An Effective Branch and Bound Algorithm for Minimax Linear Fractional Programming

Abstract

An effective branch and bound algorithm is proposed for globally solving minimax linear fractional programming problem (MLFP). In this algorithm, the lower bounds are computed during the branch and bound search by solving a sequence of linear relaxation programming problems (LRP) of the problem (MLFP), which can be derived by using a new linear relaxation bounding technique, and which can be effectively solved by the simplex method. The proposed branch and bound algorithm is convergent to the global optimal solution of the problem (MLFP) through the successive refinement of the feasible region and solutions of a series of the LRP. Numerical results for several test problems are reported to show the feasibility and effectiveness of the proposed algorithm.

1. Introduction

Minimax linear fractional programming problem (MLFP) has become a subject of wide interest for practitioners and scientists [1–3], which has broad applications in various disciplines, for examples, system engineering [3], electronic science [4], and management science [5], and whose mathematical modeling can be formulated as follows: (MLFP):min_𝑥maxΨ₁(𝑥),Ψ₂(𝑥),…,Ψ_𝑝(𝑥)s.t.𝑥∈𝐷=𝑥∈𝑅^𝑛∣𝐴𝑥≤𝑏,𝑥≥0, (1) where Ψ_𝑗(𝑥)=∑^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗∑^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗,𝑗=1,2,…,𝑝, (2)∑^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗 and ∑^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗 are all affine functions such that ∑^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗>0 and ∑^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗>0, 𝐴∈𝑅^𝑚×𝑛, 𝑏∈𝑅^𝑚, and 𝐷 is a nonempty compact set.

The minimax linear fractional programming problems (MLFP) pose significant theoretical and computational challenges. This is mainly because the problems (MLFP) possess multiple local optima that are not globally optimal. Therefore, it is necessary to put forward an effective global optimization algorithm for solving the minimax linear fractional programming problem (MLFP). During the past years, some algorithms have been proposed for solving the minimax linear fractional programming problem (MLFP), for instance, dual methods [2, 6, 7], parametric programming methods [8, 9], interior-point algorithms [10, 11], monotonic optimization approach [12], exact method [13], approximation algorithm [14], cutting plane algorithm [15], method of centers [16], inexact proximal point method [17], interval-type algorithm [18], and so on. Recently, based on the linear relaxation and branch and bound scheme, a solution algorithm has been developed for solving globally the minimax linear fractional programming problem (MLFP) [19]. Up to now, although there has been significant progress in the development of algorithms for solving the minimax linear fractional programming, to our knowledge, less work has been still done for globally solving the minimax linear fractional programming problem (MLFP).

The purpose of this paper is to present a new global optimization algorithm for solving the minimax linear fractional programming problem (MLFP), and the goals of this research are threefold. First, we present a transformation of the problem; thus, the original problem (MLFP) is reformulated as an equivalent problem (EP). Second, in order to design an effective branch and bound algorithm for the equivalent problem (EP), a new linear relaxation bounding technique is presented, and, by utilizing this technique, the nonconvex programming problem (EP) is reduced to a sequence of linear relaxation programming problems, which can provide reliable lower bounds for the optimal value of the problem (EP) and are embedded into the branch and bound framework. The main computational operation in the algorithm only involves solving a sequence of linear relaxation programming subproblems that do not grow in size from iteration to iteration. Third, compare our algorithm (using new linear relaxation bounding technique) with the known algorithms (recent literatures) with respect to robustness (finding the optimum) and efficiencies (number of function evaluations), and the numerical experimental results show that the proposed algorithm is robust and effective.

This paper is organized as follows. The next section firstly converts the problem (MLFP) into an equivalent problem (EP); a new linear relaxation bound method is presented, and then the linear relaxation programming of the problem (EP) is established. Section 3 a branch and bound algorithm is proposed for globally solving the problem (MLFP), and the convergence property of the algorithm is given. In Section 4, we report the numerical results for solving some examples with the proposed algorithm. Finally, a few concluding remarks are given in Section 5.

