A New Global Optimization Algorithm for Solving Generalized Geometric Programming

Abstract

A global optimization algorithm for solving generalized geometric programming (GGP) problem is developed based on a new linearization technique. Furthermore, in order to improve the convergence speed of this algorithm, a new pruning technique is proposed, which can be used to cut away a large part of the current investigated region in which the global optimal solution does not exist. Convergence of this algorithm is proved, and some experiments are reported to show the feasibility of the proposed algorithm.

1. Introduction

This paper considers generalized geometric programming GGP problem in the following form: (GGP)min𝜙₀(𝑥)s.t.𝜙_𝑗(𝑥)≤𝛽_𝑗𝑋,𝑗=1,…,𝑚,₀=𝑥∶0<𝑙⁰≤𝑥≤𝑢⁰, (1.1) where 𝜙_𝑗(𝑥)=∑^𝑇𝑗_𝑡=1𝑐_𝑗𝑡∏^𝑛_𝑖=1𝑥^{𝛾𝑗𝑡𝑖}_𝑖,𝑐_𝑗𝑡,𝛽_𝑗,𝛾_𝑗𝑡𝑖∈𝑅,𝑡=1,…,𝑇_𝑗,𝑖=1,2,…,𝑛,𝑗=0,1,…,𝑚.

Generally speaking, GGP problem is a non convex programming problem, which has a wide variety of applications, such as in engineering design, economics and statistics, manufacturing, and distribution contexts in risk management problems [1–4].

During the past years, many local optimization approaches for solving GGP problem have been presented [5, 6], but global optimization algorithms based on the characteristics of GGP problem are scarce. Maranas and Floudas [7] proposed a global optimization algorithm for solving GGP problem based on convex relaxation. Shen and Zhang [8] presented a method to globally solve GGP problem by using linear relaxation. Recently, several branch and bound algorithms have been developed [9, 10].

The purpose of this paper is to introduce a new global optimization algorithm for solving GGP problem. In this algorithm, by utilizing the special structure of GGP problem, a new linear relaxation technique is presented. Based on this technique, the initial GGP problem is systematically converted into a series of linear programming problems. The solutions of these converted problems can approximate the global optimal solution of GGP problem by successive refinement process.

The main features of this algorithm are: (1) a new linearization technique for solving GGP problem is proposed, which applies more information of the functions of GGP problem, (2) the generated relaxation linear programming problems are embedded within a branch and bound algorithm without increasing the number of variables and constraints, (3) a new pruning technique is presented, which can be used to improve the convergence speed of the proposed algorithm, and (4) numerical experiments are given, which show that the proposed algorithm can treat all of the test problems in finding global optimal solution within a prespecified tolerance.

The structure of this paper is as follows. In Section 2, first, we construct the lower approximate linear functions for the objective function and the constraint functions of GGP problem; then, we derive the relaxation linear programming (RLP) problem of GGP problem; finally, to improve the convergence speed of our algorithm, we present a new pruning technique. In Section 3, the proposed branch and bound algorithm is described, and the convergence of the algorithm is established. Some numerical results are reported in Section 4.

2. Linear Relaxation and Pruning Technique

The principal structure in the development of a solution procedure for solving GGP problem is the construction of lower bounds for this problem, as well as for its partitioned subproblems. A lower bound of GGP problem and its partitioned subproblems can be obtained by solving a linear relaxation problem. The proposed strategy for generating this linear relaxation problem is to underestimate every nonlinear function 𝜙_𝑗(𝑥)(𝑗=0,…,𝑚) with a linear function. In what follows, all the details of this procedure will be given.

Let 𝑋=[𝑥,𝑥] represents either the initial box 𝑋⁰, or modified box as defined for some partitioned subproblem in a branch and bound scheme.

Consider term ∏^𝑛_𝑖=1𝑥^{𝛾𝑗𝑡𝑖}_𝑖 in 𝜙_𝑗(𝑥)(𝑗=0,…,𝑚). Let ∏^𝑛_𝑖=1𝑥^{𝛾𝑗𝑡𝑖}_𝑖=exp(𝑦_𝑗𝑡), then, we have 𝑦_𝑗𝑡=^𝑛_𝑖=1𝛾_𝑗𝑡𝑖ln𝑥_𝑖. (2.1)

From (2.1), we can obtain the lower bound and upper bound of 𝑦_𝑗𝑡 as follows: 𝑦_𝑗𝑡=^𝑛_𝑖=1min𝛾_𝑗𝑡𝑖ln𝑥_𝑖,𝛾_𝑗𝑡𝑖ln𝑥_𝑖,𝑦_𝑗𝑡=^𝑛_𝑖=1max𝛾_𝑗𝑡𝑖ln𝑥_𝑖,𝛾_𝑗𝑡𝑖ln𝑥_𝑖. (2.2)

To derive the linear relaxation problem, we will use a convex separation technique and a two-part relaxation technique.

