Global Optimization for Sum of Linear Ratios Problem Using New Pruning Technique

Abstract

A global optimization algorithm is proposed for solving sum of general linear ratios problem (P) using new pruning technique. Firstly, an equivalent problem (P1) of the (P) is derived by exploiting the characteristics of linear constraints. Then, by utilizing linearization method the relaxation linear programming (RLP) of the (P1) can be constructed and the proposed algorithm is convergent to the global minimum of the (P) through the successive refinement of the linear relaxation of feasible region and solutions of a series of (RLP). Then, a new pruning technique is proposed, this technique offers a possibility to cut away a large part of the current investigated feasible region by the optimization algorithm, which can be utilized as an accelerating device for global optimization of problem (P). Finally, the numerical experiments are given to illustrate the feasibility of the proposed algorithm.

1. Introduction

Consider the following sum of general linear ratios problem:(P)∶minℎ(𝑥)=^𝑝_𝑗=1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥𝑑_𝑗0+𝑑^𝑇_𝑗𝑥,s.t.𝐴𝑥≤𝑏,𝑥∈𝑅^𝑁, (1.1)where 𝐴∈𝑅^𝑀×𝑁,𝑏∈𝑅^𝑀, 𝑝 is a natural number, 𝑐_𝑗0+𝑐^𝑇_𝑗𝑥and𝑑_𝑗0+𝑑^𝑇_𝑗𝑥 are all finite affine functions on 𝑅^𝑁 such that 𝑑_𝑗0+𝑑^𝑇_𝑗𝑥≠0 for all 𝑥∈𝑋={𝑥∣𝐴𝑥≤𝑏},𝑗=1,…,𝑝.

Problem (P) has attracted the interest of researchers for many years. This is because problem (P) has a number of important applications, including multistage shipping problems, cluster analysis, and multiobjective bond portfolio [1, 2]. However, some computational difficulties can be encountered, since multiple local optima of problem (P) that are not globally optimal exist.

In the last decades, many solution algorithms have been proposed for globally solving special cases of the (P), which are intended only for the sum of positive linear ratios problem with assumption that 𝑐_𝑗0+𝑐^𝑇_𝑗𝑥≥0 and 𝑑_𝑗0+𝑑^𝑇_𝑗𝑥>0 for all 𝑥∈𝑋={𝑥∣𝐴𝑥≤𝑏} [2, 3]. In [4], Kuno proposed a new method for solving the maximization of sum of linear ratios, which used a concave function to overestimate the optimal value of the original problem. A global optimization method was considered by Jiao et al. [5] by introducing parameters, then the global optimal solution can be derived using linear relaxation and branch and bound algorithm. Recently, in [6], Ji et al. presented a deterministic global optimization algorithm for the linear sum-of-ratios problem, and Jiao and Chen [7] give a short extensive application for the algorithm proposed in [6]. Though optimization methods for special forms of the (P) are ubiquitous, to our knowledge, little work has been done in the literature for globally solving the sum of general linear ratios problem (P) which the numerators and denominators of the ratios may be arbitrary value except that the denominators of the ratios are nonzero over the feasible region considered in this paper.

The purpose of this paper is to develop a deterministic algorithm for solving sum of general linear ratios problem (P) which the numerators and denominators of the ratios may be arbitrary value except that the denominators of the ratios are nonzero over the feasible region. The main feature of the algorithm is described as follows. (1) An equivalent optimization problem (P1) of the (P) is derived by exploiting the characteristics of this linear constraints. (2) A new linearization method is proposed to linearize the objective function of the (P1), and the linear relaxation of the (P1) is easier to be obtained and need not introduce new variables and constraints compared with the method in [5, 8]. (3) A new pruning technique is given, and this technique offers the possibility to cut away a large part of the current investigated feasible region. Using the new technique as an accelerating device and applying it to the proposed algorithm, we can largely reduce current investigated feasible region to improve the convergence of the algorithm. (4) The proposed algorithm is convergent to the global minimum through the successive refinement of the linear relaxation of feasible region of the objective function and solutions of a series of (RLP). Finally, the numerical results show the feasibility and effectiveness of the proposed algorithm.

The organization of this article is as follows. In Section 2, we show how to convert the (P) into equivalent problem (P1), and generate the relaxed linear programing (RLP) of the (P1). In Section 3, the proposed branch-and-bound algorithm in which the relaxed subproblems are embedded is described, and its convergence is shown. Some numerical results are reported in Section 4 and Section 5 provides some concluding remarks.

2. Linear Relaxation Programing

In this section, first we convert the (P) into an equivalent nonconvex programing problem (P1). In order to globally solve the (P), the branch and bound algorithm to be presented can be applied to the (P1).