2. Linear Relaxation Programming

In order to globally solve the problem (MLFP), we first compute the initial lower bound 𝑥_𝑖=min_𝑥∈𝐷𝑥_𝑖 and upper bound 𝑥_𝑖=max_𝑥∈𝐷𝑥_𝑖 of each variable 𝑥_𝑖 and denote the initial rectangle by 𝑋⁰=𝑥∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖,𝑖=1,…,𝑛. (3)

Next, we can convert the problem (MLFP) into the following equivalent problem (EP), which has the same global optimal solution and optimal value as the problem (MLFP): (EP):min_𝑥,𝑡𝑡s.t.Ψ_𝑗(𝑥)−𝑡≤0,𝑗=1,…,𝑝,𝐴𝑥≤𝑏,𝑥≥0,𝑥∈𝑋⁰. (4)

In the following, we only consider solving the equivalent problem (EP); the important step in the construction of a solution procedure for globally solving the problem (EP) is the establishment of a linear relaxation programming for computing the lower bounds of the optimal value for this problem. Here, we only need to construct linear lower bounding function of Ψ_𝑗(𝑥) in constraint function. For each 𝑗∈{1,…,𝑝}, we can let ℎ_𝑗(𝑥)=ln^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗,𝑔_𝑗(𝑥)=ln^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗,Ψ_𝑗(𝑥)=expℎ_𝑗(𝑥)−𝑔_𝑗(𝑥). (5)

The detailed new linear relaxation bounding technique can be given as follows.

First, we consider the function ℎ_𝑗(𝑥)  (𝑗=1,…,𝑝). For convenience in expression, for ∀𝑥∈𝑋^𝑘=[𝑥^𝑘,𝑥^𝑘]⊆𝑋⁰, some notations are introduced as follows: 𝑋_𝑗=^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗,𝑋_𝑗=^𝑛_𝑖=1min𝑐_𝑗𝑖𝑥^𝑘_𝑖,𝑐_𝑗𝑖𝑥^𝑘_𝑖+𝑑_𝑗,𝑋_𝑗=^𝑛_𝑖=1max𝑐_𝑗𝑖𝑥^𝑘_𝑖,𝑐_𝑗𝑖𝑥^𝑘_𝑖+𝑑_𝑗,𝐾_𝑗=ln𝑋_𝑗−ln𝑋_𝑗𝑋_𝑗−𝑋_𝑗,ℎ^𝑙_𝑗(𝑥)=ln𝑋_𝑗+𝐾_𝑗^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗−𝑋_𝑗. (6) Obviously, by the characteristic of the concave function, we have ℎ^𝑙_𝑗(𝑥)≤ℎ_𝑗(𝑥). (7) Second, we consider the concave function 𝑔_𝑗(𝑥)=ln(∑^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗)  (𝑗=1,…,𝑝) about the whole variable (∑^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗); by the property of the concave function, we have 𝑔_𝑗(𝑥)≤𝑔_𝑗𝑥_mid+∇𝑔_𝑗𝑥_mid^𝑇𝑥−𝑥_mid=𝑔^𝑢_𝑗(𝑥), (8) where 𝑥_mid=12𝑥^𝑘+𝑥^𝑘,∇𝑔_𝑗(𝑥)=𝑒_𝑗1∑^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗⋮𝑒_𝑗𝑝∑^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗. (9) From (7) and (8), for all 𝑥∈𝑋^𝑘, we have ℎ_𝑗(𝑥)−𝑔_𝑗(𝑥)≥ℎ^𝑙_𝑗(𝑥)−𝑔^𝑢_𝑗(𝑥). (10) Let 𝐻_𝑗(𝑥)=ℎ^𝑙_𝑗(𝑥)−𝑔^𝑢_𝑗(𝑥),𝐺_𝑗(𝑥)=exp𝐻_𝑗(𝑥). (11) By (5), (10), and (11), we have 𝐺_𝑗(𝑥)=exp𝐻_𝑗(𝑥)=expℎ^𝑙_𝑗(𝑥)−𝑔^𝑢_𝑗(𝑥)≤expℎ_𝑗(𝑥)−𝑔_𝑗(𝑥)=Ψ_𝑗(𝑥). (12)