2.1. First-Part Relaxation

Let 𝑦_𝑗=(𝑦_𝑗1,…,𝑦_𝑗𝑇𝑗), then 𝜙_𝑗(𝑥) can be expressed in the following form: 𝜙_𝑗(𝑥)=^𝑇𝑗_𝑡=1𝑐^𝑛_{𝑗𝑡𝑖=1}𝑥^{𝛾𝑗𝑡𝑖}_𝑖=^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗𝑡≜𝑓_𝑗𝑦_𝑗. (2.3) For 𝑓_𝑗(𝑦_𝑗), we can derive its gradient and the Hessian matrix: ∇𝑓_𝑗𝑦_𝑗=𝑐_𝑗1exp𝑦_𝑗1⋮𝑐_𝑗𝑇𝑗exp𝑦_𝑗𝑇𝑗,∇²𝑓_𝑗𝑦_𝑗=𝑐_𝑗1exp𝑦_𝑗1𝑐_𝑗2exp𝑦_𝑗2⋱𝑐_𝑗𝑇𝑗exp𝑦_𝑗𝑇𝑗. (2.4) Therefore, by (2.4), the following relation holds: ∇²𝑓_𝑗𝑦_𝑗≤max_{1≤𝑡≤𝑇𝑗}𝑐_𝑗𝑡max_{1≤𝑡≤𝑇𝑗}exp𝑦_𝑗𝑡. (2.5) Let 𝜆_𝑗=max_{1≤𝑡≤𝑇𝑗}|𝑐_𝑗𝑡|max_{1≤𝑡≤𝑇𝑗}exp(𝑦_𝑗𝑡)+0.1, then, for all 𝑦_𝑗∈𝑌_𝑗≜[𝑦_𝑗,𝑦_𝑗], we have ∇²𝑓_𝑗𝑦_𝑗<𝜆_𝑗. (2.6) Thus, the function (1/2)𝜆_𝑗‖𝑦_𝑗‖²+𝑓_𝑗(𝑦_𝑗) is a convex function on 𝑌_𝑗. Consequently, the function 𝑓_𝑗(𝑦_𝑗) can be decomposed into the difference of two convex functions, that is, 𝑓_𝑗(𝑦_𝑗) is a d.c. function, which admits the following d.c. decomposition:𝑓_𝑗𝑦_j=𝑔_𝑗𝑦_𝑗−ℎ_𝑗𝑦_𝑗, (2.7) where 𝑔_𝑗𝑦_𝑗=12𝜆_𝑗𝑦_𝑗²+𝑓_𝑗𝑦_𝑗,ℎ_𝑗𝑦_𝑗=12𝜆_𝑗𝑦_𝑗². (2.8) Let 𝑦_𝑗mid=(1/2)(𝑦_𝑗+𝑦_𝑗). Since 𝑔_𝑗(𝑦_𝑗) is a convex function, we have 𝑔_𝑗𝑦_𝑗≥𝑔_𝑗𝑦_𝑗mid+∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗−𝑦_𝑗mid≜𝑔^𝑙_𝑗𝑦_𝑗. (2.9) In addition, for 𝑦_𝑗𝑡∈[𝑦_𝑗𝑡,𝑦_𝑗𝑡], it is not difficult to show 𝑦_𝑗𝑡+𝑦_𝑗𝑡𝑦_𝑗𝑡−𝑦_𝑗𝑡𝑦_𝑗𝑡≥𝑦²_𝑗𝑡. (2.10) Furthermore, we can obtain ^𝑇𝑗_𝑡=1𝑦_𝑗𝑡+𝑦_𝑗𝑡𝑦_𝑗𝑡−𝑦_𝑗𝑡𝑦_𝑗𝑡≥𝑦_𝑗². (2.11) Since 𝜆_𝑗>0, it follows that ℎ^𝑙_𝑗𝑦_𝑗≜12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡+𝑦_𝑗𝑡𝑦_𝑗𝑡−𝑦_𝑗𝑡𝑦_𝑗𝑡≥12𝜆_𝑗𝑦_𝑗²=ℎ_𝑗𝑦_𝑗. (2.12) Thus, from (2.7), (2.9), and (2.12), we have 𝑓^𝑙_𝑗𝑦_𝑗≜𝑔^𝑙_𝑗𝑦_𝑗−ℎ^𝑙_𝑗𝑦_𝑗≤𝑓_𝑗𝑦_𝑗. (2.13) Hence, by (2.1), (2.3), and (2.13), the first-part relaxation 𝐿_𝑗(𝑥) of 𝜙_𝑗(𝑥) can be obtained as follows 𝜙_𝑗(𝑥)=𝑓_𝑗𝑦_𝑗=𝑔_𝑗𝑦_𝑗−ℎ_𝑗𝑦_𝑗≥𝑔^𝑙_𝑗𝑦_𝑗−ℎ^𝑙_𝑗𝑦_𝑗=𝑔_𝑗𝑦_𝑗mid+∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗−𝑦_𝑗mid−12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡+𝑦_𝑗𝑡𝑦_𝑗𝑡−𝑦_𝑗𝑡𝑦_𝑗𝑡=^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗mid(𝑡)+𝜆_𝑗𝑦_𝑗mid(𝑡)−12𝜆_𝑗𝑦_𝑗𝑡+𝑦_𝑗𝑡𝑦_𝑗𝑡+𝑔_𝑗𝑦_𝑗mid−∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗mid+12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡𝑦_𝑗𝑡=^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗mid(𝑡)+𝜆_𝑗𝑦_𝑗mid(𝑡)−12𝜆_𝑗𝑦_𝑗𝑡+𝑦_𝑗𝑡^𝑛_𝑖=1𝛾_𝑗𝑡𝑖ln𝑥_𝑖+𝑔_𝑗𝑦_𝑗mid−∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗mid+12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡𝑦_𝑗𝑡=^𝑛_𝑖=1^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗mid(𝑡)+𝜆_𝑗𝑦_𝑗mid(𝑡)−12𝜆_𝑗𝑦_𝑗𝑡+𝑦_𝑗𝑡𝛾_𝑗𝑡𝑖ln𝑥_𝑖+𝑔_𝑗𝑦_𝑗mid−∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗mid+12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡𝑦_𝑗𝑡≜𝐿_𝑗(𝑥). (2.14)