Firstly, we solve the following 2𝑁+𝑝 linear programing problems:𝑥_𝑖=min𝑥_𝑖,s.t.𝐴𝑥≤𝑏,𝑥_𝑖=min𝑥_𝑖,𝑟s.t.𝐴𝑥≤𝑏,_𝑗=min𝑐_𝑗0+𝑐^𝑇_𝑗𝑥,s.t.𝐴𝑥≤𝑏, (2.1)where 𝑖=1,2,…,𝑁,𝑗=1,…,𝑝. Then, we can get initial partition rectangle 𝑋⁰={𝑥∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖,𝑖=1,2,…,𝑁}. Without loss of generality, assume 𝑟_𝑗≥0(𝑗=1,…,𝑞),𝑟_𝑗<0(𝑗=𝑞+1,…,𝑝), the problem (P) can be rewritten in the following form:(P1)∶minℎ(𝑥)=^𝑞_𝑗=1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥𝑑_𝑗0+𝑑^𝑇_𝑗𝑥+^𝑝_𝑗=𝑞+1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥−𝑟_𝑗𝑑_𝑗0+𝑑^𝑇_𝑗𝑥−^𝑝_𝑗=𝑞+1−𝑟_𝑗𝑑_𝑗0+𝑑^𝑇_𝑗𝑥,s.t.𝐴𝑥≤𝑏,𝑥∈𝑋⁰. (2.2)Obviously, we have 𝑐_𝑗0+𝑐^𝑇_𝑗𝑥≥0 for each 𝑗∈{1,…,𝑞}, and 𝑐_𝑗0+𝑐^𝑇_𝑗𝑥−𝑟_𝑗≥0 and −𝑟_𝑗>0 for each 𝑗∈{𝑞+1,…,𝑝}.

Theorem 2.1. Problems (P) and the (P1) have the same global optimal solution.

Proof. Obviously, if 𝑥 is feasible to the (P), then 𝑥∈𝑋⁰. Conversely, if 𝑥 is feasible to the (P1), then 𝐴𝑥≤𝑏. So they have the same feasible region, then conclusion is followed.
The linear relaxation of the (P1) can be realized by underestimating function ℎ(𝑥) with a linear function ℎ^𝑙(𝑥). All the details of this linearization technique for generating relaxations will be given in the following theorems.
Given any 𝑋=[𝑥,𝑥]⊆𝑋⁰ and for all 𝑥=(𝑥_𝑖)_𝑛×1∈𝑋, the following notations are introduced:ℎ^𝑙(𝑥)=^𝑞_𝑗=1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥𝑑^𝑢_𝑗+^𝑝_𝑗=𝑞+1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥−𝑟_𝑗𝑑^𝑢_𝑗−^𝑝_𝑗=𝑞+1−𝑟_𝑗𝑑^𝑙_𝑗,ℎ^𝑢(𝑥)=^𝛾_𝑗=1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥𝑑^𝑙_𝑗+^𝑝_𝑗=𝑞+1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥−𝑟_𝑗𝑑^𝑙_𝑗−^𝑝_𝑗=𝑞+1−𝑟_𝑗𝑑^𝑢_𝑗,𝑑^𝑙_𝑗=min_𝑥∈𝑋(𝑑_𝑗0+𝑑^𝑇_𝑗𝑥),𝑑^𝑢_𝑗=max_𝑥∈𝑋(𝑑_𝑗0+𝑑^𝑇_𝑗𝑥),𝑗=1,…,𝑝. (2.3)(2.4)(2.5)

Theorem 2.2. Let 𝑑_𝑗(𝑥)≠0 for any 𝑥∈𝑋 and let 𝑑^𝑙_𝑗 and 𝑑^𝑢_𝑗 defined in (2.5). Then, 1/𝑑^𝑢_𝑗≤1/𝑑_𝑗(𝑥)≤1/𝑑^𝑙_𝑗(𝑗=1,…,𝑝), for all 𝑥∈𝑋.

Proof. Obviously, by the continuity of the function 𝑑_𝑗(𝑥), there must exist some 𝑥¹∈𝑋 and 𝑥²∈𝑋, such that 𝑑_𝑗(𝑥¹)=𝑑^𝑙_𝑗 and 𝑑_𝑗(𝑥²)=𝑑^𝑢_𝑗. If 𝑑^𝑢_𝑗𝑑^𝑙_𝑗<0, then 𝑑^𝑙_𝑗<0 and 𝑑^𝑢_𝑗>0, that is, 𝑑_𝑗(𝑥¹)<0 and 𝑑_𝑗(𝑥²)>0. Then, there must exist one positive number 𝜎∈(0,1) such that 𝑑_𝑗(𝑥¹+𝜎(𝑥²−𝑥¹))=0, where 𝑥¹+𝜎(𝑥²−𝑥¹)∈𝑋, which contradicts 𝑑_𝑗(𝑥)≠0 for any 𝑥∈𝑋. Therefore, we have 𝑑^𝑢_𝑗𝑑^𝑙_𝑗>0, that is, 𝑑^𝑢_𝑗≥𝑑^𝑙_𝑗>0 or 𝑑^𝑙_𝑗≤𝑑^𝑢_𝑗<0. Obviously, for ∀𝑥∈𝑋, the following two conclusions hold.
(i) If 𝑑^𝑢_𝑗≥𝑑^𝑙_𝑗>0, then 𝑑^𝑢_𝑗≥𝑑_𝑗(𝑥)≥𝑑^𝑙_𝑗>0, we have 0<1/𝑑^𝑢_𝑗≤1/𝑑_𝑗(𝑥)≤1/𝑑^𝑙_𝑗.(ii) If 𝑑^𝑙_𝑗≤𝑑^𝑢_𝑗<0, then 𝑑^𝑙_𝑗≤𝑑_𝑗(𝑥)≤𝑑^𝑢_𝑗<0, we have 1/𝑑^𝑢_𝑗≤1/𝑑_𝑗(𝑥)≤1/𝑑^𝑙_𝑗<0. By (i) and (ii), for for all 𝑥∈𝑋, we have 1/𝑑^𝑢_𝑗≤1/𝑑_𝑗(𝑥)≤1/𝑑^𝑙_𝑗.

