An Efficient Branch-and-Bound Algorithm for Globally Solving Minimax Linear Fractional Programming Problem

,is paper presents an efficient outer space branch-and-bound algorithm for globally solving a minimax linear fractional programming problem (MLFP), which has a wide range of applications in data envelopment analysis, engineering optimization, management optimization, and so on. In this algorithm, by introducing auxiliary variables, we first equivalently transform the problem (MLFP) into the problem (EP). By using a new linear relaxation technique, the problem (EP) is reduced to a sequence of linear relaxation problems over the outer space rectangle, which provides the valid lower bound for the optimal value of the problem (EP). Based on the outer space branch-and-bound search and the linear relaxation problem, an outer space branch-and-bound algorithm is constructed for globally solving the problem (MLFP). In addition, the convergence and complexity of the presented algorithm are given. Finally, numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.


Introduction
is paper considers the following minimax linear fractional programming problem: where p ≥ 2, A ∈ R m×n , b ∈ R m , D is a nonempty bounded polyhedral set, and φ i (x) � n j�1 c ij x j + d i and ψ i (x) � n j�1 e ij x j + f i are all bounded affine functions over D, and for ∀x ∈ D, we have that φ i (x) ≥ 0 and ψ i (x) > 0, i � 1, . . . , p.
e problem (MLFP) has aroused the interest of many practitioners and researchers. On the one hand, it has a wide range of applications, such as electronic circuit design [1], system identification [2][3][4], and finance and investment [5]. On the other hand, because it has multiple local optimal solutions that are not globally optimal, it is still challenging to solve the problem (MLFP). So, it is necessary to come up with an efficient algorithm to globally solve the problem (MLFP).
In the past few decades, many algorithms have been proposed for solving some special forms of the problem (MLFP). In general, these algorithms can be classified into the following categories: interior-point algorithm [6], parameter programming method [7], partial linearization algorithm [8], monotonic optimization method [9], cutting plane algorithm [10], and branch-and-bound algorithm [11][12][13][14][15][16][17][18][19]. In addition to the above algorithms, in recent years, Chen et al. [20] proposed a unified framework to study various versions of Dinkelbach-type algorithms for solving the generalized fractional programming problem; based on a proximal bundle method, Boualam and Roubi [21] designed a dual algorithm for solving the convex minimax fractional programming problem; by analyzing the dual method of centers, Boufi and Roubi [22] presented an algorithm for solving the generalized fractional programs; Lai and Huang [23] presented a duality programming problem for solving a complex nondifferentiable minimax fractional programming with complex variables. However, the above algorithms can only deal with some specific cases of the problem (MLFP). erefore, it is necessary to propose an efficient algorithm for globally solving the general form of the problem (MLFP).
In this paper, we present an efficient outer space branchand-bound algorithm for globally solving the minimax linear fractional programming problem (MLFP). In this algorithm, we first transform the problem (MLFP) into an outer space equivalent problem (EP) by introducing auxiliary variables. en, in order to obtain the lower bound of the optimal value of the problem (EP), a new linear relaxation technique is proposed to construct the linear relaxation problem (LRP) of the problem (EP) over the outer space rectangle. And we can solve the problem (EP) by solving a sequence of linear relaxation problems over the outer space rectangle. Based on the outer space branch-and-bound search and the new linear relaxation problem over the outer space rectangle, we present an outer space branch-andbound algorithm for globally solving the problem (MLFP). In addition, the global convergence and the computational complexity of the proposed algorithm are given. e numerical results demonstrate that the proposed algorithm can effectively find the global optimal solutions for all tested instances. e numerical comparisons among our algorithm and the algorithms of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24] indicate that our algorithm outperforms the algorithms presented in Feng et al. [8], Jiao and Liu [12], and Wang et al. [24]. is paper is organized as follows. In Section 2, the equivalent problem (EP) of the problem (MFLP) is obtained, and by utilizing the new linear relaxation technique, the linear relaxation problem (LRP) of the problem (EP) over the outer space rectangle is constructed. In Section 3, an efficient outer space branch-and-bound algorithm is described, and the global convergence and the computational complexity of the proposed algorithm are given. Numerical results are reported to show the feasibility and efficiency of the proposed algorithm in Section 4. Finally, some conclusions are presented in Section 5.