Third, we consider the function 𝐺_𝑗(𝑥)(𝑗=1,…,𝑝). By the property of the convex function, we have 𝐺_𝑗(𝑥)≥𝐺_𝑗𝑥_mid+∇𝐺_𝑗𝑥_mid^𝑇𝑥−𝑥_mid, (13) where ∇𝐺_𝑗𝑥_mid=𝐾_𝑗𝑐_𝑗1⋮𝑐_𝑗𝑛−∇𝑔_𝑗𝑥_midexp𝐻_𝑗𝑥_mid. (14) Let 𝐺^𝑙_𝑗(𝑥)=𝐺_𝑗𝑥_mid+∇𝐺_𝑗𝑥_mid^𝑇𝑥−𝑥_mid, (15) and, by (12) and (13), we have 𝐺^𝑙_𝑗(𝑥)≤𝐺_𝑗(𝑥)≤Ψ_𝑗(𝑥). (16) Therefore, we can follow the linear lower bounding function 𝐺^𝑙_𝑗(𝑥) of Ψ_𝑗(𝑥) for each 𝑗∈{1,…,𝑝}, which underestimates the function Ψ_𝑗(𝑥) as follows: 𝐺^𝑙_𝑗(𝑥)=𝐺_𝑗𝑥_mid+∇𝐺_𝑗𝑥_mid^𝑇𝑥−𝑥_mid. (17)

According to the above discussion, for ∀𝑋^𝑘⊆𝑋⁰, we can construct the linear relaxation programming (LRP) of the problem (EP) in 𝑋^𝑘 as follows: (LRP):min𝑡s.t.𝐺^𝑙_𝑗(𝑥)−𝑡≤0,𝑗=1,…,𝑝,𝐴𝑥≤𝑏,𝑥∈𝑋^𝑘. (18)

Based on the above construction method of the linear relaxation programming problem, for ∀𝑋^𝑘⊆𝑋⁰, LRP(𝑋^𝑘) provides a valid lower bound for the optimal value of EP(𝑋^𝑘).

The following theorem will ensure that each 𝐺^𝑙_𝑗(𝑥) will approximate the corresponding function Ψ_𝑗(𝑥) as ‖𝑥^𝑘−𝑥^𝑘‖→0; that is, the optimal solution of the LRP(𝑋^𝑘) will approximate the optimal solution of EP(𝑋^𝑘).

Theorem 1. For ∀𝑥∈𝑋^𝑘=[𝑥^𝑘,𝑥^𝑘]⊆𝑋⁰, for each 𝑗=1,…,𝑝, and then the error Θ_𝑗=Ψ_𝑗(𝑥)−𝐺^𝑙_𝑗(𝑥)→0 as ‖𝑥^𝑘−𝑥^𝑘‖→0.

