A Global Optimization Algorithm for Sum of Linear Ratios Problem

Abstract

We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

1. Introduction

We consider the sum of linear ratios programming problem as the following form: ()∶(𝑥)=GFPmin𝑓∑^𝑝_𝑖=1𝑓_𝑖(𝑥)=∑^𝑝_𝑖=1𝑛_𝑖(𝑥)𝑑_𝑖(𝑥),s.t.𝐴𝑥≤𝑏,𝑥≥0, (1) where the feasible domain 𝐷≜{𝑥∈𝑅^𝑛∣𝐴𝑥≤𝑏,𝑥≥0} is 𝑛-dimensional, nonempty, and bound, 𝐴∈𝑅^𝑚×𝑛,𝑏∈𝑅^𝑚. Assume that 𝑛_𝑖(𝑥)=𝑐_𝑖^𝑇𝑥+𝑑_𝑖≥0 and 𝑑_𝑖(𝑥)=𝑒_𝑖^𝑇𝑥+𝑟_𝑖>0 in some rectangle 𝑋 which contains 𝐷, where 𝑐_𝑖,𝑒_𝑖∈𝑅^𝑛,𝑑_𝑖,𝑟_𝑖∈𝑅, 𝑖=1,2,…,𝑝, and 2≤𝑝≪𝑛.

Fractional programming is an important branch of nonlinear optimization and it has attracted many researchers’ concern for several decades. Sum of linear ratios problem is a special class of fractional programming problem; it has wide applications, such as investment, transportation scheme, and economic benefits [1–3]. From a research point view, sum of ratios problems challenge theoretical analysis and computation because these problems possess multiple local optima that are not globally optimal solutions; it is difficult to solve the global solution.

At present there exist a number of algorithms for globally solving sum of linear ratios problems. When 𝑝=2, Konno et al. [4] constructed a similar parametric simplex algorithm which can solve large-scale optimization problems; when 𝑝=3, Konno and Abe [5] developed parametric simplex algorithm and constructed an effected heuristic algorithm; when 𝑝>3, the literature [6] is a sum of linear ratios problem with coefficients; by using an equivalent transformation and linearization technique, the original nonconvex programming problem reduces to a series of linear programming problems to achieve the purpose of solving it. To minimize the problem, Yanjun et al. [7] use the linearization technique twice by the nature of exponential and logarithmic functions to achieve a linear relaxation programming of the original problem. Benson [8] put forward a new branch and bound algorithm to solve the equivalent concave minimum problem of the original problem. Jiao and Feng [9] present a new pruning technique. In the literature [10], the numerator and denominator of the ratios are not necessarily positive. In this paper, we present a new branch and bound algorithm for solving the sum of linear ratios problems, and the convergence of the algorithm is proved. At last, the numerical experiments are carried out.

This paper is organized as follows. In Section 2, we show how to convert the problem (GFP) into an equivalent problem (EP) by a transformed technique. In Section 3, the linear relaxation programming problem of (EP) is constructed. The branching process of the rectangle is given in Section 4. In Section 5, the branch and bound algorithm for globally solving (EP) is presented and the convergence of the algorithm is proved. In Section 6, some numerical results are given to show the effectiveness of the present algorithm. Finally, the conclusion is given.

2. Equivalent Transformation

Because the set 𝐷 is nonempty and bound, we can construct the rectangle 𝑋=[𝑙,𝑢], which contains the feasible region of the problem (GFP), 𝑙=(𝑙₁,𝑙₂,…,𝑙_𝑛)^𝑇,𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇,𝑙_𝑗 and 𝑢_𝑗 is the optimal value of the linear programming problem (2) and (3), respectively. min𝑙𝑥_𝑗=𝑥_𝑗,𝑥s.t.𝐴𝑥≤𝑏,𝑥≥0,max𝑢_𝑗=𝑥_𝑗,s.t.𝐴𝑥≤𝑏,𝑥≥0. (2)(3)