Equivalent Problem and Its Linear Relaxation
For globally solving the problem (MLFP), we need to establish its equivalent problem (EP). For this purpose, we first compute the initial lower bound β 0 i � min x∈D ψ i (x) and the upper bound β 0 i � max x∈D ψ i (x) of the function ψ i (x) over D so that we can obtain the initial outer space rectangle: By introducing a new variable r, we can establish the outer space equivalent problem (EP) of the problem (MLFP) as follows: ≤ r, i � 1, 2, . . . , p, Proof. e conclusion of the theorem is obvious; therefore, the proof is omitted.
From eorem 1, for globally solving the problem (MLFP), we can globally solve its equivalence problem (EP) instead; they have the same optimal solutions and optimal value. In order to globally solve the problem (EP), we need to construct its linear relaxation problem of the problem (EP) over the outer space rectangle, which can provide a reliable lower bound for the optimal value of the problem (EP). And the next main task is to construct the linear relaxation problem of the problem (EP).
First of all, let . . , p} denote the initial rectangle Ω 0 or a subrectangle of Ω 0 that is generated by the branching operation.
en, for convenience in expression, some notations are introduced as follows: From the above formulas, it is easy to see that Expanding the above formula, we have that Since β 2 i ψ i (x) > 0, we divide both ends of the above inequality by β 2 i ψ i (x) at the same time, and we can obtain that Let we have the following relation: rough the above discussions, the linear relaxation problem of the problem (EP) over the outer space rectangle Ω is constructed as follows: From the above results, the optimal value of the problem (LRP) is less than or equal to that of the problem (EP) over Ω. erefore, the problem (LRP) can provide a valid lower bound for the optimal value of the problem (EP) over Ω. eorem 2 will show that the problem (LRP) will infinitely approximate the problem (EP) over Ω as ‖β − β ‖ ⟶ 0. □

Theorem 2. For any ψ
Proof. From the definitions Ψ i (x, r) and Ψ l i (x, r), we have erefore, we have that and the proof of the theorem is completed. From eorem 2, we can know that the linear relaxation function Ψ l i (x, r) will infinitely approximate the function Ψ i (x, r) as ‖β − β ‖ ⟶ 0, which guarantees the global convergence of the proposed algorithm.

Algorithm, Global Convergence, and Complexity Analysis
In this section, we first describe the outer space branching rule. en, based on the outer space partitioning search and the linear relaxation problem, an efficient outer space branch-and-bound algorithm is developed to globally solve the problem (MLFP). Finally, the global convergence and the computational complexity of the presented algorithm are derived.

Outer Space Partitioning
Rule. e important factor to ensure that the proposed algorithm in this paper can converge to the global minimum of the problem (MLFP) is to choose a suitable outer space partitioning strategy. In this paper, we choose an outer space rectangle bisection method, which can ensure the global convergence of the proposed outer space branch-and-bound algorithm. e proposed outer space partitioning rule is given as follows.
Consider any node subproblem identified by the sub- . . , p}⊆ Ω 0 ; the branching rule is described as follows: From the outer space partitioning rule, it can be seen that the outer space rectangle Ω is partitioned into two outer space subrectangles Ω ′ and Ω ″ .

Algorithm Statement.
By the above discussions, the basic steps of the proposed algorithm are given as follows. Let LB(Ω k ) and (x(Ω k ), r(Ω k )) be the optimal objective functional value of the problem (LRP) and an element of the corresponding argmin over the subrectangle Ω k , respectively.
Step 1: given the convergence tolerance ε ≥ 0, the set of feasible points F � ∅ and the upper bound UB 0 � +∞.

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If (x 0 , r 0 ) is feasible to the problem (EP), then we update F and UB 0 . If UB 0 − LB 0 ≤ ε, then the algorithm stops with that (x 0 , r 0 ) is an ε-global optimal solution for the problem (EP); otherwise, set Θ 0 � Ω 0 be the set of all active notes, k � 1, and go to step 2.
Update the upper bound: If UB k � r(Ω k,s ), let (x k , r k ) � (x(Ω k,s ), r(Ω k,s )), and it is obvious that (x k , r k ) is the best feasible solution for the problem (EP).
If Θ k+1 � ∅, then the algorithm stops, and (x k , r k ) is an ϵ-global optimal solution of the problem (EP).

Convergence Analysis.
e following theorem gives the proof of the convergence of the above algorithm.