Proof. Let Θ_𝑗=Ψ_𝑗(𝑥)−𝐺_𝑗(𝑥)+𝐺_𝑗(𝑥)−𝐺^𝑙_𝑗(𝑥)=Θ_𝑗1+Θ_𝑗2, (19) and then it is obvious that Θ_𝑗1≥0 and Θ_𝑗2≥0.
First, we consider the difference Θ_𝑗1. Let Ψ_𝑗(𝑥)=lnℎ_𝑗(𝑥)−ln𝑔_𝑗(𝑥)=ln^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗−ln^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗, (20) and it follows that Θ_𝑗1=expln^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗−ln^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗−exp𝐻_𝑗(𝑥)=expΨ_𝑗(𝑥)−exp𝐻_𝑗(𝑥)≤Ψ_𝑗(𝑥)−𝐻_𝑗(𝑥)sup_{𝜎𝑗∈𝐿}Ψ_{𝑗(𝑥),𝐻𝑗(𝑥)}exp𝜎_𝑗, (21) where 𝐿Ψ_𝑗(𝑥),𝐻_𝑗(𝑥)=𝛼Ψ_𝑗(𝑥)+(1−𝛼)𝐻_𝑗(𝑥),for∀𝛼∈[0,1]. (22)
Let Θ_𝑗1.1=ln^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗−ln𝑋_𝑗+𝐾_𝑗^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗−𝑋_𝑗,Θ_𝑗1.2=𝑔_𝑗𝑥_mid+∇𝑔_𝑗𝑥_mid^𝑇𝑥−𝑥_mid−𝑔_𝑗(𝑥), (23) We have Ψ_𝑗(𝑥)−𝐻_𝑗(𝑥)=ln^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗−ln^𝑛_𝑖=1𝑒_𝑗𝑖𝑥_𝑖+𝑓_𝑗−ln𝑋_𝑗+𝐾_𝑗^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗−𝑋_𝑗−𝑔_𝑗𝑥_mid−∇𝑔_𝑗𝑥_mid^𝑇𝑥−𝑥_mid=Θ_𝑗1.1+Θ_𝑗1.2. (24) Since Θ_𝑗1.1 is a concave function about (∑^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗), we can know that Θ_𝑗1.1 can attain the maximum Θ^max_𝑗1.1 at the point ∑^𝑛_𝑖=1𝑐_𝑗𝑖𝑥_𝑖+𝑑_𝑗=1/𝐾_𝑗. Let 𝑢_𝑗=𝑋_𝑗/𝑋_𝑗. Then, through computing, we derive Θ^max_𝑗1.1=ln𝑢_𝑗𝑢_𝑗−1−1−lnln𝑢_𝑗𝑢_𝑗.−1 (25) Since 𝑢_𝑗→1 as ‖𝑥^𝑘−𝑥^𝑘‖→0, we have Θ^max_𝑗1.1→0 as ‖𝑥^𝑘−𝑥^𝑘‖→0.
One has Θ_𝑗1.2=𝑔_𝑗𝑥_mid+∇𝑔_𝑗𝑥_mid^𝑇𝑥−𝑥_mid−𝑔_𝑗(𝑥)=∇𝑔_𝑗𝑥_mid^𝑇𝑥−𝑥_mid−∇𝑔_𝑗𝜂^𝑇𝑥−𝑥_mid≤∇²𝑔_𝑗𝜌𝜂−𝑥_mid𝑥−𝑥_mid, (26) where 𝜂 and 𝜌 are constant vectors, which satisfy 𝑔_𝑗(𝑥)−𝑔_𝑗𝑥_mid=∇𝑔_𝑗𝜂^𝑇𝑥−𝑥_mid,∇𝑔_𝑗𝑥_mid−∇𝑔_𝑗𝜂=∇²𝑔_𝑗𝜌𝜂−𝑥_mid. (27) Since ∇²𝑔_𝑗(𝑥) is continuous, and 𝑋 is a compact set, there exists some 𝑀>0 such that ∇²𝑔_𝑗(𝑥)≤𝑀. (28) By (26), we have Θ_𝑗1.2≤𝑀𝑥^𝑘−𝑥^𝑘2, (29) Furthermore, we have Θ_𝑗1.2→0 as ‖𝑥^𝑘−𝑥^𝑘‖→0.
Therefore, we have Ψ_𝑗(𝑥)−𝐻_𝑗(𝑥)=Θ_𝑗1.1+Θ_𝑗1.2⟶0as𝑥^𝑘−𝑥^𝑘⟶0. (30)
Since exp(𝜎_𝑗) is a continuous and bounded function about 𝑥, there exists some 𝑀>0 such that |exp(𝜎_𝑗)|≤𝑀. Therefore, by (30), we have Θ_𝑗1≤𝑀Ψ_𝑗(𝑥)−𝐻_𝑗(𝑥)⟶0as𝑥^𝑘−𝑥^𝑘⟶0. (31) Second, we consider the difference Θ_𝑗2, and it follows that Θ_𝑗2=𝐺_𝑗(𝑥)−𝐺^𝑙_𝑗(𝑥)=𝐺_𝑗(𝑥)−𝐺_𝑗𝑥_mid+∇𝐺_𝑗𝑥_mid^𝑇𝑥−𝑥_mid=∇𝐺_𝑗𝛾^𝑇𝑥−𝑥_mid−∇𝐺_𝑗𝑥_mid^𝑇𝑥−𝑥_mid≤∇²𝐺_𝑗𝛽𝛾−𝑥_mid𝑥−𝑥_mid, (32) where 𝛾 and 𝛽 are constant vectors, which satisfy 𝐺_𝑗(𝑥)−𝐺_𝑗𝑥_mid=∇𝐺_𝑗𝛾^𝑇𝑥−𝑥_mid,∇𝐺_𝑗𝛾−∇𝐺_𝑗𝑥_mid=∇²𝐺_𝑗𝛽𝛾−𝑥_mid. (33) Since ∇²𝐺_𝑗(𝑥) is a continuous function, and 𝑋 is a compact set, there exists some 𝑀>0 such that ∇²𝐺_𝑗(𝑥)≤𝑀. (34) By (32), we have Θ_𝑗2≤𝑀𝑥^𝑘−𝑥^𝑘2. (35) Furthermore, we have Θ_𝑗2⟶0as𝑥^𝑘−𝑥^𝑘⟶0. (36) By (19), (31), and (36), we can derive that Θ_𝑗=Ψ_𝑗(𝑥)−𝐺_𝑗(𝑥)+𝐺_𝑗(𝑥)−𝐺^𝑙_𝑗(𝑥)=Θ_𝑗1+Θ_𝑗2⟶0as𝑥^𝑘−𝑥^𝑘⟶0. (37) By the above discussion, it is obvious that the conclusion is followed.