Firstly, we solve the following 2𝑝 linear programming problems: min𝑑_𝑖(𝑥),s.t.𝑥∈𝐷,max𝑑_𝑖(𝑥),s.t.𝑥∈𝐷,𝑖=1,2,…,𝑝. (4) The optimal solutions of (4) are 𝑥¹_𝑖 and 𝑥²_𝑖 (𝑖=1,2,…,𝑝), and the optimal value is denoted by 𝑙_𝑖 and 𝑢_𝑖 (𝑖=1,2,…,𝑝) respectively. Obviously, 𝑥¹_𝑖 and 𝑥²_𝑖 are feasible to (GFP). Set 𝑊=𝑊∪{𝑥¹_𝑖,𝑥²_𝑖∶𝑖=1,2,…,𝑝}, where 𝑊 represent the set of the current feasible solution of the problem (GFP). Set 𝐻⁰=𝑦∈𝑅^𝑝∣𝑙⁰_𝑖≤𝑦_𝑖≤𝑢⁰_𝑖,𝑦,𝑖=1,2,…,𝑝𝑦=₁,𝑦₂,…,𝑦_𝑝^𝑇, (5) where 𝑙⁰_𝑖=1/𝑢_𝑖, 𝑢⁰_𝑖=1/𝑙_𝑖. Then the problem (GFP) is converted into an equivalent nonconvex programming problem: EP𝐻⁰∶min𝜑₀𝑥,𝑦=∑^𝑝_𝑖=1𝑦_𝑖𝑛_𝑖(𝑥)=∑^𝑝_𝑖=1𝑦∑_𝑖^𝑛_𝑗=1𝑐_𝑖𝑗𝑥_𝑗+𝑑_𝑖,s.t.𝜑_𝑖𝑥,𝑦=𝑦_𝑖𝑑_𝑖(𝑥)=𝑦∑_𝑖^𝑛_𝑗=1𝑒_𝑖𝑗𝑥_𝑗+𝑟_𝑖≥1,𝑖=1,2,…,𝑝,𝑥∈𝐷∩𝑋,𝑦∈𝐻⁰. (6)

Theorem 1 (see [10]). If (𝑥^∗,𝑦^∗) is a global optimal solution of the problem 𝐸𝑃(𝐻⁰), then 𝑥^∗is a global optimal solution of the problem (GFP), and for every 𝑖=1,2,…,𝑝, when 𝑛_𝑖(𝑥^∗)≥0, 𝑦^∗_𝑖=1/𝑑_𝑖(𝑥^∗); conversely, if 𝑥^∗ is a global optimal solution of the problem (GFP), then (𝑥^∗,𝑦^∗) is a global optimal solution of the problem 𝐸𝑃(𝐻⁰), where 𝑦^∗_𝑖=1/𝑑_𝑖(𝑥^∗),𝑖=1,2,…,𝑝.

From Theorem 1, the problems (GFP) and EP(𝐻⁰) are equivalent; their global optimal values are equal. Therefore, in order to solve (GFP), we only need to solve EP(𝐻⁰) instead.

3. Linear Relaxation Technique

From Section 2, 𝑋=[𝑙,𝑢] and 𝐻⁰=[𝑙⁰,𝑢⁰] are rectangles; set Ω_𝑖=𝑥,𝑦_𝑖∣𝑙≤𝑥≤𝑢,𝑙⁰_𝑖≤𝑦_𝑖≤𝑢⁰_𝑖=Ω_1𝑖×Ω_2𝑖×⋅⋅⋅Ω_𝑛𝑖, (7) where Ω_𝑗𝑖=𝑥_𝑗,𝑦_𝑖∣𝑙_𝑗≤𝑥_𝑗≤𝑢_𝑗,𝑙⁰_𝑖≤𝑦_𝑖≤𝑢⁰_𝑖,𝑗=1,2,…,𝑛. (8) Because in Ω_𝑗𝑖 we have 𝑥_𝑗−𝑙_𝑗≥0,𝑦_𝑖−𝑙⁰_𝑖≥0, so 𝑥_𝑗−𝑙_𝑗𝑦_𝑖−𝑙⁰_𝑖≥0,𝑗=1,2,…,𝑛, (9) expanding it, then we have 𝑥_𝑗𝑦_𝑖≥𝑙⁰_𝑖𝑥_𝑗+𝑙_𝑗𝑦_𝑖−𝑙_𝑗𝑙⁰_𝑖,𝑗=1,2,…,𝑛.