2.2. Second-Part Relaxation

Consider the function ln𝑥_𝑖 on the interval [𝑥_𝑖,𝑥_𝑖]. As we all know, its linear lower bound function and upper bound function can be derived as follows: 𝐾_𝑖𝑥_𝑖−𝑥_𝑖+ln𝑥_𝑖≤ln𝑥_𝑖≤𝐾_𝑖𝑥_𝑖−1−ln𝐾_𝑖, (2.15) where 𝐾_𝑖=(ln𝑥_𝑖−ln𝑥_𝑖)/(𝑥_𝑖−𝑥_𝑖).

In 𝐿_𝑗(𝑥)(𝑗=0,…,𝑚), let 𝛼_𝑗𝑖=^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗mid(𝑡)+𝜆_𝑗𝑦_𝑗mid(𝑡)−12𝜆_𝑗𝑦_𝑗𝑡+𝑦_𝑗𝑡,𝑖=1,…,𝑛. (2.16) Then, from (2.14) and (2.15),we can obtain the linear lower bound function of 𝜙_𝑗(𝑥) denoted by 𝜙^𝑙_𝑗(𝑥) as follows: 𝜙^𝑙_𝑗(𝑥)≜^𝑛_𝑖=1𝛼_𝑗𝑖𝜑_𝑖𝑥_𝑖+𝑔_𝑗𝑦_𝑗mid+∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗mid+12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡𝑦_𝑗𝑡, (2.17) where 𝜑_𝑖𝑥_𝑖=𝐾_𝑖𝑥_𝑖−𝑥_𝑖+ln𝑥_𝑖,if𝛼_𝑗𝑖𝐾≥0,_𝑖𝑥_𝑖−1−ln𝐾_𝑖,else. (2.18) Obviously, 𝜙^𝑙_𝑗(𝑥)≤𝐿_𝑗(𝑥)≤𝜙_𝑗(𝑥).

Consequently, the corresponding approximation relaxation linear programming (RLP) problem of GGP problem on 𝑋 can be obtained (RLP)min𝜙^𝑙₀(𝑥)s.t.𝜙^𝑙_𝑗(𝑥)≤𝛽_𝑗,𝑗=1,…,𝑚,𝑥∈𝑋. (2.19)

Theorem 2.1. Let 𝛿_𝑖=𝑥_𝑖−𝑥_𝑖,𝑖=1,…,𝑛. Then, for all 𝑥∈𝑋, the difference of 𝜙_𝑗(𝑥) and 𝜙^𝑙_𝑗(𝑥) satisfies 𝜙_𝑗(𝑥)−𝜙^𝑙_𝑗(𝑥)→0 as 𝛿_𝑖→0, 𝑖=1,…,𝑛,𝑗=0,…,𝑚.