3. Algorithm and Its Convergence

In this section, based on the former linear relaxation method, we will present a branch and bound algorithm for globally solving problem (EP). There are three fundamental operations in the proposed algorithm: a branching operation, an updating upper bounds operation, and an updating lower bounds operation.

The first fundamental operation iteratively subdivides the investigated rectangle 𝑋 into two subrectangles. During the process of iteration of the algorithm, the branching operation produces a more refined partition that cannot yet be excluded from further consideration in finding the global optimum for the problem (EP). In this paper, we choose a simple and standard branching rule. This branching rule is enough to ensure the convergence of the algorithm since it drives the intervals shrinking to a singleton for all variables along any infinite branch of the branch and bound tree. Consider any node subproblem identified by the hyperrectangle 𝑋=[𝑥,𝑥]⊆𝑋⁰. This branching rule is as follows.(a)Let 𝑞=argmax{𝑥_𝑖−𝑥_𝑖∶𝑖=1,…,𝑛}.(b)Let 𝑋¹=𝑥∈𝑅^𝑛∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖,𝑖≠𝑞,𝑥_𝑞≤𝑥_𝑞≤𝑥_𝑞+𝑥_𝑞2,𝑋²=𝑥∈𝑅^𝑛∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖𝑥,𝑖≠𝑞,_𝑞+𝑥_𝑞2≤𝑥_𝑞≤𝑥_𝑞. (38)By this branching rule, the rectangle 𝑋 is subdivided into two subrectangles 𝑋¹ and 𝑋².

The second fundamental operation is to update the lower bounds of the optimal value of the problem (EP). This main computation needs to solve a sequence of linear relaxation programming problems, which can be easily solved by using the simplex method.

The third fundamental operation is to update the upper bounds of the optimal value of the problem (EP). The upper bounds can be updated by computing the objective function values of the original problem (MLFP) and the equivalent problem (EP) which corresponds to optimal solution of each linear relaxation programming problem, respectively.

The set 𝐹 in the algorithm is the set of fathomed subrectangles 𝑋 of 𝑋⁰. Let LB(𝑋^𝑘) refer to the optimal objective function value of the problem (LRP) on the sub-hyper-rectangles 𝑋^𝑘 and 𝑥^𝑘=𝑥(𝑋^𝑘) refer to an element of corresponding argmin. The basic steps of the proposed algorithm are summarized as follows.

Algorithm Statement. Consider the following

Step   0. Choose 𝜖≥0. Let 𝑋⁰={𝑥∣𝑥_𝑖≤𝑥_𝑖≤𝑥_𝑖,𝑖=1,…,𝑛},𝑈𝐵₀=+∞. Find an optimal solution (𝑥⁰,𝑡⁰) and the optimal value 𝜐(𝑋⁰) of the LRP(𝑋⁰). Set LB₀∶=LB𝑋⁰=𝜐𝑋⁰,𝑥^𝑐,𝑡^𝑐=𝑥⁰,𝑡⁰. (39) For the given feasible tolerance 𝜖₁, if (𝑥^𝑐,𝑡^𝑐) is feasible to the problem EP(𝑋⁰), update 𝑈𝐵₀=𝑡^𝑐, if necessary. Set 𝑈𝐵₀=min𝑈𝐵₀,maxΨ₁𝑥^𝑐,Ψ₂𝑥^𝑐,…,Ψ_𝑝𝑥^𝑐. (40) If 𝑈𝐵₀−LB₀≤𝜖, stop; 𝑥^𝑐 is a global 𝜖-optimal solution for the problem (MLFP). Otherwise, let the set of active node Ω₀=𝑋⁰ and 𝐹∶=0/,  𝑘=1, and go to Step 𝑘.

Step  𝑘. 𝑘≥1.

Step  𝑘1. Set 𝑈𝐵_𝑘=𝑈𝐵_𝑘−1. (41) Subdivide 𝑋^𝑘−1 into two 𝑛-dimensional rectangles 𝑋^𝑘,1, 𝑋^𝑘,2 via the rectangular bisection process. Denote the set of new partitioned rectangles by 𝑋^𝑘. Set 𝐹=𝐹∪𝑋^𝑘−1.

