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

Abstract

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.

Image space branch-reduction-bound algorithm for globally minimizing a class of multiplicative problems

“…In this section, we numerically compare our algorithm with the software BARON [18] and the known branchand-bound algorithms [5,8,22,32,34,39]. All numerical tests are implemented in MATLAB R2014a and run on a microcomputer with Intel(R) Core(TM) i5-7200U CPU @2.50 GHz processor and 16 GB RAM.…”

“…In the past several decades, some algorithms have been proposed for solving such problems, which can be classified into the following categories: branch-and-bound algorithms [7,9,12,23,25,29,[35][36][37][38], level set algorithm [22], outer-approximation [21], cutting-plane method [2], finite algorithms [20,27], etc. In recent years, Shen et al [30] proposed an outer space branch-and-bound algorithm for linear multiplicative programming problem by combining the linear relaxation method with the rectangular branching technique and the outer space region reduction technique; by using the decomposability of the problem, Shen and Huang [28] proposed a decomposition branch-and-bound algorithm for linear multiplicative problem; Jiao et al [13] presented an efficient outer space branch-and-bound algorithm for generalized linear multiplicative programming problem based on the outer space search and the branch-and-bound framework; Zhang et al [39] presented a new relaxation bounding method based on the search of the output space; based on the characteristics of the initial problem, Shen et al [31] proposed a branch-and-bound algorithm for globally solving the linear multiplicative problem. Zhang et al [40] proposed an efficient polynomial time algorithm for a class of generalized linear multiplicative programs with positive exponents by utilizing a new two-stage acceleration technique; Jiao and Shang [11] gave a two-Level linear relaxation method for generalized linear fractional programming problem, which includes linear multiplicative problem; by using new affine relaxed technique, Jiao et al [16] formulated a novel branch-and-bound algorithm for solving generalized polynomial problem.…”

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

“…Recently, Shen and Wang [18] also proposed a full polynomial time approximation algorithm for resolving the problem GLMP globally, but there is no acceleration technique. Moreover, for a more comprehensive overview of the GLMP, we encourage the readers to go through the more detailed literature [8,[19][20][21].…”

“…Further, through formula (19) and combined with the definition of δ, we can understand that formula (16) is formed, and formula (15) is of course also true.…”

An Efficient Algorithm for Solving a Class of Fractional Programming Problems

“…Over the two decades, a variety of algorithms have been developed for solving the special forms of the problem (FP). For example, for the linear sum-of-ratios problem, several algorithms can be obtained, such as simplex and parametric simplex methods [2,3], image space approach [4], branchand-bound methods [5][6][7][8][9][10][11][12], trapezoidal algorithm [13,14], and monotonic optimization algorithm [15]; for the linear multiplicative programming problem, some algorithms can be also found in literatures, such as branch-and-bound algorithms [16][17][18], polynomial time approximation algorithm [19], outcome space algorithms [20,21], level set algorithm [22], heuristic method [23], and monotonic optimization algorithm [15]. Furthermore, Jiao et al [24,25] and Chen and Jiao [26] presented three different algorithms for solving the linear multiplicative programming problem.…”