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

Abstract

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.

An outcome space branch and bound algorithm for affine multiplicative problems using piecewise linear approximation technique

“…[26].Taking into account the linear relaxation technique and the separability characteristics of the linear relaxation problem, Shen and Huang proposed a solution to the linear multiplication problem under the framework of the rectangular branch and bound algorithm. [27] Combining polynomial rectangular partitioning and second-order acceleration technology, Zhang et al [28] presented a polynomial-time algorithm for generalized linear multiplication problems with positive exponents.Under the framework of the branch-and-bound algorithm, Zhao et al [29] established a global algorithm for solving generalized linear multiplicative programming by comprehensively using the convex relaxation method and some region reduction techniques.Based on using the differential mean value theorem to construct a linear relaxation problem, Jiao et al [30] proposed an image space branch-and-bound algorithm for global minimization of generalized linear multiplication problems. Jiao et al [31] first derived the linear relaxation problem by using two-stage linearization technology, and based on this, proposed a branch-and-bound reduction algorithm for the generalized linear fractional programming problem.By combining some relaxation techniques and machine learning strategies, Khajavirad and Sahinidis [32] proposed a global optimization relaxation of a hybrid LP/NLP paradigm, which has been incorporated into the BARON software.…”

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

“…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. However, although many scholars have proposed some algorithms, these algorithms either can only solve a certain special form of the problem (MP), or it is difficult to solve the problem (MP) with large-size variables.…”