Outer space branching search method for solving generalized a ffi ne fractional optimization problem

: This paper proposes an outer space branching search method, which is used to globally solve the generalized a ffi ne fractional optimization problem (GAFOP). First, we will convert the GAFOP into an equivalent problem (EP). Next, we structure the linear relaxation problem (LRP) of the EP by using the linearization technique. By subsequently partitioning the initial outer space rectangle and successively solving a series of LRPs, the proposed algorithm globally converges to the optimum solution of the GAFOP. Finally, comparisons of numerical results are reported to show the superiority and the e ff ectiveness of the presented algorithm


Introduction
The considered generalized affine fractional optimization problem is as follows: (GAFOP): where A ∈ R m×n , b ∈ R m , e i , d i ∈ R n , and g i , f i are arbitrary real numbers.p ≥ 2, Y is a nonempty compact set, n j=1 e i j y j + f i and n j=1 c i j y j + h i are all bounded linearity functions defined on Y, and for any y ∈ Y, the denominator n j=1 c i j y j + h i 0, i = 1, 2, . . ., p.
As a class of special fractional optimization problems, the GAFOP has attracted the attention of many researchers and practitioners for decades.It has a variety of applications in many fields, including finance and investment [1][2][3], transportation planning [4,5], optimal design [6], estimation of iterative parameters [7], signal processing [8], data envelopment analysis and others [9][10][11][12][13][14][15][16][17].Furthermore, since the GAFOP may be not (quasi)convex, there may exist many local optimal solutions, many of which fail to be global solutions.Hence, it is still of great significance to propose an effective global algorithm to solve the GAFOP.Some algorithms have been presented to globally solve the GAFOP over the past several decades: for instance, the cutting plane algorithm [18], branch-relaxation-bound methods [19,20], the interior-point algorithm [21], the partial linearization algorithm [22], the monotonic optimization method [23], the method of centers [24] and the prox-regularization method [25].Recently, based on the Dinkelbach type algorithm, Ghazi and Roubi [26] presented a difference of convex functions (DC) method for globally solving the generalized convex fractional optimization problem.By utilizing the proximal bundle theory, Boualam and Roubi [27] proposed a dual method for the generalized convex fractional optimization problem.Jiao et al. [28] designed an image space branch-and-bound algorithm for solving minimax linear fractional programs.Haffari and Roubi [29] described a prox-dual regularization method for globally solving generalized fractional programs.By utilizing convex hull and concave hull approximation of a bilinear function, Jiao and Li [30] put forward a novel algorithm for globally addressing min-max linear fractional programs.However, the previous reviewed methods can only solve a particular form of the GAFOP, or they are difficult to use to solve large-scale practical problems.Therefore, there remains the necessity to propose a practical algorithm to solve the GAFOP.
In addition to the methods reviewed above, some theoretical progress on the generalized fractional optimization problem (GFOP) has also been made.For example, Ahmad and Husain [31] gave a duality theory for a non-differentiable GFOP with generalized convexity.Schmitendorf [32] presented some optimality conditions for the GFOP.By utilizing the optimality condition, Tanimoto [33] gave a dual problem for a class of non-differentiable GFOP and derived the duality theorems.Yadav and Mukherjee [34] gave a duality theory for GFOP.When the data in the system are uncertain, Jeyakumar et al. [35] put forward a strong duality theorem for the robust GFOP.Based on unconstrained conditions, Lai et al. [36] gave the duality theorem for the GFOP.For a detailed review of the methods and theories for the GFOP, readers can refer to Stancu-Minasian [37,38].
In this article, an outer space branching search method is designed for globally solving the GAFOP.We first convert the GAFOP into the EP.Next, by utilizing the structural characteristics of the EP, we construct a new linearizing method for establishing the LRP of the EP.Compared with the known existing algorithms, the branching search of the presented algorithm occurs in the outer space R p rather than the variable dimension space R n , which provides the possibility of mitigating the required computational efforts of the algorithm.In addition, the numerical computational results are reported, indicating that the proposed algorithm has higher efficiency and notable superiority compared to the known existing algorithms [19,39,40].
The remainder of this article is organized as follows.We derive the EP of the GAFOP and establish the LRP of the EP in Section 2. In Section 3, we give an outer space branching search method for globally solving the GAFOP, and we also analyze the global convergence of the algorithm.Numerical results for some test examples from recent studies are presented in Section 4. Finally, Section 5 gives some conclusions.