Similarly, we can obtain that 𝑥_𝑗−𝑢_𝑗≤0,𝑦_𝑖−𝑢⁰_𝑖≤0, so 𝑥_𝑗−𝑢_𝑗𝑦_𝑖−𝑢⁰_𝑖≥0,𝑗=1,2,…,𝑛, (10) expanding it, then we have 𝑥_𝑗𝑦_𝑖≥𝑢⁰_𝑖𝑥_𝑗+𝑢_𝑗𝑦_𝑖−𝑢_𝑗𝑢⁰_𝑖,𝑗=1,2,…,𝑛. Let 𝜃¹¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖=𝑙⁰_𝑖𝑥_𝑗+𝑙_𝑗𝑦_𝑖−𝑙_𝑗𝑙⁰_𝑖𝜃,𝑗=1,2,…,𝑛,_12𝑗𝑖𝑥_𝑗,𝑦_𝑖=𝑢⁰_𝑖𝑥_𝑗+𝑢_𝑗𝑦_𝑖−𝑢_𝑗𝑢⁰_𝑖,𝑗=1,2,…,𝑛. (11) Because 𝑥_𝑗𝑦_𝑖≥𝜃¹¹_𝑗𝑖(𝑥_𝑗,𝑦_𝑖),𝑥_𝑗𝑦_𝑖≥𝜃¹²_𝑗𝑖(𝑥_𝑗,𝑦_𝑖),𝑗=1,2,…,𝑛, we have the following result: 𝑥_𝑗𝑦_𝑖≥max𝜃¹¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃¹²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑗=1,2,…,𝑛. (12)

Similarly, we have (𝑥_𝑗−𝑙_𝑗)(𝑦_𝑖−𝑢⁰_𝑖)≤0,(𝑥_𝑗−𝑢_𝑗)(𝑦_𝑖−𝑙⁰_𝑖)≤0,𝑗=1,2,…,𝑛, expanding them, then we have 𝑥_𝑗𝑦_𝑖≤𝑢⁰_𝑖𝑥_𝑗+𝑙_𝑗𝑦_𝑖−𝑙_𝑗𝑢⁰_𝑖,𝑥_𝑗𝑦_𝑖≤𝑙⁰_𝑖𝑥_𝑗+𝑢_𝑗𝑦_𝑖−𝑢_𝑗𝑙⁰_𝑖; let 𝜃²¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖=𝑢⁰_𝑖𝑥_𝑗+𝑙_𝑗𝑦_𝑖−𝑙_𝑗𝑢⁰_𝑖𝜃,𝑗=1,2,…,𝑛,_22𝑗𝑖𝑥_𝑗,𝑦_𝑖=𝑙⁰_𝑖𝑥_𝑗+𝑢_𝑗𝑦_𝑖−𝑢_𝑗𝑙⁰_𝑖,𝑗=1,2,…,𝑛. (13) Consequently, 𝑥_𝑗𝑦_𝑖≤min𝜃²¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃²²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑗=1,2,…,𝑛. (14) From formulae (12) and (14), the following formula is obtained: max𝜃¹¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃¹²_𝑗𝑖𝑥_𝑗,𝑦_𝑖≤𝑥_𝑗𝑦_𝑖≤min𝜃²¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃²²_𝑗𝑖𝑥_𝑗,𝑦_𝑖. (15)

In the problem EP(𝐻⁰), let LB(𝑥) and UB(𝑥), respectively, represent the lower bound and upper bound of 𝑥; then LB𝑐_𝑖𝑗𝑥_𝑗𝑦_𝑖=𝑐_𝑖𝑗⋅max𝜃¹¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃¹²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑐_𝑖𝑗𝑐≥0,_𝑖𝑗⋅min𝜃²¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃²²𝑥_𝑗,𝑦_𝑖,𝑐_𝑖𝑗𝑒<0,UB_𝑖𝑗𝑥_𝑗𝑦_𝑖=𝑒_𝑖𝑗⋅min𝜃²¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃²²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑒_𝑖𝑗𝑒≥0,_𝑖𝑗⋅max𝜃¹¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝜃¹²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑒_𝑖𝑗<0. (16) From formula (16), we can obtain the linear relaxation programming problem REP(𝐻⁰) of the problem EP(𝐻⁰): REP𝐻⁰∶min𝜑^𝑙₀𝑥,𝑦=∑^𝑝∑_𝑖=1^𝑛_𝑗=1LB𝑐_𝑖𝑗𝑥_𝑗𝑦_𝑖+𝑑_𝑖𝑦_𝑖,s.t.𝜑^𝑢_𝑖𝑥,𝑦=∑^𝑛_𝑗=1UB𝑒_𝑖𝑗𝑥_𝑗𝑦_𝑖+𝑟_𝑖𝑦_𝑖≥1,𝑖=1,2,…,𝑝,𝑥∈𝐷∩𝑋,𝑦∈𝐻⁰. (17) The optimal value of the problem REP(𝐻⁰) is a lower bound of the optimal value of the problem EP(𝐻⁰) in the feasible region 𝐷.