Theorem 2.3. For any 𝑥∈𝑋, consider the functions ℎ(𝑥),ℎ^𝑙(𝑥), and ℎ^𝑢(𝑥) defined in (2.3) and (2.4). Then, the following two statements hold.
(i) The functions ℎ(𝑥),ℎ^𝑙(𝑥), and ℎ^𝑢(𝑥) satisfy ℎ^𝑙(𝑥)≤ℎ(𝑥)≤ℎ^𝑢(𝑥).(ii)The maximal errors of bounding ℎ(𝑥) using ℎ^𝑙(𝑥) and ℎ^𝑢(𝑥) satisfylim_{‖𝑥−𝑥‖→0}𝐸𝑟𝑟𝑜𝑟^𝐿_max=lim_{‖𝑥−𝑥‖→0}𝐸𝑟𝑟𝑜𝑟^𝑈_max=0, (2.6)where𝐸𝑟𝑟𝑜𝑟^𝐿_max=max_𝑥∈𝑋(ℎ(𝑥)−ℎ^𝑙(𝑥)),𝐸𝑟𝑟𝑜𝑟^𝑈_max=max_𝑥∈𝑋(ℎ^𝑢(𝑥)−ℎ(𝑥)). (2.7)

Proof. The proof the theorem can be found in [7].

For convenience in exposition, in the following we assume that 𝑋^𝑘=[𝑥^𝑘,𝑥^𝑘] represents either the initial bounds on the variables of the problem (P1), or modified bounds as defined for some partitioned subproblem in a branch-and-bound scheme. By means of Theorem 2.3, we can give the linear relaxation of the (P1). Let 𝑋^𝑘=[𝑥^𝑘,𝑥^𝑘]⊆𝑋⁰, consequently we construct the corresponding approximation relaxation linear programing (RLP) of (P1) in 𝑋^𝑘 as follows:RLP(𝑋^𝑘)∶minℎ^𝑙(𝑥)=^𝑞_𝑗=1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥𝑑^𝑢_𝑗+^𝑝_𝑗=𝑞+1𝑐_𝑗0+𝑐^𝑇_𝑗𝑥−𝑟_𝑗𝑑^𝑢_𝑗−^𝑝_𝑗=𝑞+1−𝑟_𝑗𝑑^𝑙_𝑗,s.t.𝐴𝑥≤𝑏,𝑥∈𝑋^𝑘. (2.8)

Based on the above linear underestimators, every feasible point of (P1) in subdomain 𝑋^𝑘 is feasible in (RLP); and the value of the objective function for (RLP) is less than or equal to that of (P1) for all points in 𝑋^𝑘. Thus, (RLP) provides a valid lower bound for the solution of (P1) over the partition set 𝑋^𝑘. It should be noted that problem (RLP) contains only the necessary constraints to guarantee convergence of the algorithm.

3. New Pruning Technique

In this section, we pay more attention to how to form the new pruning technique to delete or reduce a large part of regions in which there exists no global optimal solution so that we can accelerate the convergence of the proposed algorithm. Let 𝑋=(𝑋_𝑖)_𝑁×1 with 𝑋_𝑖=[𝑥_𝑖,𝑥_𝑖] be a subrectangle of 𝑋⁰, that is, 𝑋⊆𝑋⁰. Moreover, assume that ℎ are currently known upper bound of optimal objective value ℎ^∗ of problem (P1). For convenience of the following discussions, we introduce some notations as follows:

𝐸=^𝑝_𝑗=1𝑐_𝑗0𝑑^𝑢_𝑗+^𝑝_𝑗=𝑞+1(𝑟_𝑗𝑑^𝑙_𝑗−𝑟_𝑗𝑑^𝑢_𝑗𝛽),_𝑖=^𝑝_𝑗=1𝑐_𝑗𝑖𝑑^𝑢_𝑗,𝑖=1,2,…,𝑁,𝑆=^𝑁_𝑖=1min{𝛽_𝑖𝑥_𝑖,𝛽_𝑖𝑥_𝑖𝐿},_𝑡=^𝑁_𝑖=1min{𝑎_𝑡𝑖𝑥_𝑖,𝑎_𝑡𝑖𝑥_𝑖𝑠},𝑡=1,2,…,𝑀,_𝑚=min^𝑁_{𝑥∈𝑋𝑖=1,𝑖≠𝑚}𝛽_𝑖𝑥_𝑖,𝑧_𝑚=ℎ−𝑠_𝑚−𝐸𝛽_𝑚,𝑧¹_𝑡𝑚=𝑏_𝑡−𝐿_𝑡+𝑎_𝑡𝑚𝑥_𝑚𝑎_𝑡𝑚,𝑧²_𝑡𝑚=𝑏_𝑡−𝐿_𝑡+𝑎_𝑡𝑚𝑥_𝑚𝑎_𝑡𝑚. (3.1)