Step  𝑘2. For each 𝑗=1,2, compute 𝜐(𝑋^𝑘,𝑗), and if 𝜐(𝑋^𝑘,𝑗)≠+∞, find an optimal solution (𝑥^𝑘,𝑗,𝑡^𝑘,𝑗) for the problem EQ(𝑋) with 𝑋=𝑋^𝑘,𝑗. For each 𝑗=1,2, set LB𝑋^𝑘,𝑗=𝜐𝑋^𝑘,𝑗. (42) Set 𝑠=0.

Step  𝑘3. Set 𝑠=𝑠+1. If 𝑠>2, go to Step 𝑘7. Otherwise, continue.

Step  𝑘4. If LB(𝑋^𝑘,𝑠)≥𝑈𝐵_𝑘, set 𝐹=𝐹∪𝑋^𝑘,𝑠 and go to Step 𝑘6. Otherwise, continue.

Step  𝑘5. For the given feasible tolerance 𝜖₁, if (𝑥^𝑘,𝑠,𝑡^𝑘,𝑠) is feasible to the EP(𝑋⁰), update 𝑈𝐵_𝑘=𝑡^𝑘,𝑠, if necessary. Let 𝑈𝐵_𝑘=min𝑈𝐵_𝑘,maxΨ₁𝑥^𝑘,𝑠,Ψ₂𝑥^𝑘,𝑠,…,Ψ_𝑝𝑥^𝑘,𝑠. (43) If 𝑈𝐵_𝑘<maxΨ₁𝑥^𝑘,𝑠,Ψ₂𝑥^𝑘,𝑠,…,Ψ_𝑝𝑥^𝑘,𝑠, (44) go to Step 𝑘6. If 𝑈𝐵_𝑘=maxΨ₁𝑥^𝑘,𝑠,Ψ₂𝑥^𝑘,𝑠,…,Ψ_𝑝𝑥^𝑘,𝑠, (45) set 𝑥^𝑐=𝑥^𝑘,𝑠,𝐹=𝐹∪𝑋∈Ω_𝑘−1∣LB(𝑋)≥𝑈𝐵_𝑘, (46) and continue.

Step  𝑘6. Go to Step 𝑘3.

Step  𝑘7. Set Ω_𝑘=𝑋∣𝑋∈Ω_𝑘−1∪𝑋^𝑘,1,𝑋^𝑘,2,𝑋∉𝐹. (47)

Step  𝑘8. Set LB_𝑘=min{LB(𝑋)∣𝑋∈Ω_𝑘}, and let 𝑋^𝑘∈Ω_𝑘 satisfying LB_𝑘=LB(𝑋^𝑘). If 𝑈𝐵_𝑘−LB_𝑘≤𝜖, stop; 𝑥^𝑐 is a global 𝜖-optimal solution for the problem (MLFP), and 𝑣=max{Ψ₁(𝑥^𝑐),Ψ₂(𝑥^𝑐),…,Ψ_𝑝(𝑥^𝑐)} is global optimal value for the problem EP(𝑋⁰). Otherwise, set 𝑘=𝑘+1 and go to Step 𝑘.

The convergence properties of the proposed algorithm are given as follows.

Theorem 2. If the proposed algorithm terminates in finite steps, then a global optimal solution of the problem (MLFP) is obtained when the algorithm is terminated.

Proof. Assume that the algorithm is terminated finitely at 𝑥^𝑘. Obviously, LB_𝑘=𝑈𝐵_𝑘 when it is terminated at the 𝑘_𝑡ℎ iteration; therefore, 𝑥^𝑘 is a global optimal solution of the problem (MLFP).

Theorem 3. If the algorithm generates an infinite sequence {𝑥^𝑘}, then every accumulation point 𝑥^∗ of this sequence is a global optimal solution of the problem (MLFP).

