multiplicative programs (GLMPs) with positive exponent. In this algorithm, the original nonconvex problem GLMP is transformed into a series of linear programming problems by dividing the outer space of the problem GLMP into finite polynomial rectangles. A new two-stage acceleration technique is put in place to improve the computational efficiency of the algorithm, which removes part of the region of the optimal solution without problems GLMP in outer space. In addition, the global convergence of the algorithm is discussed, and the computational complexity of the algorithm is investigated. It demonstrates that the algorithm is a complete polynomial time approximation scheme. Finally, the numerical results show that the algorithm is effective and feasible.

Output-Space Branch-and-Bound Reduction Algorithm for a Class of Linear Multiplicative Programs

In this paper, a new relaxation bounding method is proposed for a class of linear multiplicative programs. Although the 2 p − 1 variable is introduced in the construction of equivalence problem, the branch process of the algorithm is only carried out in p − dimensional space. In addition, a super-rectangular reduction technique is also given to greatly improve the convergence rate. Furthermore, we construct an output-space branch-and-bound reduction algorithm based on solving a series of linear programming sub-problems, and prove the convergence and computational complexity of the algorithm. Finally, to verify the feasibility and effectiveness of the algorithm, we carried out a series of numerical experiments and analyzed the advantages and disadvantages of the algorithm by numerical results.

An Out Space Accelerating Algorithm for Generalized Affine Multiplicative Programs Problem

This paper presents an out space branch-and-bound algorithm for solving generalized affine multiplicative programs problem. Firstly, by introducing new variables and constraints, we transform the original problem into an equivalent nonconvex programs problem. Secondly, by utilizing new linear relaxation technique, we establish the linear relaxation programs problem of the equivalent problem. Thirdly, based on the out space partition and the linear relaxation programs problem, we construct an out space branch-and-bound algorithm. Fourthly, to improve the computational efficiency of the algorithm, an out space reduction operation is employed as an accelerating device for deleting a large part of the investigated out space region. Finally, the global convergence of the algorithm is proved, and numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.