Theorem 3.1. Consider the subrectangle 𝑋=(𝑋_𝑖)_𝑁×1⊆𝑋⁰, we have the following conclusions.
(i)If 𝑆+𝐸>UB, then min_𝑥∈𝑋ℎ(𝑥)>ℎ^∗.(ii)If 𝑆+𝐸≤UB, then if there exists some index 𝑚(𝑚∈{1,2,…,𝑁}) such that 𝛽_𝑚>0 and 𝑧_𝑚<𝑥_𝑚, then min_𝑥∈𝑋¹ℎ(𝑥)>ℎ^∗; if there exists some index 𝑚(𝑚∈{1,2,…,𝑁}) such that 𝛽_𝑚<0 and 𝑧_𝑚>𝑥_𝑚, then min_𝑥∈𝑋²ℎ(𝑥)>ℎ^∗, where𝑋¹=(𝑋¹_𝑖)_𝑁×1⊆𝑋,𝑋²=(𝑋²_𝑖)_𝑁×1𝑋⊆𝑋,_1𝑖=𝑋_𝑖,if𝑖≠𝑚,𝑖=1,2,…,𝑁,(𝑧_𝑚,𝑥_𝑚]∩𝑋_𝑚𝑋,if𝑖=𝑚,_2𝑖=𝑋_𝑖,if𝑖≠𝑚,𝑖=1,2,…,𝑁,[𝑥_𝑚,𝑧_𝑚)∩𝑋_𝑚,if𝑖=𝑚. (3.2)

Proof. (i) By assumption and definitions of 𝛽_𝑖,𝑆,𝐸 in (3.1), if 𝑆+𝐸>UB, then ℎ^𝑙(𝑥)≥𝑆+𝐸>UB≥ℎ^∗ for any 𝑥∈𝑋, that is, min_𝑥∈𝑋ℎ(𝑥)>ℎ^∗ for any 𝑥∈𝑋.
(ii) By assumption and definitions of 𝛽_𝑖,𝑆,𝐸 in (3.1), if 𝑆+𝐸≤UB, then we have the following two conclusions:
(a)If there exists some index 𝑚(𝑚∈{1,2,…,𝑁}) such that 𝛽_𝑚>0 and 𝑧_𝑚<𝑥_𝑚, then for any 𝑥=(𝑥_𝑖)_𝑁×1∈𝑋¹, let 𝛽_𝑚>0 and 𝑧_𝑚<𝑥_𝑚 for some 𝑚, we first show that ℎ(𝑥)≥ℎ. By assumption and definitions of 𝛽_𝑚,𝑠_𝑚, and 𝑧_𝑚 in (3.1), we have 𝑥_𝑚>𝑧_𝑚, that is, 𝑥_𝑚>(ℎ−𝑠_𝑚−𝐸)/𝛽_𝑚. Furthermore, from the definitions of ℎ^𝑙(𝑥), we getℎ<𝛽_𝑚𝑥_𝑚+𝑠_𝑚+𝐸=𝛽_𝑚𝑥_𝑚+min^𝑁_{𝑥∈𝑋𝑖=1,𝑖≠𝑚}𝛽_𝑖𝑥_𝑖≤+𝐸_𝑁𝑖=1𝛽_𝑖𝑥_𝑖=+𝐸_{𝑁𝑝𝑖=1𝑗=1}𝑐_𝑗𝑖𝑑^𝑢_𝑗𝑥_𝑖+^𝑝_𝑗=1𝑐_𝑗0𝑑^𝑢_𝑗+^𝑝_𝑗=𝑞+1(𝑟_𝑗𝑑^𝑙_𝑗−𝑟_𝑗𝑑^𝑢_𝑗)=^𝑝𝑁_𝑗=1𝑖=1𝑐_𝑗𝑖𝑑^𝑢_𝑗𝑥_𝑖+^𝑝_𝑗=1𝑐_𝑗0𝑑^𝑢_𝑗+^𝑝_𝑗=𝑞+1(𝑟_𝑗𝑑^𝑙_𝑗−𝑟_𝑗𝑑^𝑢_𝑗)=ℎ^𝑙(𝑥). (3.3)By the above discussion and assumption, it follows that ℎ^∗≤ℎ<ℎ^𝑙(𝑥)≤ℎ(𝑥) for any 𝑥∈𝑋¹. Therefore, min_𝑥∈𝑋¹ℎ(𝑥)>ℎ≥ℎ^∗, that is, there exists no global optimal solution in 𝑋¹. This proof of part (a) is completed.(b)If there exists some index 𝑚(𝑚∈{1,2,…,𝑁}) such that 𝛽_𝑚<0 and 𝑧_𝑚>𝑥_𝑚, then min_𝑥∈𝑋²ℎ(𝑥)>ℎ^∗, since the proof of the part (b) is similar to the part (a) of the theorem, it is omitted here.

Theorem 3.2. Consider the subrectangle 𝑋=(𝑋_𝑖)_𝑁×1⊆𝑋⁰, we have the following conclusions.
(i)If there exists some 𝑡(𝑡∈{1,…,𝑀}) such that 𝐿_𝑡>𝑏_𝑡 on the subrectangle 𝑋, then there exists no global optimization solution in 𝑋.(ii)If 𝐿_𝑡≤𝑏_𝑡(𝑡∈{1,…,𝑀) on the subrectangle 𝑋, then if there exists some index 𝑚(𝑚∈{1,2,…,𝑁}) such that 𝑎_tm>0 and 𝑧¹_tm<𝑥_𝑚, then there exists no global optimization solution in 𝑋³; if there exists some index 𝑚(𝑚∈{1,2,…,𝑁}) such that 𝑎_tm<0 and 𝑧²_tm>𝑥_𝑚, then there exists no global optimization solution in 𝑋⁴, where𝑋³=(𝑋³_𝑖)_𝑁×1⊆𝑋,𝑋⁴=(𝑋⁴_𝑖)_𝑁×1𝑋⊆𝑋,_3𝑖=𝑋_𝑖,if𝑖≠𝑚,𝑖=1,2,…,𝑁,(𝑧¹_𝑡𝑚,𝑥_𝑚]∩𝑋_𝑚𝑋,if𝑖=𝑚,_4𝑖=𝑋_𝑖,if𝑖≠𝑚,𝑖=1,2,…,𝑁,[𝑥_𝑚,𝑧²_𝑡𝑚)∩𝑋_𝑚,if𝑖=𝑚. (3.4)

Proof. Since the proof of Theorem 3.2 is similar to that of Theorem 3.1, it is omitted here.