Proof. For all 𝑥∈𝑋, let Δ=𝜙_𝑗(𝑥)−𝜙^𝑙_𝑗(𝑥)=𝜙_𝑗(𝑥)−𝐿_𝑗(𝑥)+𝐿_𝑗(𝑥)−𝜙^𝑙_𝑗(𝑥), (2.20) and let Δ¹=𝜙_𝑗(𝑥)−𝐿_𝑗(𝑥),Δ²=𝐿_𝑗(𝑥)−𝜙^𝑙_𝑗(𝑥). Then, it is obvious that we only need to prove Δ¹⟶0,Δ²⟶0,as𝛿_𝑖⟶0. (2.21) To this end, first, we consider the difference Δ¹=𝜙_𝑗(𝑥)−𝐿_𝑗(𝑥). By (2.7), (2.13), and (2.14), it follows that Δ¹=𝜙_𝑗(𝑥)−𝐿_𝑗(𝑥)=^𝑇𝑗_𝑡=1𝑐^𝑛_{𝑗𝑡𝑖=1}𝑥^{𝛾𝑗𝑡𝑖}_𝑖−^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗mid(𝑡)+𝜆_𝑗𝑦_𝑗mid(𝑡)−12𝜆_𝑗𝑦_𝑗𝑡+𝑦_𝑗𝑡×^𝑛_𝑖=1𝛾_𝑗𝑡𝑖ln𝑥_𝑖+𝑔_𝑗𝑦_𝑗mid−∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗mid+12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡𝑦_𝑗𝑡=^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗𝑡−^𝑇𝑗_𝑡=1𝑐_𝑗𝑡exp𝑦_𝑗mid(𝑡)+𝜆_𝑗𝑦_𝑗mid(𝑡)−12𝜆_𝑗𝑦_𝑗𝑡+𝑦_𝑗𝑡𝑦_𝑗𝑡+𝑔_𝑗𝑦_𝑗mid−∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗mid+12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡𝑦_𝑗𝑡=𝑓_𝑗𝑦_𝑗−𝑓^𝑙_𝑗𝑦_𝑗=𝑔_𝑗𝑦_𝑗−𝑔^𝑙_𝑗𝑦_𝑗+12𝜆_𝑗𝑦_𝑗²−12𝜆_𝑗^𝑇𝑗_𝑡=1𝑦_𝑗𝑡+𝑦_𝑗𝑡𝑦_𝑗𝑡−𝑦_𝑗𝑡𝑦_𝑗𝑡≤∇𝑔_𝑗(𝜉)−∇𝑔_𝑗𝑦_𝑗mid^𝑇𝑦_𝑗−𝑦_𝑗mid+12𝜆_𝑗𝑦_𝑗−𝑦_𝑗²≤2𝜆_𝑗𝜉−𝑦_𝑗mid𝑦_𝑗−𝑦_𝑗mid+12𝜆_𝑗𝑦_𝑗−𝑦_𝑗²≤52𝜆_𝑗𝑦_𝑗−𝑦_𝑗², (2.22) where 𝜉 is a constant vector, and satisfies 𝑔_𝑗(𝑦_𝑗)−𝑔_𝑗(𝑦_𝑗mid)=∇𝑔_𝑗(𝜉)^𝑇(𝑦_𝑗−𝑦_𝑗mid).
By the definition of 𝑦_𝑗𝑡, we have ‖𝑦_𝑗−𝑦_𝑗‖→0 as 𝛿_𝑖→0(𝑖=1,…,𝑛). Thus, we have Δ¹→0 as 𝛿_𝑖→0(𝑖=1,…,𝑛).
Second, we consider the difference Δ²=𝐿_𝑗(𝑥)−𝜙^𝑙_𝑗(𝑥). From the definitions of 𝐿_𝑗(𝑥) and 𝜙^𝑙_𝑗(𝑥), it follows that Δ²=𝐿_𝑗(𝑥)−𝜙^𝑙_𝑗(𝑥)=^𝑛_𝑖=1𝛼_𝑗𝑖ln𝑥_𝑖−𝜑_𝑖𝑥_𝑖=_{𝛼𝑗𝑖≥0}𝛼_𝑗𝑖ln𝑥_𝑖−𝐾_𝑖𝑥_𝑖−𝑥_𝑖−ln𝑥_𝑖+_{𝛼𝑗𝑖<0}𝛼_𝑗𝑖ln𝑥_𝑖−𝐾_𝑖𝑥_𝑖+1+ln𝐾_𝑖. (2.23)
By [8], we know that |ln𝑥_𝑖−𝐾_𝑖(𝑥_𝑖−𝑥_𝑖)−ln𝑥_𝑖|→0 and |ln𝑥_𝑖−𝐾_𝑖𝑥_𝑖+1+ln𝐾_𝑖|→0, as 𝛿_𝑖→0(𝑖=1,…,𝑛). Thus, we have Δ²→0 as 𝛿_𝑖→0(𝑖=1,…,𝑛).
Taken togethe the above, it implies that Δ=Δ¹+Δ²⟶0,as𝛿_𝑖⟶0(𝑖=1,…,𝑛), (2.24) and this completes the proof.

