A novel algorithm for solving sum of several a ffi ne fractional functions

: By using the outer space branch-and-reduction scheme, we present a novel algorithm for globally optimizing the sum of several a ffi ne fractional functions problem (SAFFP) over a nonempty compact set. For providing the reliable lower bounds in the searching process of iterations, we devise a novel linearizing method to establish the a ffi ne relaxation problem (ARP) for the SAFFP. Thus, the main computational work involves solving a series of ARP. For improving the convergence speed of the algorithm, an outer space region reduction technique is proposed by utilizing the objective function characteristics. Through computational complexity analysis, we estimate the algorithmic maximum iteration times. Finally, numerical comparison results are given to reveal the algorithmic computational advantages.


Introduction
We investigate globally optimizing the sum of several affine fractional functions problem defined by (SAFFP) : where h m j , d m , g m j , f m ∈ R, m = 1, 2, . . ., q, j = 1, 2, . . ., n, A ∈ R µ×n , b ∈ R m , Λ is a nonempty compact set, and n j=1 g m j z j + f m 0 for any z ∈ Λ.
From 1990s, the SAFFP has attracted a lot of attention attentions of many practitioners and researchers.The SAFFP is widely used in computer vision, investment and portfolio optimization, optimal strategy in supply chain, risk-averse and so on, see Refs.[1][2][3][4][5][6].Besides, the SAFFP is nonconvex optimization problem, which usually contains many locally minimum solutions that are not globally minimum.
Additionally, there are some theoretical progress on the generalized SAFFP, for example, Saxena and Jain [32] presented an dual problem for the linear fractional programming problem under fuzzy environment.Based on the membership function of the target multiplied by the appropriate weights, Borza and Rambely [33] proposed a set of linear inequalities.Goli and Nasseri [34] investigate for linear programming problems with intuitionistic fuzzy variables and proposed its pairwise results with a generalization of the pairwise simplex method.
In this article, by using the outer space branch-and-reduction scheme, we propose a global algorithm to effectively solve the SAFFP.We first convert the SAFFP into an equivalent bilinear optimization problem (EBOP).Next, by utilizing new linearizing method, we establish the ARP of the EBOP.To improve the running speed of the outer space searching algorithm, an outer space region reduction method is proposed.By iteratively subdividing the initial outer space region and computing a sequence of LRP, the presented algorithm is globally convergent to the minimum point of the SAFFP.By analysing the algorithmic complexity, we give an estimation for the maximum number of iterations of the proposed algorithm in this paper.Finally, numerical comparisons are reported to reveal the computational superiority and higher efficiency of the algorithm.
The rests of this article are organized as below.In Section 2, the EBOP and its ARP of the SAFFP are derived.In Section 3, based the outer space branch-and-reduction scheme, we construct a global algorithm for the SAFFP, prove and analyse the algorithmic convergence and complexity, and estimate the algorithmic maximum iteration times.Numerical examples and their computational comparisons are reported in Section 4. Finally, we give some conclusions in Section 5.

Equivalence problem and its affine relaxation
In this section, we firstly equivalently convert the SAFFP into the EBOP.Since the denominator n j=1 g m j z j + f m 0 for any z ∈ Λ, by the continuity of the function , without losing generality, we can always assume n j=1 g m j z j + f m > 0.
Without losing generality, let s m = 1 n j=1 g m j z j + f m , and define and construct the initial outer space rectangle then the SAFFP may be changed to the following equivalent bilinear optimization problem: Obviously, (z * , s * ) is a globally optimum solution to the EBOP S 0 if and only if z * is a globally optimum solution to the SAFFP, where Therefore, we may consider globally solving the EBOP S 0 instead of globally solving the SAFFP.
Next, we will give the detailed process for constructing the ARP of the EBOP S 0 as below.

Global algorithm, convergence, and its complexity
In this part, for globally solving the SAFFP, combining the previous affine relaxation problem, we design an outer space region reduction operation, and based on the branch-and-bound searching framework, a global algorithm is designed.

Outer space region reduction operation
To enhance convergence speed of the presented algorithm, we construct a new outer space region reduction operation as follows.
For any investigated rectangles where Theorem 2. Denote U B k as the known best upper bound at the k th iteration, for any investigated rectangle S ⊆ S 0 , we get the following several conclusions: , then the rectangle S contains no globally optimum point to the EBOP(S 0 ).

