Linearization method of global optimization for generalized geometric programming

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Abstract

Many methods for solving generalized geometric programming (GGP) problem can only find locally optimal solutions. But up to now, less work has been devoted to solving global optimization of GGP due to the inherent difficulty. This paper gives a method for finding the globally optimal solutions of GGP. Utilizing an exponentially variable transformation and some other techniques the initial nonlinear and nonconvex GGP problem is reduced to a sequence of linear programming problems. The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Several GGP examples in the literatures are tested to demonstrate that the proposed method can systematically solve these examples to find the global optimum within a prespecified error.

Introduction

In this paper, we consider the global optimization of generalized geometric programming problem of the following form,GGP0):MinimizeΦ0(x)subjecttoΦj(x)⩽0,j=1,…,m,Ω0:={x∈Rn:0<xl⩽x⩽xu},whereΦj(x)=∑t∈Tjαjti∈Ijtxiγjti,j=0,1,…,m,and the index set of the terms in Φj(x) is represented by Tj, here, each term tTj has some nonzero real coefficient αjt, and is composed of products of monomials xγjtii for i belong to some subset Ijt of N={1,…,n}. The indices in Ijt are assumed to be distinct, each γjti is assumed to be nonzero real number. In general, formulation GGP corresponds to a nonlinear optimization problem with nonconvex objective function and constraint set. Note that if we set αjt>0, for all tTj, j=0,1,…,m, then the GGP reduces to the classical posynomial geometric programming formulation which laid the foundation for the theory of GGP problem.

GGP has found a wide range of applications in production planning, location, and distribution contexts in risk management problems, and in various chemical process design and engineering design situations [1], [2], [12], [17], [18], [19], [20], [21]. Though GGP is a special class of nonlinear programming, as noted by Ref. [15], [16], many nonlinear programming may be restated as geometric programming with very little additional effort by simple techniques such as a change of variables or by straightforward algebraic manipulation of terms. Hence, it is necessary that present good GGP algorithms.

Though local optimization methods [7], [8], [9] for solving GGP problem are ubiquitous, the global optimization algorithm based on the characteristics of GGP problem is scarce. Particular cases of GGP problem, such as bilinear programming and quadratic programming, have attracted much attention in finding globally optimal solutions. Work on these problems was reviewed by Pardalos and Rosen [3]. Sherali and Tuncbilek [4], [5], [6] derived a reformulation linearization technique (RLT) for global solution of polynomial programming problems, and subsequently linearized the resulting problem by defining new variables. Although the RLT process is promising with respect to converging to a global solution, the process is in practice very difficult to implement owing to the following reasons: (i) several types of implied constraints, or subsets need to be generated in a linearized form. Tightening its representation at the expense of an exponential constraint step by step is a long trial and error process, (ii) there are considerable variants in designing a RLT process, depending on the actual structure of the problem being solved. A user needs to formulate a special RLT scheme corresponding to each of his programs. Summarizing, the above approaches exploit the special structure of the GGP problem and therefore they are not directly applicable to GGP problem discussed in this paper. Maranas and Floudas [12] proposed a global optimization algorithm of GGP based on the convex relaxation and branch-and-bound on some hyperrectangle region.

This paper presents a new global optimization method for GGP problem by solving a sequence of linear programming problems over partitioned subsets, and the proposed method uses a convenient linearization technique to systematically convert a GGP problem into a series of linear programming problems. The solutions of these converted problems can be as close as possible to the global optimum of the original GGP problem by a successive refinement process. A comparison of this method with other methods reviewed above is given below. First, the proposed method can solve general GGP problem, but bilinear programming and quadratic programming methods [3] or RLT approach [4], [5], [6] can only treat problems with specific objective functions and constraints. Secondly, the generated relaxed linear programming problems are embedded within a branch-and-bound algorithm without increasing new variables and constraints, however the RLT algorithm always needs to generate a huge amount of bounded constraints, many of these constraints are redundant [6]. Thirdly, the proposed linear programming method for GGP problem is more convenient in the computation than the convex programming of Maranas and Floudas [12], thus any effective linear programming algorithm can be used to solve this nonlinear programming problem. This result will also promote the research on linear programming. Finally, numerical experiments are presented, which show that the proposed method stable treats all of the test problems in finding globally optimal solutions within a prespecified tolerance, and indicate some improvement over another method and the high potential of our algorithm.

This paper is organized as follows. In Section 2 the linearization technique is presented for generating the relaxed linear programming of GGP. In Section 3 the global optimization algorithm for GGP and its convergence are discussed. Numerical results are considered in Section 4 to illustrate the feasibility and effectiveness of the present algorithm, and Section 5 provides a summary.

Section snippets

Linearization technique for GGP

The principal structure in the development of a solution procedure for solving GGP is the construction of lower bounds for this problem, as well as for its partitioned subproblems. A lower bound on the solution of GGP and its partitioned subproblems can be obtained by solving a relaxation linear programming of GGP. Toward this end, we apply the exponent transformation xi=exp(yi) (i=1,…,n) for the original formulation GGP0), and can obtain the following equivalent programming problem:GGP(Ω):

Global optimization algorithm and its convergence

In this section a branch-and-bound algorithm is developed to solve the GGP based on the former linear relaxation method. This method needs to solve a sequence of relaxation linear programming over partitioned subsets of Ω in order to find a global optimum solution. Furthermore, in order to ensure convergence to a global optimum, some bound tightening strategies can be applied in order to enhance the solution procedure.

The branch-and-bound approach is based on partitioning the set Ω into

Numerical experiments

To illustrate the proposed global optimization algorithm, consider the following examples adopted from the Refs. [11], [12]. The algorithm is coded in C++ and uses the elementary simplex method to solve the relaxation linear programming. Computational experiments were made on Celeron (433 MHz) microcomputer. For the first two problems, the global minimum points are achieved in less than a second CPU time. For the last problem, the numerical results are compared with those of Ref. [11], and it

Summary

In this paper, a global optimization algorithm is presented for generalized geometric programming (GGP) problem which arise in various engineering design and robust stability problems. An exponentially variable transformation and interval method are employed to the initial nonconvex problem (GGP), and a relaxation linear programming problem of GGP is then obtained based on the linear lower bounding of the objective function and constraints. The algorithm is shown to attain finite ϵ convergence

Acknowledgements

This paper is supported by the National Natural Science Foundations of China and The Aid Financially Plan for the young skeleton teacher of Education Department of Henan Province.

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