Global Algorithm for Generalized Affine Multiplicative Programming Problem

Abstract

In this paper, a new outer space rectangle branch and bound algorithm is proposed for globally solving generalized affine multiplicative programming problem. By applying the equivalent transformations and affine approximations of bilinear function, the initial generalized affine multiplicative programming problem can be reduced to a linear relaxed programming problem. By subsequently refine the initial outer space rectangle, and by means of the subsequent solutions of a series of linear relaxed programming problems, the algorithm is convergent to the global optimal solution of the primal problem, and numerical results show the feasibility and effectiveness of the algorithm. INDEX TERMS Generalized affine multiplicative programming, global optimization, linear relaxed programming, branch and bound.

[Retracted] Branch‐and‐Reduction Algorithm for Indefinite Quadratic Programming Problem

“…On the one hand, it is because the IQPP has a wide range of applications in management science, optimal control, financial optimization, engineering design, production plan, and so on [1,2]. On the other hand, it is because many nonlinear and nonconvex optimization problems can be transformed into this form of the IQPP [3][4][5][6], such as linear multiplicative programs problem, generalized linear multiplicative programs problem, and 0-1 programs problem. In addition, since the IQPP usually produces many local optimum solutions which are not global optimum, which puts forward many important theories and computational difficulties, that it is very necessary to propose a feasible and effective algorithm for globally solving the IQPP.…”

An Efficient Outer Space Algorithm for Generalized Linear Multiplicative Programming Problem

“…Since we set the initial iteration number k = 0 in Step 1, so that the output iteration number is 0. Obviously, compared with the numerical results of Problem 11 in [18], this shows that the proposed algorithm in this paper has higher computational efficiency than that of [18].…”

“…From numerical results of Problem 11 in Table 2, by solving randomly generated large-scale generalized linear multiplicative programming problem, and with the increase of the scale of the Problem 11, our algorithm has the higher computational efficiency than that of Ref. [18]. On the whole, numerical results of Tables 1 and 2 show that the algorithm can globally solve all test problems 1-11 with the robustness and effectiveness.…”

“…. , p, Yin et al [18] proposed a new outcome space branch-and-bound algorithm for solving the general form of problem (MP); by using new linearizing technique, Wang et al [19] presented a novel algorithm for the problem (MP); based on simplex partition and search, wang et al [20] proposed a simplex branch-and-bound algorithm for the problem (MP). In addition to the branch and bound algorithm mentioned above, some other types of algorithms are also proposed to solve the problem (MP), such as, Phuong and Tuy [21] and Chen and Jiao [22] proposed two monotonic optimization methods for generalized linear fractional programming problem and generalized linear multiplicative programming problems; by utilizing the concept of level set, Liu and Zhao [10] gave a level set method for solving the linear multiplicative programming problem with exponents.…”

“…But as far as we know, apart from [18], most of the algorithms mentioned above require that acquire arbitrary values has been still little studied in the literatures. Thus, it is still very necessary to propose a new practical algorithm for the general form of the problem (MP).…”