Novel algorithm base on the outer space branch-and-reduction scheme
By combining the above affine relaxation problem and outer space region reduction technique, a novel algorithm to globally solve the SAFFP can be described as below: Step 0. Letting . ., q , and setting ϵ ∈ [0, 1), solve the ARP S 0 to achieve its optimum solution (z 0 , ŝ0 ) and optimum value LB S 0 , respectively.Simultaneously, let If U B 0 − LB 0 ≤ ϵ, then the presented algorithm will finish with obtaining the ϵ-globally optimum solution (z c , s c ) to the EBOP(S 0 ) and the ϵ-globally optimum solution z c to the SAFFP.
Step 1.Let U B k = U B k−1 , by using the dichotomy method to segment the largest edge of the selected rectangle, and subdivide S k−1 into two q-dimensional sub-rectangles S k,1 and S k,2 .Let F = F ∪ S k−1 .
Step 2. Use the proposed outer space region reduction technique to compress the range of the rectangle S k,α , where α = 1, 2, solve the ARP(S k,α ) to obtain LB S k,α and its optimum solution (z k,α , ŝk,α ).Set η = 0.
and proceed with Step 5.
Step 5. Let Step 6.Let If P k+1 = ∅, then the presented algorithm will finish with obtaining the ϵ-globally optimum solution (z c , s c ) to the EBOP(S 0 ) and the ϵ-globally optimum solution z c to the SAFFP.Otherwise, select S k+1 satisfying that S k+1 = arg min S ∈P k+1 LB (S ) , set k = k + 1, and go back Step 1.

Convergence analysis
In this sub-section, we will prove the convergence of the proposed algorithm by the following theorem.Theorem 3.For any given ϵ ∈ [0, 1).We denote z k as the obtained best solution z c of the SAFFP at the k th iteration.If the presented algorithm finitely terminates after k iterations, then we can obtain an ϵ-globally optimum solution (z c , s c ) to the EBOP(S 0 ) and an ϵ-globally optimum solution z c to the SAFFP.Otherwise, the presented algorithm will generate an infinite feasible solution sequence {z k } with that its each gathering point is a globally optimum solution to the SAFFP.Proof.Assume that the presented algorithm finitely finishes at the k th iteration, then: when the algorithm terminates, (z c , ŝc ) can be obtained by solving the ARP(S ) for some S ⊆ S 0 , and let Obviously, z c and (z c , s c ) are the feasible solutions for the SAFFP and EBOP S 0 , respectively.Upon termination of the presented algorithm, we have From Steps 0 and 4, this implies that By the bounding method, it can follow that Since (z c , s c ) is feasible to the EBOP S 0 , it follows that v ≤ Ψ 0 (z c , s c ) .
Combine the above several inequalities, we can get that , m = 1, 2, . . ., q, we can follow that Combine the above several inequalities, we have that If the presented method does not terminate in finite step, then it will create a best feasible solution sequence z k , s k to the EBOP S 0 .
For each k ≥ 1, for some a rectangle S k ⊆ S 0 , suppose that z k , ŝk is obtained by solving the problem ARP S k , and let Obviously, z k , s k is a feasible solution sequence to the EBOP S 0 .
Without losing generality, we assume that z is an accumulation point of the sequence z k with that lim k→∞ z k = z, then, due to the fact that z k is always feasible solution to the SAFFP and Λ is a nonempty bounded compact set, we must have z ∈ Λ.
Furthermore, when the presented algorithm is infinite, without loss of generality, for each k ≥ 1, assume that S k+1 ⊆ S k .For each k ≥ 1, since the rectangles S k are generated by rectangular bisection, by Horst and Tuy [35], then there must exist some a point s ∈ R q such that lim Let S = { s} and , . . ., q for each k ≥ 1, since S k+1 ⊂ S k ⊂ S 0 , and from Step 4 of the algorithm and Remark 2, this indicates that LB S k is a nondecreasing bounded sequence satisfying that LB S k ≤ v. Thus, lim k→∞ LB S k exists and meets that lim From Step 2 of the algorithm, for each k ≥ 0, LB S k is the optimum value to the ARP S k , and (z k , ŝk ) is the optimal solution for this problem.
From (2), it follows that lim By the continuity of the function n j=1 g m j z j + f m , lim k→∞ z k = z, and This indicates that (z, s) is a feasible solution to the EBOP S 0 ).Thus, Combine ( 3) and ( 4) together, it follows that lim k→∞ LB S k ≤ v ≤ Ψ 0 (z, s).
( 5) From ( 4), ( 5), and the former discussions, it can follow that Hence, (z, s) is a globally optimum solution to the EBOP S 0 .From equivalent conclusions of the EBOP S 0 and SAFFP, this indicates that z is also a global optimal solution to the SAFFP.
For each k ≥ 1, since z k is the best feasible solution to the SAFFP at the k th iteration, then the upper bound satisfies that By the function continuity of F(z), we can follow that Since z is a globally optimum solution to the SAFFP, we have F(z) = v.Thus, we have that and the proof of the theorem is completed.□ By the above theorem, the algorithm is convergent, then, we will analyze the computational efficiency of the algorithm in the worst case.