From Theorem 2.1, it follows that 𝜙^𝑙_𝑗(𝑥) will approximate the function 𝜙_𝑗(𝑥) as 𝛿_𝑖→0(𝑖=1,…,𝑛).

Based on the above discussion, it can be seen that the optimal value of RLP problem is smaller than or equal to that of GGP problem for all feasible points, that is, the optimal value of RLP problem provides a valid lower bound for the optimal value of GGP problem. Thus, for any problem (P), let us denote the optimal value of (P) by 𝑉(P), then we have 𝑉(RLP)≤𝑉(GGP).

2.3. Pruning Technique

In order to improve the convergence speed of this algorithm, we present a new pruning technique, which can be used to eliminate the region in which the global optimal solution of GGP problem does not exist.

Assume that UB is current known upper bound of the optimal value 𝜙^∗₀ of GGP problem. Let 𝜏_𝑖=𝛼_0𝑖𝐾_𝑖,𝑖=1,…,𝑛,𝑇=^𝑛_𝑖=1𝑇_𝑖,where𝑇_𝑖=𝛼_0𝑖ln𝑥_𝑖−𝐾_𝑖𝑥_𝑖,if𝛼_0𝑖𝛼≥0,_0𝑖−1−ln𝐾_𝑖,else,𝑇=𝑇+𝑔₀𝑦_0mid−∇𝑔₀𝑦_0mid^𝑇𝑦_0mid+12𝜆₀^𝑇0_𝑡=1𝑦_0𝑡𝑦_0𝑡. (2.25)

Theorem 2.2. For any subrectangle 𝑋=(𝑋_𝑖)_𝑛×1⊆𝑋⁰ with 𝑋_𝑖=[𝑥_𝑖,𝑥_𝑖]. Let 𝜌_𝑘=UB−^𝑛_{𝑖=1,𝑖≠𝑘}min𝜏_𝑖𝑥_𝑖,𝜏_𝑖𝑥_𝑖−𝑇,𝑘=1,…,𝑛. (2.26) If there exists some index 𝑘∈{1,…,𝑛} such that 𝜏_𝑘>0 and 𝜌_𝑘<𝜏_𝑘𝑥_𝑘, then there is no globally optimal solution of GGP problem on 𝑋¹; if 𝜏_𝑘<0 and 𝜌_𝑘<𝜏_𝑘𝑥_𝑘, then there is no globally optimal solution of GGP problem on 𝑋², where 𝑋¹=𝑋¹_𝑖𝑛×1⊆𝑋,with𝑋¹_𝑖=𝑋_𝑖𝜌,𝑖≠𝑘,_𝑘𝜏_𝑘,𝑥_𝑘𝑋_𝑖𝑋,𝑖=𝑘,₂=𝑋²_𝑖𝑛×1⊆𝑋,with𝑋²_𝑖=𝑋_𝑖𝑥,𝑖≠𝑘,_𝑘,𝜌_𝑘𝜏_𝑘𝑋_𝑖,𝑖=𝑘. (2.27)

Proof. First, we show that for all 𝑥∈𝑋¹, 𝜙₀(𝑥)>UB. When 𝑥∈𝑋¹, consider the 𝑘th component 𝑥_𝑘 of 𝑥. Since 𝑥_𝑘∈(𝜌_𝑘/𝜏_𝑘,𝑥_𝑘]⋂𝑋_𝑘, it follows that 𝜌_𝑘𝜏_𝑘<𝑥_𝑘≤𝑥_𝑘. (2.28) Note 𝜏_𝑘>0, we have 𝜌_𝑘<𝜏_𝑘𝑥_𝑘. From the definition of 𝜌_𝑘 and the above inequality, we obtain UB<^𝑛_{𝑖=1,𝑖≠𝑘}min𝜏_𝑖𝑥_𝑖,𝜏_𝑖𝑥_𝑖+𝜏_𝑘𝑥_𝑘≤+𝑇_𝑛𝑖=1𝜏_𝑖𝑥_𝑖=+𝑇_𝑛𝑖=1𝜏_𝑖𝜑_𝑖𝑥_𝑖+𝑔₀𝑦_0mid−∇𝑔₀𝑦_0mid^𝑇𝑦_0mid+12𝜆₀^𝑇0_𝑡=1𝑦_0𝑡𝑦_0𝑡.=𝜙^𝑙₀(𝑥). (2.29) This implies that, for all 𝑥∈𝑋¹, 𝜙₀(𝑥)≥𝜙^𝑙₀(𝑥)>UB≥𝜙^∗₀. In other words, for all 𝑥∈𝑋¹, 𝜙₀(𝑥) is always greater than the optimal value of GGP problem. Therefore, there is no globally optimal solution of GGP problem on 𝑋¹.
For all 𝑥∈𝑋², if 𝜏_𝑘<0 and 𝜌_𝑘<𝜏_𝑘𝑥_𝑘 with some 𝑘, by arguments similar to the above, we can derive that there is no globally optimal solution of GGP problem on 𝑋².

