Branch-and-Reduction Algorithm for Indefinite Quadratic Programming Problem

Yongjian Qiu,Yuming Zhu,Jingben Yin

doi:10.1155/2021/5578427

Abstract

This paper presents a rectangular branch-and-reduction algorithm for globally solving indefinite quadratic programming problem (IQPP), which has a wide application in engineering design and optimization. In this algorithm, first of all, we convert the IQPP into an equivalent bilinear optimization problem (EBOP). Next, a novel linearizing technique is presented for deriving the linear relaxation programs problem (LRPP) of the EBOP, which can be used to obtain the lower bound of the global optimal value to the EBOP. To obtain a global optimal solution of the EBOP, the main computational task of the proposed algorithm involves the solutions of a sequence of LRPP. Moreover, the global convergent property of the algorithm is proved, and numerical experiments demonstrate the higher computational performance of the algorithm.

Highlights

E indefinite quadratic programming problem (IQPP) is a class of important nonlinear and nonconvex optimization problems; it has attracted the attention of many scholars for many years
In a branch-and-bound procedure, to globally solving the equivalent bilinear optimization problem (EBOP), the principal operation is the computation of lower bounds for the EBOP and its subproblems. e lower bounds of the global optimal values of the EBOP and its subproblems can be obtained by solving a series of linear relaxation programs problem (LRPP) of the EBOP, which can be established by the following linear relaxation technique
Based on the above constructing method of the LRPP, it is obvious that each feasible solution of the IQPP over subrectangle H is feasible to the LRPP, and the global optimal value of the LRPP is less than or equal to that of the EBOP over subrectangle H. us, the LRPP can provide a reliable lower bound for the global optimal value of the IQPP over subrectangle H

Summary

Linear Relaxation Programming Problems

Let Θi be the ith row of the matrix Θ, and let n ti Θis 􏽘 Θiksk, i 1, 2, . . . , n, k 1. E lower bounds of the global optimal values of the EBOP and its subproblems can be obtained by solving a series of LRPP of the EBOP, which can be established by the following linear relaxation technique. We can get variable, (si − ti) is a convex function about (si − ti) that si − ti + si − ti􏼁 si − ti􏼁 −. Us, based on the former discussions, we can construct the corresponding LRPP of the EBOP as follows: fL(s, t) 􏽘 cisi + 􏽘 gl si, ti􏼁, i 1 i 1. Based on the above constructing method of the LRPP, it is obvious that each feasible solution of the IQPP over subrectangle H is feasible to the LRPP, and the global optimal value of the LRPP is less than or equal to that of the EBOP over subrectangle H. us, the LRPP can provide a reliable lower bound for the global optimal value of the IQPP over subrectangle H

New Rectangular Reduction Technique

New Branch-and-Reduction Algorithm

Computational Complexity of the Algorithm

Numerical Experiment

Methods

Concluding Remarks