Complexity results
In this sub-part, by analyzing the algorithmic complexity, we give a maximum estimation of iterations of the outer space algorithm.First of all, for convenience, we denote the maximum size ∆(S ) of the sub-rectangle In addition, we denote , where δ 0 j = max{z j | z ∈ Λ}.Theorem 3.For any setting convergence error ϵ > 0, at iteration k, when the sub-rectangle S k generated by the outer space branching process satisfies where LB(S k ) is the optimum value to the ARP(S k ), and U B is the currently known upper bound of the global optimum value to the EBOP S 0 .Proof.Denote (z k , ŝk ) as the optimum solution to the ARP(S k ), and let then (z k , s k ) is a feasible point of the EBOP(S k ).By the updating and computing methods of U B and LB(S k ), we have that Thus, from (1), (6), and the definitions of ∆(S k ) and β, it follows that Further, from the previous inequalities and ∆(S k ) ≤ ϵ qβ , we can follow that and the proof of the theorem is completed.⋄ By the above Theorem 4 and Step 6 of the presented algorithm, when ∆(S k ) ≤ ϵ qβ , S k will be deleted.Hence, when the sizes of all refined subdivision rectangle S produced by the outer space bisection operation satisfy ∆(S ) ≤ ϵ qβ , the proposed algorithm will be terminated.According to Theorem 4, we may give a maximum estimation of iteration times for the proposed algorithm in this article, see the following Theorem 4 for details.Theorem 4. For arbitrary ϵ > 0, the presented algorithm can seek out an ϵ-globally optimum solution to the SAFFP in at most iterations, where β is defined in the former, and According to Theorem 4 and the partitioning process of the algorithm, the conclusion of the Theorem can be easily concluded, so it is omitted.⋄

Numerical experiments
In this part, we give numerical comparison results among the BARON solver [36], the algorithm proposed in Jiao and Liu [12] which works by globally addressing an equivalent bilinear programming problem, and our algorithm.All algorithms are coded in the software MATLAB R2014a and run on a microcomputer with 2.50 GHz i5-7200U processor and 16 GB RAM.The maximum CPU running time limit for all test problems is set at 3800 s.We reported the numerical result statistics for all test Problems 1 and 2. For each randomly generated test problem, we all solved ten randomly generated test examples and recorded their best results, their worst results and their average results, and highlighted the winner of comparisons of their average results in bold.In the following, we firstly present these test problems and then report their numerical comparisons.
For Problem 1 with the large-size number of variables, with the convergent tolerance ϵ = 10 −2 , numerical comparisons among algorithm of Jiao and Liu [12], our algorithm and BARON are reported in Table 1.For each random example, we solve ten independently generated instances and record the best, the worst and the average results among these ten tests, and we highlight in bold the winner of average results in comparison.
For Problem 2 with the large-size number q, with the convergence tolerance ϵ = 10 −3 , numerical comparisons between our algorithm and BARON are reported in Table 2.In Tables 1 and 2, "−" stand for the condition that the used algorithm failed to seek out the globally optimum solution to some of ten random examples in 3800s.
From Table 1, for Problem 1 with large-size number of variables, we firstly can observe that the BARON solver takes more time than our algorithm proposed in this article, despite its number of iterations for the BARON solver is smaller.Secondly, our algorithm is obviously better than the BARON solver and the algorithm of Jiao and Liu [12].The iteration number of our algorithm proposed in this article is much less than the algorithm of Jiao and Liu [12].Especially, when q = 2 and n = 8000, the BARON solver failed to seek out the globally optimum solution to each of ten random examples in 3800s, but our outer space searching algorithm can achieve the globally optimum solution to all ten random examples of Problem 1 with higher computational efficiency and performance.
From Table 2, for Problem 2 with the large-size number q, we observe that, when q = 10, 15 and n = 500, 600, and q = 20 and n = 400, 500, the BARON solver failed to terminate in 3800s for each one of ten independently generated instances, but our outer space searching algorithm in this paper can seek out the globally optimum solution to all ten independently generated instances within a reasonable time, this demonstrate the strong robustness and reliable stability of our algorithm.

Conclusions
By combining the outer space branch searching scheme, the constructed affine relaxation problem, and the outer space region reduction technique, we design a novel algorithm to efficiently solve the SAFFP.In contrast to the known existing algorithms, by analysing the algorithmic complexity, we can get that the proposed algorithm in this paper can achieve an ϵ-global optimum solution of the SAFFP after at most 2 q m=1 ⌈log 2 qβ(s 0 m −s 0 m ) ϵ ⌉ − 1 iterations.Finally, numerical comparison results are given to demonstrate better computational performance of the proposed algorithm in this paper.In the future work, we will extend our algorithm to globally solve generalized linear fractional programming problem.

Table 1 .
[12]arisons of numerical results among the algorithm of Jiao and Liu[12], the BARON solver and our algorithm in this article on Problem 1 with q = 2 and n = 100.

Table 2 .
Comparisons of numerical results between the BARON solver and our algorithm on Problem 2.