3. Algorithm and Its Convergence

In this section, based on the former relaxation linear programming (RLP) problem, a branch and bound algorithm is presented to globally solve GGP problem. In order to ensure convergence to the global optimal solution, this algorithm needs to solve a sequence of (RLP) problems over partitioned subsets of 𝑋⁰.

In this algorithm, the set 𝑋⁰ will be partitioned into subrectangles. Each subrectangle is concerned with a node of the branch and bound tree, and is associated with a relaxation linear subproblem.

At stage 𝑘 of the algorithm, suppose that we have a collection of active nodes denoted by 𝑄_𝑘. For each node 𝑋∈𝑄_𝑘, we will have computed a lower bound for the optimal value of GGP problem via solution LB(𝑋) of RLP problem, so that the lower bound of the optimal value of GGP problem on the whole initial box region 𝑋⁰ at stage 𝑘 is given by LB_𝑘=min{LB(𝑋),forall𝑋∈𝑄_𝑘}. Whenever the solution of RLP problem for any node subproblem turns out to be feasible to GGP problem, we update the upper bound UB if necessary. Then, for each stage 𝑘, the active nodes collection 𝑄_𝑘 will satisfy LB(𝑋)<UB, for all 𝑋∈𝑄_k. We now select an active node to partition its associated rectangle into two subrectangles according to the following branching rule. For such two subrectangles, the fathoming step is applied in order to identify whether the subrectangles should be eliminated. In the end, we obtain a collection of active nodes for the next stage. This process is repeated until convergence is obtained.

3.1. Branching Rule

As we all know, the critical element in guaranteeing convergence to the global optimal solution is the choice of a suitable partitioning strategy. In this paper, we choose a simple and standard bisection rule. This rule is sufficient to ensure convergence since it drives all the intervals to a singleton for all variables. Consider any node subproblem identified by rectangle 𝑋=𝑥∈𝑅^𝑛∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖,𝑖=1,…,𝑛⊆𝑋⁰. (3.1)

The branching rule is as follows.(a)Let 𝑠=argmax{𝑥_𝑖−𝑥_𝑖,𝑖=1,…,𝑛}.(b) Let 𝛾_𝑠 satisfy 𝛾_𝑠=(1/2)(𝑥_𝑠+𝑥_𝑠).(c)Let 𝑋¹=𝑥∈𝑅^𝑛∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖,𝑖≠𝑠,𝑥_𝑠≤𝑥_𝑠≤𝛾_𝑠,𝑋²=𝑥∈𝑅^𝑛∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖,𝑖≠𝑠,𝛾_𝑠≤𝑥_𝑠≤𝑥_𝑠. (3.2) Through this branching rule, the rectangle 𝑋 is partitioned into two subrectangles 𝑋¹ and 𝑋².

3.2. Algorithm Statement

Based on the former results, the basic steps of the proposed global optimization algorithm are summarized as follows. Let LB(𝑋) refer to the optimal value of RLP problem on the rectangle 𝑋.

Step 1. Choose 𝜖>0. Find an optimal solution 𝑥⁰ and the optimal value LB(𝑋⁰) for RLP problem with 𝑋=𝑋⁰. Set LB₀=LB(𝑋⁰). If 𝑥⁰ is feasible to GGP problem, then update the upper bound UB₀=𝜙₀(𝑥⁰). If UB₀−LB₀≤𝜖, then stop: 𝑥⁰ is a global ?-optimal solution of GGP problem. Otherwise, set 𝑄₀={𝑋⁰},𝐹=∅,𝑘=1.

Step 2. Set LB_𝑘=LB_𝑘−1. Subdivide 𝑋^𝑘−1 into two rectangles 𝑋^𝑘,1,𝑋^𝑘,2 via the branching rule. Let 𝐹=𝐹⋃{𝑋^𝑘−1}.