Proof. Let 𝑥^∗ be an accumulation point of the sequence {𝑥^𝑘}, and let {𝑥^𝑘𝑞} be a subsequence {𝑥^𝑘𝑞} of the sequence {𝑥^𝑘} which is convergent to 𝑥^∗. Obviously, in the proposed algorithm, the lower bound sequence {LB_𝑘} is monotonic increasing and the upper bound sequence {𝑈𝐵_𝑘} is monotonic decreasing, so that {LB_𝑘} and {𝑈𝐵_𝑘} are convergent and lim_𝑘→∞𝑈𝐵_𝑘=lim_𝑞→∞𝑈𝐵_𝑘𝑞=lim_𝑞→∞maxΨ₁𝑥^𝑘𝑞,Ψ₂𝑥^𝑘𝑞,…,Ψ_𝑝𝑥^𝑘𝑞=maxΨ₁𝑥^∗,Ψ₂𝑥^∗,…,Ψ_𝑝𝑥^∗. (48)
Without loss of generality, we assume that 𝑥^𝑘𝑞 is the solution of the problem (LRP) on 𝑋^𝑘𝑞 which satisfies 𝑋^𝑘𝑞+1⊆𝑋^𝑘𝑞,  𝑞=1,2,…. Because the proposed rectangle partition is exhaustive; that is, lim_𝑞→∞𝑋^𝑘𝑞=𝑥^∗, and, from Theorem 1, we have 0≤𝑈𝐵_𝑘𝑞−𝐿𝐵_𝑘𝑞≤maxΨ₁𝑥^𝑘𝑞−𝐺^𝑙₁𝑥^𝑘𝑞,…,Ψ_𝑝𝑥^𝑘𝑞−𝐺^𝑙_𝑝𝑥^𝑘𝑞⟶0as𝑞⟶∞. (49) Thus, lim_𝑞→∞𝑈𝐵_𝑘𝑞−LB_𝑘𝑞=0. (50) Hence, lim_𝑘→∞LB_𝑘=lim_𝑞→∞𝑈𝐵_𝑘𝑞−𝑈𝐵_𝑘𝑞−LB_𝑘𝑞=lim_𝑞→∞𝑈𝐵_𝑘𝑞=maxΨ₁𝑥^∗,Ψ₂𝑥^∗,…,Ψ_𝑝𝑥^∗. (51) Therefore, 𝑥^∗ is a global optimal solution of the problem (MLFP).

4. Numerical Experiments

To verify the performance of our algorithm, several test examples in recent literatures are implemented on on a Intel(R) Core(TM)2 Duo CPU (1.58 GHZ) microcompute; the algorithm program is coded in C++, and each linear relaxation programming problem is solved by using simplex method, and the convergence tolerance is set to 𝜖=5×10⁻⁸ in our experiment. For the test problems, numerical results are illustrated in Table 1. For Examples 1–5, feasible errors 𝜖₁ are set by 0.005, 0.005, 0,0.005, and 0.001, respectively.

Table 1 

Computational results for test Examples 1–4.

In Tables 1 and 2, the notations have been used for column headers: Iter: number of algorithm iteration; 𝐿_max: the maximal number of algorithm active nodes necessary; time: execution time of algorithm in seconds.

Table 2 

Numerical results for Example 5.

In Table 1, optimal value is denoted by objective function value of the optimal solution in computational procedure of [19] and this paper, respectively.

Example 1 (see [12, 19]). Consider minmax3𝑥₁+𝑥₂−2𝑥₃+0.82𝑥₁−𝑥₂+𝑥₃,4𝑥₁−2𝑥₂+𝑥₃7𝑥₁+3𝑥₂−𝑥₃s.t.𝑥₁+𝑥₂−𝑥₃≤1,−𝑥₁+𝑥₂−𝑥₃≤−1,12𝑥₁+5𝑥₂+12𝑥₃≤34.8,12𝑥₁+12𝑥₂+7𝑥₃≤29.1,−6𝑥₁+𝑥₂+𝑥₃≤−4.1,1.0≤𝑥₁≤1.1,0.55≤𝑥₂≤0.65,1.35≤𝑥₃≤1.45. (52)

Example 2 (see [12, 19]). Consider maxmin37𝑥₁+73𝑥₂+1313𝑥₁+13𝑥₂,+1363𝑥₁−18𝑥₂+3913𝑥₁+26𝑥₂+13s.t.5𝑥₁−3𝑥₂=3,0≤𝑥₁≤3. (53)
This example is originally presented in Phuong and Tuy [12], which has the same global optimal solution as the following problem 2^′′: minmax13𝑥₁+13𝑥₂+1337𝑥₁+73𝑥₂,+1313𝑥₁+26𝑥₂+1363𝑥₁−18𝑥₂+39s.t.5𝑥₁−3𝑥₂=3,0≤𝑥₁≤3. (54) Therefore, we can use the proposed algorithm to globally solve Example 2 by solving the problem 2^′′.