Based on Theorems 3.1 and 3.2, we can give new pruning technique to cut away or reduce a large part of region in which there exists no optimal solution.

Next, we will show how this new pruning technique is formed, that is, we provide a process that show how a subrectangle 𝑋⊆𝑋⁰ can be deleted or reduced, where 𝑋=(𝑋_𝑖)_𝑁×1 with 𝑋_𝑖=[𝑥_𝑖,𝑥_𝑖]. Let 𝑌_𝑚 denote the discarded interval in 𝑋_𝑚. First, we calculate 𝐸,𝛽_𝑚,𝑆,𝑠_𝑚,𝑧_𝑚,𝐿_𝑡,𝑧¹_𝑡𝑚,𝑧²_𝑡𝑚 according to (3.1). Then, the eliminated interval 𝑌_𝑚 can be determined according to the following rules, which is called new pruning technique.

Rule 1
If 𝑆+𝐸>𝑈𝐵, then𝑌_𝑚=[𝑥_𝑚,𝑥_𝑚],𝑚=1,…,𝑁. (3.5)

Rule 2
If 𝑆+𝐸≤𝑈𝐵, then
(a)If 𝛽_𝑚>0 and 𝑧_𝑚<𝑥_𝑚, then 𝑌_𝑚=(𝑧_𝑚,𝑥_𝑚],if𝑥_𝑚≤𝑧_𝑚,[𝑥_𝑚,𝑥_𝑚],if𝑧_𝑚≤𝑥_𝑚, (3.6)(b) If 𝛽_𝑚<0 and 𝑧_𝑚>𝑥_𝑚, then𝑌_𝑚=[𝑥_𝑚,𝑧_𝑚),if𝑧_𝑚≤𝑥_𝑚,[𝑥_𝑚,𝑥_𝑚],if𝑧_𝑚≥𝑥_𝑚. (3.7)

Rule 3
If 𝐿_𝑡>𝑏_𝑡 for some 𝑡(𝑡∈{1,…,𝑀}), then 𝑌_𝑚=[𝑥_𝑚,𝑥_𝑚],𝑚=1,…,𝑁. (3.8)

Rule 4
If 𝐿_𝑡≤𝑏_𝑡 for some 𝑡(𝑡∈{1,…,𝑀}), then
(c)If 𝑎_𝑡𝑚>0 and 𝑧¹_𝑡𝑚<𝑥_𝑚, then 𝑌_𝑚=[𝑥_𝑚,𝑥_𝑚],if𝑧¹_𝑡𝑚<𝑥_𝑚,(𝑧¹_𝑡𝑚,𝑥_𝑚],if𝑧¹_𝑡𝑚≥𝑥_𝑚, (3.9)(d)If 𝑎_𝑡𝑚<0 and 𝑧²_𝑡𝑚>𝑥_𝑚, then 𝑌_𝑚=[𝑥_𝑚,𝑥_𝑚],if𝑧²_𝑡𝑚>𝑥_𝑚,[𝑥_𝑚,𝑧²_𝑡𝑚),if𝑧²_𝑡𝑚≤𝑥_𝑚. (3.10)

Consequently, under some assumption the discarded part of 𝑋_𝑚 is 𝑋_𝑚∩𝑌_𝑚 for some 𝑚(𝑚∈{1,2,…,𝑁}), the new rectangle reducing tactics provides the possibility to cut away or reduce a large part of the rectangle that is currently investigated by the procedure. The rest part of 𝑋 denoted by 𝑋=(𝑋_𝑖)_𝑁×1 is left for further considering, where 𝑋_𝑖=𝑋_𝑖 if 𝑖≠𝑚(𝑖=1,2,…,𝑁);𝑋_𝑚=𝑋_𝑚⧵𝑌_𝑚 if 𝑖=𝑚.

4. Algorithm and Its Convergence

In this section, by connecting the former branch-and-bound algorithm with new pruning technique a global optimization algorithm is proposed for solving problem (P1). This algorithm needs to solve a sequence of relaxation linear programing over partitioned subsets of 𝑋⁰ in order to find a global optimum solution.