Linear relaxation programming problem
In the following, to solve the GAFOP, we first transform the GAFOP into the EP.Then, we present a novel linearization technique and construct the LRP of the EP.For this purpose, for each i = 1, . . ., p, we introduce the additional variables z i = n j=1 c i j y j + h i .By computing the minimum value  , we can simplify the objective function of the original problem GAFOP to r, so that we can get the EP of the GAFOP as below: (EP) : Theorem 1. y * is a global optimum solution of the GAFOP if and only if (y * , z * , r * ) is a global optimum solution of the EP, with Additionally, the global optimal values of the GAFOP and EP are equal.
Proof.By the above discussion, the conclusions are obvious, and thus we omit the proof.□ By Theorem 1, to globally solve the GAFOP, we can instead solve the EP.In the following, we only consider solving the EP.
For globally solving the EP, we need to establish its LRP for providing the lower bound in the branch-and-bound process.The detailed derivation process of the LRP is as follows.
For any Obviously, for each i = 1, . . ., p, we can see that Based on (1), for any Z ⊆ Z 0 , we can construct the LRP of the EP as below.
(LRP) : From the above discussion, it is known that all feasible points of the EP over the sub-rectangle Z are also feasible for the LRP.Let v(EP) and v(LRP) be the global optimal values of the LRP and EP, respectively, and we have v(LRP) ≤ v(EP) over Z k .Thus, the optimal value of the LRP will provide a valid lower bound for that of the EP over Z.
Next, we will prove that the optimal solution of the LRP will infinitely approximate the optimal solution of the EP over Z as ∥z − z∥ → 0, as detailed in Theorem 2. Theorem 2. For each i = 1, 2, . . ., p, consider the functions Φ i (y, z i ) and Φ i (y, z i , z i ).We have the following: Proof.By the definitions of the functions Φ i (y, z i ) and Φ i (y, z i , z i ), for any y ∈ Y, z i ∈ [z i , z i ], we have e i j y j z i − e i j y j z i e i j y j z i − e i j y j z i e i j y j z i − e i j y j z i e i j y j + e i j y j e i j y j + e i j y j and complete the proof of the Theorem.□ The above Theorem ensures that the function Φ i (y, z i ) will be infinitely approximated by the function Φ i (y, z i , z i ) as ∥z − z∥ → 0, so the global optimal solution of the LRP will infinitely approximate the global optimal solution of the EP over Z as ∥z − z∥ → 0.

Global algorithm and its convergence
In this section, we first put forward an outer space rectangle bisection method.Next, by combining the previous LRP and the branch-and-bound framework, an outer space branching search method is designed to globally solve the GAFOP.In addition, we derive the global convergence of the outer space branching search method.

Outer space rectangle bisection method
The outer space rectangle bisection method iteratively subdivides the currently investigated rectangle into two sub-rectangles.Consider any selected sub-rectangle Z = {z ∈ R p |z i ≤ z i ≤ z i , i = 1, 2, . . ., p} ⊆ Z 0 .The outer space rectangle bisection method is given as follows: (i) Let q = arg max{z i − z i |i = 1, 2, . . ., p}; , . . ., p, i q; z q ≤ z q ≤ (z q + z q )/2} and Through utilizing the proposed outer space rectangle bisection method, the selected sub-rectangle Z can be subdivided into two sub-rectangles Z 1 and Z 2 .

Outer space branching search method
In this subsection, the basic steps of the proposed outer space branching search method are formulated as follows.
Step 0. Let the convergence error ϵ ≥ 0, and let the initial outer space rectangle Denote F = ∅ as the set of the initial feasible points, let k = 0, and let the set of all active nodes Ω 0 = {Z 0 }.
Step 1. Solve the LRP over Z 0 , and define (y 0 , z 0 , r 0 ) and LB 0 as its optimal solution and optimal value.Let If U B 0 − LB 0 ≤ ϵ, then the proposed algorithm stops.y 0 and (y 0 , z 0 , r0 ) are ϵ-optimal solutions of the GAFOP and EP over (Z 0 ), respectively.Otherwise, proceed with Step 2.
Step 2. Use the proposed rectangle bisection method to subdivide Z k−1 into two sub-rectangles Z k,1 and Z k,2 .Let Q = {Z k,1 , Z k,2 }.
Step 3.For each Z k,t , t = 1, 2, compute the lower bound LB(Z k,t ) and (y(Z k,t ), z(Z k,t ), r(Z k,t )) by solving the LRP over Z k,t , and let Step 5. Set LB k = min{LB(Z)|Z ∈ Ω k }, and let Z k be the sub-rectangle which satisfies LB k = LB(Z k ).
If U B k −LB k ≤ ϵ, then the proposed algorithm stops.y k and (y k , z k ) are the ϵ-global optimal solutions of the GAFOP and EP, respectively.
Otherwise, set k = k + 1, and go back to Step 2.