Example 3 (see [19]). Considerminmax2𝑥₁+2𝑥₂−𝑥₃+0.9𝑥₁−𝑥₂+𝑥₃,3𝑥₁−𝑥₂+𝑥₃8𝑥₁+4𝑥₂−𝑥₃s.t.𝑥₁+𝑥₂−𝑥₃≤1,−𝑥₁+𝑥₂−𝑥₃≤−1,12𝑥₁+5𝑥₂+12𝑥₃≤34.8,12𝑥₁+12𝑥₂+7𝑥₃≤29.1,−6𝑥₁+𝑥₂+𝑥₃≤−4.1,1.0≤𝑥₁≤1.2,0.55≤𝑥₂≤0.65,1.35≤𝑥₃≤1.45. (55)

Example 4 (see [19]). Consider minmax3𝑥₁+𝑥₂−2𝑥₃+0.82𝑥₁−𝑥₂+𝑥₃,4𝑥₁−2𝑥₂+𝑥₃7𝑥₁+3𝑥₂−𝑥₃,3𝑥₁+2𝑥₂−𝑥₃+1.9𝑥₁−𝑥₂+𝑥₃,4𝑥₁−𝑥₂+𝑥₃8𝑥₁+4𝑥₂−𝑥₃s.t.𝑥₁+𝑥₂−𝑥₃≤1,−𝑥₁+𝑥₂−𝑥₃≤−1,12𝑥₁+5𝑥₂+12𝑥₃≤34.8,12𝑥₁+12𝑥₂+7𝑥₃≤29.1,−6𝑥₁+𝑥₂+𝑥₃≤−4.1,1.0≤𝑥₁≤1.2,0.55≤𝑥₂≤0.65,1.35≤𝑥₃≤1.45. (56)

Example 5. Consider ∑minmax^𝑛𝑖=1𝑐_1𝑖𝑥_𝑖+𝑑₁∑^𝑛_𝑖=1𝑒_1𝑖𝑥_𝑖+𝑓₁,∑^𝑛_𝑖=1𝑐_2𝑖𝑥_𝑖+𝑑₂∑^𝑛_𝑖=1𝑒_2𝑖𝑥_𝑖+𝑓₂,∑…,^𝑛𝑖=1𝑐_𝑝𝑖𝑥_𝑖+𝑑_𝑝∑^𝑛_𝑖=1𝑒_𝑝𝑖𝑥_𝑖+𝑓_𝑝s.t.𝐴𝑥≤𝑏,0≤𝑥_𝑖≤3,𝑖=1,…,𝑛, (57) where 𝐴 is an 𝑚×𝑛 matrix, 𝑏 is an 𝑚 dimension vector, all elements of 𝑐_𝑗𝑖, 𝑒_𝑗𝑖, 𝑗=1,…,𝑝, 𝑖=1,…,𝑛, are randomly generated between 0 and 1; all elements of 𝑑_𝑗, 𝑓_𝑗, 𝑗=1,…,𝑝 are randomly generated between 0 and 𝑝; all elements of 𝐴 are randomly generated between 0 and 1; all elements of 𝑏 are randomly generated between 0 and 16.

In Table 2, the notations have been also used for column headers: 𝑝: the number of linear fractional functions in the objective function; 𝑚: represents the number of rows for 𝐴; 𝑛: stands for the dimension of considered problem. From the experimental results in Table 2, it is seen that our algorithm can effectively solve the minimax linear fractional programming problem (MLFP) with large scale number of fractional functions.

5. Concluding Remarks

In this paper, an effective branch and bound algorithm is proposed for globally solving the minimax linear fractional programming problem (MLFP). In this algorithm, the lower bound is computed during the branch and bound search by solving linear relaxation programming (LRP) of the problem (MLFP), which can be derived by using the new linear relaxation bounding technique and can be effectively solved by using the simplex method or interior point algorithm. The proposed algorithm is convergent to the global minimum of the problem (MLFP) through the successive refinement of the feasible region and solutions of a series of the LRP. Numerical results for several test problems have been reported to show the feasibility and effectiveness of the proposed algorithm. It is hoped that the ideas and methods used to construct the algorithm will offer useful tools for solving the minimax linear fractional programming problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China under Grant (11171094) and the Science and Technology Key Project of Education Department of Henan Province (14A110024).