The branch and bound approach is based on partitioning the set 𝑋⁰ into sub-hyperrectangles, each concerned with a node of the branch and bound tree, and each node is associated with a relaxation linear subproblem in each sub-hyperrectangle. Hence, at any stage 𝑘 of the algorithm, suppose that we have a collection of active nodes denoted by 𝑄_𝑘, say, each associated with a hyperrectangle 𝑋⊆𝑋⁰, for all 𝑋∈𝑄_𝑘. For each such node 𝑋, we will have computed a lower bound of the optimal value of (P1) via the solution LB(𝑋) of the RLP, so that the lower bound of optimal value of (P1) on the whole initial box region 𝑋⁰ at stage 𝑘 is given by LB_𝑘=min{LB(𝑋), for all 𝑋∈𝑄_𝑘. Whenever the solution of the relaxation linear programing (RLP) turns out to be feasible to the problem (P1), we update the upper bound of incumbent solution UB if necessary. Then, the active nodes collection 𝑄_𝑘 will satisfy LB(𝑋)<UB, for all 𝑋∈𝑄_𝑘, for each stage 𝑘. We now select an active node to partition its associated hyperrectangle into two sub-hyperrectangles as described below, computing the lower bounds for each new node as before. Upon fathoming any nonimproving nodes, we obtain a collection of active nodes for the next stage, and this process is repeated until convergence is obtained.

The critical element in guaranteeing convergence to a global minimum is the choice of a suitable partitioning strategy. In our paper, we choose a simple and standard bisection rule. This method is sufficient to ensure convergence since it drives all the intervals to zero for all variables. This branching rule is given as follows.

Assume that the sub-hyperrectangle 𝑋^′=[𝑥^′,𝑥^′]⊆𝑋⁰ is going to be divided. Then, we select the branching variable 𝑥^′_𝛾, satisfying 𝛾=argmax{𝑥^′_𝑖−𝑥^′_𝑖∶𝑖=1,…,𝑁} and partition 𝑋^′ by bisection the interval [𝑥^′_𝛾,𝑥^′_𝛾] into the subintervals [𝑥^′_𝛾,(𝑥^′_𝛾+𝑥^′_𝛾)/2] and [(𝑥^′_𝛾+𝑥^′_𝛾)/2,𝑥^′_𝛾].

The basic steps of the proposed algorithm are summarized as follows. Let 𝐿𝐵(𝑋^𝑘) refer to the optimal objective function value of P1 for the sub-hyperrectangles 𝑋^𝑘 and 𝑥^𝑘=𝑥(𝑋^𝑘) refer to an element of corresponding argmin.

Algorithm Statement
Step 1 (initialization). Initialize the iteration counter 𝑘∶=0, the set of all active node 𝑄₀={𝑋⁰}, the upper bound UB=∞, and the set of feasible points 𝐹∶=∅. Solve the problem (RLP) for 𝑋=𝑋⁰, obtaining LB₀∶=LB(𝑋) and 𝑥⁰∶=𝑥(𝑋). If 𝑥⁰ is feasible to (P1) update 𝐹 and UB, if necessary. If UB≤LB₀+𝜖, where 𝜖>0 is some accuracy tolerance, then stop with 𝑥⁰ as the prescribed solution to problem (P1). Otherwise, proceed to Step 2.Step 2 (midpoint check). Select the midpoint 𝑥^𝑚 of 𝑋^𝑘, if 𝑥^𝑚 is feasible to the (P1), then 𝐹∶=𝐹∪{𝑥^𝑚}. Define the upper bound UB∶=min_𝑥∈𝐹ℎ(𝑥). If 𝐹≠∅, the best known feasible point is denoted 𝑏∶=argmin_𝑥∈𝐹ℎ(𝑥).Step 3 (branching). Choose a branching variable 𝑥_𝛾 to partition 𝑋^𝑘 to get two new sub-hyperrectangles according to the above selected branching rule. Call the set of new partition rectangles as 𝑋^𝑘.Step 4 (pruning). (1) If 𝑋^𝑘≠∅, then calculate 𝐸,𝛽_𝑖,𝑆,𝐿_𝑡,𝑠_𝑚,𝑧_𝑚,𝑧¹_𝑡𝑚, and 𝑧²_𝑡𝑚 for each 𝑋∈𝑋^𝑘. If one of assumption conditions in rules 1–4 is satisfied, then the proposed new pruning technique can be applied into each 𝑋∈𝑋^𝑘, the rest parts of 𝑋 and 𝑋^𝑘 are denoted by 𝑋 and 𝑋^𝑘, respectively.
(2) If 𝑋^𝑘≠∅, solve RLP to obtain LB(𝑋) and 𝑥(𝑋), for each 𝑋∈𝑋^𝑘. If LB(𝑋)>UB, set 𝑋^𝑘∶=𝑋^𝑘⧵𝑋, otherwise, update the best available solution UB,𝐹 and 𝑏 if possible, as in Step 2.Step 5 (updating lower bound). The partition set remaining is now 𝑄_𝑘∶=(𝑄_𝑘⧵𝑋^𝑘)∪𝑋^𝑘 giving a new lower bound LB_𝑘∶=inf_{𝑋∈𝑄𝑘}LB(𝑋).Step 6 (convergence check). Fathom any nonimproving nodes by setting 𝑄_𝑘+1=𝑄_𝑘⧵{𝑋∶UB−LB(𝑋)≤𝜖,𝑋∈𝑄_𝑘}. If 𝑄_𝑘+1=∅, then stop with UB is the solution of (P1), and 𝑏 is an optimal solution. Otherwise, 𝑘∶=𝑘+1, and select an active node 𝑋^𝑘 such that 𝑋^𝑘=argmin_{𝑋∈𝑄𝑘}LB(𝑋), 𝑥^𝑘∶=𝑥(𝑋^𝑘), and return to Step 2.

Convergence of The Algorithm
Let 𝑋^𝑎 be the set of accumulation points of {𝑥^𝑘}, and let 𝑋^∗ be argmin_𝑥∈𝐺ℎ(𝑥), where 𝐺≠∅ is the feasible space of the (P1).