Global convergence analysis
In this part, first of all, we define Let v be the global optimal value of the EP over Θ 0 , and define r(y k , z k ) as the objective functional value of the EP corresponding to the feasible solution (y k , z k ).The global convergence analysis of the proposed algorithm can be given by the following theorem.Theorem 3. Given any ϵ ≥ 0, if the proposed algorithm finitely terminates after k iterations, then y k is a global ϵ-optimal solution to the GAFOP in the sense that Otherwise, the proposed algorithm will generate an infinite sequence {y k }, whose accumulation point will be a global optimum solution to the GAFOP.
Proof .If the presented algorithm finitely terminates after k iterations, according to the termination of the algorithm, it follows that By Step 3 of the presented algorithm, we can find a feasible solution (y k , z k ) to the EP such that Since (y k , z k ) is feasible for the EP, we have By using the above conclusions, we have So, (y k , z k ) is a global ϵ-optimal solution of the EP, with Thus, y k is a global ϵ-optimum solution to the GAFOP.If the presented algorithm does not finitely terminate, then it must produce an infinite feasible solution sequence {(y k , z k )}, and the sequence {(y k , z k )} has a convergence subsequence.Therefore, we can let lim So, we have From the branch-and-bound structure of the algorithm, we also get Since y * is a feasible solution of the GAFOP over Z 0 , and due to Theorem 2, we can get v ≤ Λ(y * ).
Combining the above inequalities, we have Furthermore, by the equivalence of the GAFOP and EP, and the continuity of the function Λ(y), we can conclude the following: Based on the above inequalities (2) and (3), we have Therefore, this implies that any accumulation point y * of the sequence {y k } is a globally optimum solution to the GAFOP.The proof is complete.□

Numerical comparisons
For verifying the computational superiority of the algorithm, the presented algorithm is implemented in the software MATLAB R2014a and solved on the same microcomputer with an Intel(R) Core(TM) i5-7200U CPU @2.50 GHz processor and 4 GB RAM.
We first tested some randomly generated Problem 1 with small-size variables, numerically compared them with the known existing algorithms [19,39,40] and listed these numerical comparison results in Table 1.Next, we tested some randomly generated Problem 1 with large-size variables to verify our algorithm further and listed the numerical results in Table 2.In Table 2, Avg.Iter represents the average iteration times and Avg.Time represents the average execution CPU time in seconds.
Before executing the algorithm, by calculating the upper and lower bounds of z, we can obtain the initial rectangle We set the approximation error as ϵ = 10 −2 , and a brief summary of the algorithm's solution steps for this problem is as follows.
Repeat the above iterative process, and the algorithm stops when U B − LB ≤ ϵ is satisfied.The optimal solution (y 1 , y 2 , y 3 ) = (1.0167,0.5500, 1.4500) and optimal value 1.34783 of the problem can be obtained after the algorithm executes 14 iterations.

Conclusions
We study the GAFOP.By exploiting equivalent conversion and a new linearizing technique, the initial GAFOP is able to be converted into a series of LRPs.By integrating the outer space branching search method and the LRP, we put forward an efficient global algorithm for the GAFOP.In contrast to the known existing algorithms, our algorithm has the following computational superiority: (i) The branching search occurs in R p outer space, which provides the possibility of mitigating the required computational efforts of the algorithm.(ii) Numerical results demonstrate that our algorithm has superior efficiency compared to the known existing algorithms.Future work is to give a further improvement of our algorithm and extend our method to deal with the general nonlinear fractional optimization problem.
y j + h p y j + h i and the maximum value z 0 i = max y∈Y n j=1 c i j y j + h i of the linear function n j=1 c i j y j + h i over Y, an initial outer space rectangle 9292, 8.8500 ≤ z 2 ≤ 9.2000 and Z 1,2 = {z ∈ R 2 | 1.7315 ≤ z 1 ≤ 1.9292, 9.2000 ≤ z 2 ≤ 9.5500}.
p}, can be constructed.

Table 1 .
Numerical comparisons among some algorithms and our algorithm on Problem 1.