Step 3. For each new subrectangle 𝑋^𝑘,𝑡,𝑡=1,2, utilize the pruning technique of Theorem 2.2 to prune box 𝑋^𝑘,𝑡,𝑡=1,2. Update the corresponding parameters 𝐾_𝑖,𝑦_𝑗,𝑦_𝑗(𝑖=1,…,𝑛,𝑗=0,…,𝑚). Compute LB(𝑋^𝑘,𝑡) and find an optimal solution 𝑥^𝑘,𝑡 for RLP problem with 𝑋=𝑋^𝑘,𝑡, where 𝑡=1,2. If possible, update the upper bound UB_𝑘=min{UB_𝑘,𝜙₀(𝑥^𝑘,𝑡)}, and let 𝑥^𝑘 denote the point which satisfies UB_𝑘=𝜙₀(𝑥^𝑘,𝑡).

Step 4. If UB_𝑘≤LB(𝑋^𝑘,𝑡), then set 𝐹=𝐹⋃{𝑋^𝑘,𝑡}.

Step 5. 𝐹=𝐹⋃{𝑋∈𝑄_𝑘−1∣UB_𝑘≤LB(𝑋)}.

Step 6. Set 𝑄_𝑘={𝑋∣𝑋∈𝑄_𝑘−1⋃{𝑋^𝑘,1,𝑋^𝑘,2},𝑋∉𝐹}.

Step 7. Set LB_𝑘=min{LB(𝑋)∣𝑋∈𝑄_𝑘}, and let 𝑋^𝑘∈𝑄_𝑘 satisfy LB_𝑘=LB(𝑋^𝑘). If UB_𝑘−LB_𝑘≤𝜖, then stop: 𝑥^𝑘 is a global 𝜖-optimal solution of GGP problem. Otherwise, set 𝑘=𝑘+1, and go to Step 2.

3.3. Convergence of the Algorithm

The convergence properties of the algorithm are given in the following theorem.

Theorem 3.1. (a) If the algorithm is finite, then upon termination, 𝑥^𝑘 is a global 𝜖-optimal solution of GGP problem.
(b) If the algorithm is infinite, then, along any infinite branch of the branch and bound tree, an infinite sequence of iterations will be generated, and any accumulation point of the sequence {𝑥^𝑘} will be a global solution of GGP problem.

Proof. (a) If the algorithm is finite, then it terminates in some stage 𝑘,𝑘≥0. Upon termination, by the algorithm, it follows that UB_𝑘−LB_𝑘≤𝜖. From Steps 1 and 3, this implies that 𝜙₀(𝑥^𝑘)−LB_𝑘≤𝜖. Let 𝜙^∗₀ denote the optimal value of GGP problem, then, by Section 2, we know that LB_𝑘≤𝜙^∗₀. Since 𝑥^𝑘 is a feasible solution of GGP problem, 𝜙₀(𝑥^𝑘)≥𝜙^∗₀.
Taken together above, this implies that 𝜙^∗₀≤𝜙₀(𝑥^𝑘)≤LB_𝑘+𝜖≤𝜙^∗₀+𝜖. Therefore, 𝜙^∗₀≤𝜙₀(𝑥^𝑘)≤𝜙^∗₀+𝜖, and the proof of part (a) is complete.
(b) Let 𝐷 denote the feasible region of GGP problem. When the algorithm is infinite, from the construction of the algorithm, we know that LB_𝑘 is a nondecreasing sequence bounded above by min_𝑥∈𝐷𝜙₀(𝑥), which guarantees the existence of the limit LB=lim_𝑘→∞LB_𝑘≤min_𝑥∈𝐷𝜙₀(𝑥). Since {𝑥^𝑘} is contained in a compact set 𝑋⁰, there exists one convergent subsequence {𝑥^𝑠}⊆{𝑥^𝑘}, and suppose lim_𝑠→∞𝑥^𝑠=𝑥. Then, from the proposed algorithm, there exists a decreasing subsequence {𝑋^𝑟}⊆{𝑋^𝑠}, where 𝑥^𝑟∈𝑋^𝑟,LB_𝑟=LB(𝑋^𝑟)=𝜙^𝑙₀(𝑥^𝑟), and lim_𝑟→∞𝑋^𝑟=𝑥. According to Theorem 2.1, we have lim_𝑟→∞𝜙^𝑙₀𝑥^𝑟=lim_𝑟→∞𝜙₀𝑥^𝑟=𝜙₀(𝑥). (3.3)
The only remaining problem is to prove that 𝑥∈𝐷. Since 𝑋⁰ is a closed set, it follows that 𝑥∈𝑋⁰. Assume that 𝑥∉𝐷, then there exists some 𝜙_𝑗(𝑥)(𝑗=1,…,𝑚) such that 𝜙_𝑗(𝑥)>𝛽_𝑗. Since 𝜙^𝑙_𝑗(𝑥) is continuous, by Theorem 2.1, the sequence {𝜙^𝑙_𝑗(𝑥^𝑟)} will convergent to 𝜙_𝑗(𝑥). By the definition of convergence, there exists 𝑟, such that for any 𝑟>𝑟, |𝜙^𝑙_𝑗(𝑥^𝑟)−𝜙_𝑗(𝑥)|<𝜙_𝑗(𝑥)−𝛽_𝑗. Therefore, for any 𝑟>𝑟, we have 𝜙^𝑙_𝑗(𝑥^𝑟)>𝛽_𝑗, which implies that RLP(𝑋^𝑟) is infeasible. This contradicts the assumption of 𝑥^𝑟=𝑥(𝑋^𝑟). Therefore, 𝑥∈𝐷, that is, 𝑥 is a global solution of GGP problem, and the proof of part (b) is complete.