Theorem 4.1. The above algorithm either terminates finitely with the incumbent solution being optimal to the (P1) or generates an infinite sequence of iteration such that along any infinite branch of the branch and bound tree, any accumulation point of the sequence{LB_𝑘} will be global minimum of the (P1).

Proof. If the above proposed algorithm terminates finitely at some iteration 𝑘, obviously, UB is global optimal value and 𝑥(𝑋^𝑘) is optimal solution for the (P1). If the algorithm is infinite, it generates at least one infinitely sequence {𝑋^𝑘} such that 𝑋^𝑘+1⊂𝑋^𝑘 for any 𝑘. In the case, since partition sets used by the proposed algorithm are all rectangular and compact, by Tuy [9], it follows that this rectangular subdivision is exhaustive. Hence, for every iteration, 𝑘=0,1,2,…, by design of the algorithm, we haveLB_𝑘≤min_𝑥∈𝐺ℎ(𝑥),𝑋^𝑘∈argmin_{𝑋∈𝑄𝑘}LB(𝑋),𝑥^𝑘=𝑥(𝑋^𝑘)∈𝑋^𝑘⊆𝑋⁰. (4.1)Horst [10] gives that {LB_𝑘} is a nondecreasing sequence bounded above by min_𝑥∈𝐺ℎ(𝑥), which guarantees the existence of the limit LB∶=lim_𝑘→∞LB_𝑘≤min_𝑥∈𝐺ℎ(𝑥).{𝑥^𝑘} is a sequence on a compact set, therefore, it has a convergent subsequence. For any 𝑥∈𝑋^𝑎, suppose that there exists a subsequence {𝑥^𝑟} of {𝑥^𝑘} with lim_𝑟→∞𝑥^𝑟=𝑥. By the proposed algorithm and [9], it follows that the subdivision of partition sets in Step 3 is exhaustive on 𝑋⁰, and the selection of elements to be partitioned in Step 3 is bound improving. Thus, there exists a decreasing subsequence {𝑋^𝑞}⊂𝑋^𝑟, where 𝑋^𝑟∈𝑄_𝑟 with 𝑥^𝑞∈𝑋^𝑞,LB_𝑞=LB(𝑥^𝑞)=ℎ^𝑙(𝑥^𝑞),lim_𝑞→∞𝑥^𝑞=𝑥. From the construction method of linear lower bound relaxation functions for objective function of the (P1), we know that the linear subfunctions ℎ^𝑙(𝑥) used in (RLP) are strongly consistent on 𝑋⁰. Thus, it follows that lim_𝑞→∞LB_𝑞=LB=ℎ(𝑥).

5. Numerical Experiments

To verify performance of the proposed algorithm, some common used test problems are implemented on Pentium IV (433 MHZ) microcomputer. The algorithm is coded in C++ language and each linear programing is solved by simplex method, and the convergence tolerance 𝜖_𝑐 set to 10⁻⁸ in our experiment. Below, we describe some of these sample problems and solution results are summarized in Table 1. In Table 1, the notations have been used for column headers. Iter: number of algorithm iteration; maxnode: the maximal number of active nodes necessary; time: execution time in seconds; 𝜖_𝑓: feasibility tolerance.

Table 1 

Computational results of test examples.

Example 5.1 (see [3]). We have min−𝑥₁+2𝑥₂+23𝑥₁−4𝑥₂++54𝑥₁−3𝑥₂+4−2𝑥₁+𝑥₂,+3s.t.𝑥₁+𝑥₂𝑥≤1.5,₁−𝑥₂≤0,0≤𝑥₁,𝑥₂≤1. (5.1)

Example 5.2 (see [2]). We have min−4𝑥₁+3𝑥₂+3𝑥₃+503𝑥₂+3𝑥₃−+503𝑥₁+4𝑥₃+504𝑥₁+4𝑥₂+5𝑥₃−𝑥+50₁+2𝑥₂+5𝑥₃+50𝑥₁+5𝑥₂+5𝑥₃−𝑥+50¹+2𝑥₂+4𝑥₃+505𝑥₂+4𝑥₃,+50s.t.2𝑥₁+𝑥₂+5𝑥₃𝑥≤10,₁+6𝑥₂+3𝑥₃≤10,5𝑥₁+9𝑥₂+2𝑥₃≤10,9𝑥₁+7𝑥₂+3𝑥₃𝑥≤10,₁,𝑥₂,𝑥₃≥0. (5.2)

If there exists sum of general linear ratios in constraint functions of problem (P1), by using the same linear relaxation method proposed in Section 2, we can construct the corresponding linear relaxation programing of problem (P1). Therefore, the above proposed algorithm can be extensively applied to solve the following sum of linear ratios problem with sum of linear ratios constraints.