Theorem 3.
e proposed algorithm either terminates finitely with the solution x k which is a global ϵ-optimal solution for the problem (MLFP) or generates an infinite sequence x k of iterations such that any infinite branch of the branch-andbound tree and any accumulation point will be a global optimal solution of the problem (MLFP).
Proof. When the algorithm is finitely terminated, the conclusion is obvious. When the algorithm is not finitely terminated, Horst and Tuy [25] point out that a sufficient condition for the algorithm to be convergent to the global minimum is that the bounding operation must be consistent and the selection operation is bound improving.
Let LB k be a lower bound computed in stage k, and let UB k be the best upper bound at iteration k not necessarily occurring inside the same subrectangle with LB k . A bounding operation is called consistent if, at every step, any unfathomed partition can be further refined, and since our subdivision process is the bisection, the branching process is exhaustive. erefore, from eorem 2, we have that en, this implies that the employed bounding operation is consistent.
A selection operation is called bound improving if at least one partition element where the actual upper bound is attained is selected for further partition after a finite number of refinements. Obviously, since the partition element where the actual upper bound is attained is selected for further partition in the immediately following iteration, the employed selection operation is bound improving.
In general, it can show that the bounding operation is consistent and that selection operation is bound improving. erefore, the proposed algorithm is convergent to the global minimum of the problem (MFLP), and the proof of the theorem is completed.

Computational Complexity Analysis.
In this section, we will analyze the complexity of the algorithm. To this end, we define the size Δ Additionally, for convenience, we denote by Theorem 4. Given the convergence tolerance ε > 0, if there exists a rectangle Ω k generated by the algorithm at k th iteration, such that Δ(Ω k ) ≤ (ε/μ), then we have where LB(Ω k ) is the optimal value to the problem (LRP (Ω k )) and UB is the best current known upper bound of the optimum value to the problem (EP).
Proof. From eorem 2, Δ(Ω k ) ≤ (ε/μ), and the definition of α 0 i , β i , and β i , we have and the proof of the theorem is completed.
□ Remark 1. Given the convergence tolerance ε > 0, the proposed algorithm finds a global ε-optimal solution to the problem (MFLP) in at most iterations: Based on the above discussions, the maximum iterations of the algorithm can be obtained by analyzing the computational complexity of the proposed algorithm.

Numerical Experiments
In this section, we numerically compare our algorithm with the existing branch-and-bound-algorithms presented in the works of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24]. All algorithms are coded in MATLAB R2014a; our algorithm is executed on the microcomputer with Intel(R) Core(TM) i7-10700K CPU @3.80 GHz processor and 32 GB RAM. All test examples and their numerical results are listed as follows.
First of all, for Examples 1-8, when n is smaller, with the given approximation error ε � 10 − 6 , numerical comparisons among our algorithm and the algorithms presented in works of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24] are given in Table 1. Next, for randomly generated large-scale Example 9, when n ≥ 1000, with the given approximation error ε � 10 − 4 , numerical computational results of our algorithm are given in Table 2. When n ≥ 1000, since the algorithms presented in works of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24] all failed to solve the randomly generated large-scale Example 9 in 3600 s, we only present the numerical computational results of our algorithm in Table 2. For all numerical tests of randomly generated Example 9, we solved arbitrary ten independently generated test problems and recorded their average numerical results among these ten test problems.
From the numerical results for Examples 1-8 in Table 1, first of all, we can observe that our algorithm can obtain the almost same optimal solution and optimal value as the algorithms presented in the works of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24]. Secondly, in terms of computational efficiency, in all cases, our algorithm outperforms the algorithms presented in the works of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24].
From the numerical results for the randomly generated large-scale Example 9 in Table 2, it is obvious that the proposed algorithm can solve randomly generated large-scale Example 9 with the large-size variables (n ≥ 1000). However, the algorithms of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24] all failed to solve the randomly generated large-scale Example 9.
erefore, this demonstrates the robustness and stability of our algorithm.

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Example 8 (see [12,14]).   Tables 1 and 2, it is seen that the proposed algorithm has higher computational efficiency than the algorithms presented in the works of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24], which can be used to globally solve the problem (MLFP) with the largesize variables.

Concluding Remarks
In this paper, based on the outer space partitioning search and the new linear relaxation problem, we present an outer space branch-and-bound algorithm for globally solving the problem (MLFP). e proposed algorithm is convergent to the global optimal value by the successive refinement of the outer space region and the subsequent solutions of a series of linear relaxation programming problems over the outer space region. e main work of the proposed algorithm involves solving a series of linear relaxation programming problems over the outer space region which does not grow in size for each iteration and which can be efficiently solved by the simplex method. Compared with the algorithms presented in the works of Feng et al. [8], Jiao and Liu [12], and Wang et al. [24], numerical results for some test examples are given to illustrate the feasibility and effectiveness of our new algorithm. It is also hoped that the ideas and methods used to create the algorithm will offer a useful tool for globally solving the problem (MLFP).

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest or personal relationships that could have appeared to influence the work reported in this paper.
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