4. Numerical Experiment

In this section, we report some numerical results to verify the performance of the proposed algorithm. The test problems are implemented on a Pentium IV (1.66?GHZ) microcomputer. The algorithm is coded in Matlab 7.1 and uses simplex method to solve the relaxation linear programming problems. In our experiments, the convergence tolerance set to 𝜖=1.0𝑒−3.

Example 4.1 (see [8]). We have the following: min0.5𝑥₁𝑥₂⁻¹−𝑥₁−5𝑥₂⁻¹s.t.0.01𝑥₂𝑥₃⁻¹+0.01𝑥₂+0.0005𝑥₁𝑥₃≤1,𝑥∈𝑋⁰=𝑥∣70≤𝑥₁≤150,1≤𝑥₂≤30,0.5≤𝑥₃≤21. (4.1)
By using the method in this paper, the optimal solution is (150,30,1.3189), the optimal value is -147.6667, and the number of algorithm iterations is 557. But using the method in [8], the optimal solution is (88.72470,7.67265,1.31786), the optimal value is -83.249728406, and the number of algorithm iterations is 1829.

Example 4.2 (see [10]). We have the following: min𝑥₁s.t.4𝑥₂−4𝑥²₁≤1,−𝑥₁−𝑥₂≤−1,𝑥∈𝑋⁰=𝑥∣0.01≤𝑥₁≤15,0.01≤𝑥₂≤15. (4.2)
Through using the method in this paper, the optimal solution (0.5, 0.5) with optimal value 0.5 is found after 16 iterations. But using the method in [10], the optimal solution (0.5, 0.5) with optimal value 0.5 is found after 96 iterations.

Example 4.3. We have the following: min−𝑥₁+𝑥₁^1.2𝑥₂^0.5−𝑥₂^0.8s.t.−6𝑥₁𝑥³₂+8𝑥₂^0.5≤3,3𝑥₁−𝑥₂≤3,𝑥∈𝑋⁰=𝑥∣0.5≤𝑥₁≤15,0.5≤𝑥₂≤15. (4.3)
By utilizing the method in this paper, the optimal value -1.3501 is found after 14 iterations at an optimal solution (0.5, 1.5).

Example 4.4. We have the following: min0.3578𝑥₃^0.1+0.8357𝑥₁𝑥₅s.t.0.00002584𝑥₃𝑥₅−0.00006663𝑥₂𝑥₅−0.0000734𝑥₁𝑥₄≤5,0.00085303𝑥₂𝑥₅+0.00009395𝑥₁𝑥₄−0.00033085𝑥₃𝑥₅≤5,1.3294𝑥₂⁻¹𝑥₅⁻¹−0.4200𝑥₁𝑥₅⁻¹−0.30586𝑥₂⁻¹𝑥²₃𝑥₅⁻¹≤5,0.00024186𝑥₂𝑥₅+0.00010159𝑥₁𝑥₂+0.00007379𝑥²₃≤5,2.1327𝑥₃⁻¹𝑥₅⁻¹−0.26680𝑥₁𝑥₅⁻¹−0.40584𝑥₄𝑥₅⁻¹≤5,0.00029955𝑥₃𝑥₅+0.00007992𝑥₁𝑥₃+0.00012157𝑥₃𝑥₄≤5,𝑥∈𝑋⁰=𝑥∣1≤𝑥₁≤60,1≤𝑥₂≤60,1≤𝑥₃≤60,1≤𝑥₄≤60,1≤𝑥₅≤60. (4.4)
Utilizing the method in this paper, we find the optimal value 1.1935 after 15 iterations at an optimal solution 𝑥^∗=(1.0,30.2734,1.0,17.4640,1.0).
In the future work, we will do more numerical experiments to test the performance of the proposed algorithm.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (60974082), and the Fundamental Research Funds for the Central Universities (K50510700004).