Example 5.3. We have min2𝑥₁+𝑥₂+𝑥₃+1𝑥₁+2𝑥₂+𝑥₃+𝑥+2¹+2𝑥₂+𝑥₃+22𝑥₁+2𝑥₂+𝑥₃−𝑥+3¹+𝑥₂+3𝑥₃+5𝑥₁+2𝑥₂+3𝑥₃−+41.5𝑥₁+𝑥₂+𝑥₃+6𝑥₁+𝑥₂+1.5𝑥₃,+5s.t.2𝑥₁+𝑥₂+𝑥₃+22𝑥₁+𝑥₂+𝑥₃−+32𝑥₁+2𝑥₂+𝑥₃+5𝑥₁+2𝑥₂+𝑥₃−+42𝑥₁+3𝑥₂+𝑥₃+6𝑥₁+2𝑥₂+2𝑥₃−+51.5𝑥₁+𝑥₂+2𝑥₃+71.5𝑥₁+2𝑥₂+𝑥₃+6≤−2.4,1.5𝑥₁+𝑥₂+𝑥₃+3𝑥₁+1.5𝑥₂+𝑥₃++42𝑥₁+𝑥₂+𝑥₃+4𝑥₁+𝑥₂+2𝑥₃+𝑥+5¹+2𝑥₂+𝑥₃+5𝑥₁+2𝑥₂+𝑥₃+𝑥+6¹+𝑥₂+3𝑥₃+6𝑥₁+3𝑥₂+𝑥₃𝑥+7≤3.8,₁+𝑥₂+𝑥₃+4𝑥₁+𝑥₂+𝑥₃+𝑥+5¹+𝑥₂+𝑥₃+5𝑥₁+𝑥₂+𝑥₃+𝑥+6¹+𝑥₂+3𝑥₃+6𝑥₁+𝑥₂+𝑥₃+𝑥+7¹+𝑥₂+𝑥₃+7𝑥₁+𝑥₂+𝑥₃𝑥+8≤3.9,₁+𝑥₂+𝑥₃+5𝑥₁+𝑥₂+𝑥₃+𝑥+6¹+𝑥₂+𝑥₃+6𝑥₁+𝑥₂+𝑥₃−𝑥+7¹+𝑥₂+𝑥₃+9𝑥₁+𝑥₂+𝑥₃−𝑥+8¹+𝑥₂+𝑥₃+10𝑥₁+𝑥₂+𝑥₃+9≤0.1,1.0≤𝑥₁≤3.0,1.0≤𝑥₂≤3.0,1.0≤𝑥₃≤3.0. (5.3)

Example 5.4. We have 𝑥min¹+𝑥₂+𝑥₃+1𝑥₁+𝑥₂+𝑥₃+𝑥+2¹+𝑥₂+𝑥₃+2𝑥₁+𝑥₂+𝑥₃−𝑥+3¹+𝑥₂+𝑥₃+5𝑥₁+2𝑥₂+2𝑥₃−𝑥+4¹+𝑥₂+𝑥₃+6𝑥₁+𝑥₂+𝑥₃,𝑥+5s.t.¹+𝑥₂+𝑥₃+2𝑥₁+𝑥₂+𝑥₃−𝑥+3¹+𝑥₂+𝑥₃+5𝑥₁+𝑥₂+𝑥₃−𝑥+4¹+𝑥₂+𝑥₃+6𝑥₁+𝑥₂+𝑥₃−𝑥+5¹+𝑥₂+𝑥₃+7𝑥₁+𝑥₂+𝑥₃𝑥+6≤−2.0,₁+𝑥₂+𝑥₃+3𝑥₁+𝑥₂+𝑥₃+𝑥+4¹+𝑥₂+𝑥₃+4𝑥₁+𝑥₂+𝑥₃+𝑥+5¹+𝑥₂+𝑥₃+5𝑥₁+𝑥₂+𝑥₃+𝑥+6¹+𝑥₂+𝑥₃+6𝑥₁+𝑥₂+𝑥₃𝑥+7≤3.59,₁+𝑥₂+𝑥₃+4𝑥₁+𝑥₂+𝑥₃+𝑥+5¹+𝑥₂+𝑥₃+5𝑥₁+𝑥₂+𝑥₃+𝑥+6¹+𝑥₂+3𝑥₃+6𝑥₁+𝑥₂+𝑥₃+𝑥+7¹+𝑥₂+𝑥₃+7𝑥₁+𝑥₂+𝑥₃𝑥+8≤3.6,₁+𝑥₂+𝑥₃+5𝑥₁+𝑥₂+𝑥₃+𝑥+6¹+𝑥₂+𝑥₃+6𝑥₁+𝑥₂+𝑥₃+𝑥+7¹+𝑥₂+𝑥₃+7𝑥₁+𝑥₂+𝑥₃+𝑥+8¹+𝑥₂+𝑥₃+8𝑥₁+𝑥₂+𝑥₃+9≤3.7,1.0≤𝑥₁≤3.0,1.0≤𝑥₂≤3.0,1.0≤𝑥₃≤3.0. (5.4)

From Table 1, numerical results show that our algorithm can globally solve sum of general linear ratios problem (P) on a microcomputer.

6. Concluding Remarks

A global optimization algorithm is proposed for solving sum of general linear ratios problem (P). To globally solve the (P), we first convert (P) into an equivalent problem (P1), then one new linearization method is proposed to construct the linear relaxation programing of the (P1). Then, a new pruning technique is proposed, this technique offers a possibility to cut away a large part of the current investigated feasible region by the algorithm, which can be utilized as an accelerating device for global optimization of problem (P). The proposed algorithm is convergent to the global minimum of (P1) through the successive refinement of linear relaxation of the feasible region and the subsequent solutions of a series of (RLP). Finally, the numerical experiments are given to illustrate the feasibility and effectiveness of the proposed algorithm.

Acknowledgment

This paper is supported by the National Science Foundation of China (10671057) and the National Science Foundation of Henan Institute of Science and Technology (06055).