Obviously, the problem REP(𝐻⁰) can equivalently be converted into the following linear programming problem LRP(𝐻⁰):LRP𝐻⁰∶min𝑓𝑥,𝑦,𝑡,𝑠=∑^𝑝∑_𝑖=1^𝑛_𝑗=1𝑡_𝑗𝑖+𝑑_𝑖𝑦_𝑖,s.t.∑^𝑛_𝑗=1𝑠_𝑗𝑖+𝑟_𝑖𝑦_𝑖𝑡≥1,𝑖=1,2,…𝑝,_𝑗𝑖≥𝑐_𝑖𝑗⋅𝜃¹¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑐_𝑖𝑗𝑡≥0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,_𝑗𝑖≥𝑐_𝑖𝑗⋅𝜃¹²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑐_𝑖𝑗𝑡≥0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,_𝑗𝑖≥𝑐_𝑖𝑗⋅𝜃²¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑐_𝑖𝑗𝑡<0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,_𝑗𝑖≥𝑐_𝑖𝑗⋅𝜃²²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑐_𝑖𝑗𝑠<0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,_𝑗𝑖≤𝑒_𝑖𝑗⋅𝜃²¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑒_𝑖𝑗𝑠≥0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,_𝑗𝑖≤𝑒_𝑖𝑗⋅𝜃²²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑒_𝑖𝑗𝑠≥0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,_𝑗𝑖≤𝑒_𝑖𝑗⋅𝜃¹¹_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑒_𝑖𝑗𝑠<0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,_𝑗𝑖≤𝑒_𝑖𝑗⋅𝜃¹²_𝑗𝑖𝑥_𝑗,𝑦_𝑖,𝑒_𝑖𝑗<0,𝑗=1,2,…,𝑛,𝑖=1,2,…,𝑝,𝑥∈𝐷∩𝑋,𝑦∈𝐻⁰. (18)

The optimal value of the problem LRP(𝐻⁰) can be obtained by solving the linear programming problem LRP(𝐻⁰), which is a lower bound of the problem EP(𝐻⁰) in feasible region 𝐷.

The Determination of Upper Bound. From the process of the determination of lower bound, by solving LRP(𝐻⁰), we can obtain a global optimal solution 𝑥^∗; let 𝑦_𝑖^∗=^𝑛_𝑗=1𝑒_𝑖𝑗𝑥_𝑗^∗+𝑟_𝑖⁻¹. (19) It is obvious that (𝑥^∗,𝑦^∗) is a feasible solution of EP(𝐻⁰). Therefore, 𝜑₀(𝑥^∗,𝑦^∗) provide an upper bound for the global optimal value 𝜈(𝐻⁰) of the problem EP(𝐻⁰).

4. Branching

In this algorithm, the branching process is executed in the space of 𝑅^𝑝 other than in 𝑅^𝑛. In general, when 𝑝≪𝑛, the amount of computation will decrease so that the efficiency of computation will improve. Therefore, we choose the rectangle 𝐻⁰ which contains 𝑦 to branch, and the subrectangle after branching is also 𝑝-dimensional. Set 𝐻=𝑦∈𝑅^𝑝∣𝐿_𝑖≤𝑦_𝑖≤𝑈_𝑖,𝑖=1,2,…,𝑝. (20) Denote the initial rectangle 𝐻⁰ or subrectangle of it. The branching rule is as follows:(i)choose the longest side of 𝐻, that is, 𝑈_𝑠−𝐿_𝑠=max{𝑈_𝑖−𝐿_𝑖∶𝑖=1,2,…,𝑝};(ii)let 𝑉_𝑠=(𝑈_𝑠+𝐿_𝑠)/2 and 𝐻¹=^𝑠−1_𝑖=1𝐿_𝑖,𝑈_𝑖×𝐿_s,𝑉_𝑠×^𝑝_𝑖=𝑠+1𝐿_𝑖,𝑈_𝑖,𝐻²=^𝑠−1_𝑖=1𝐿_𝑖,𝑈_𝑖×𝑉_𝑠,𝑈_𝑠×^𝑝_𝑖=𝑠+1𝐿_𝑖,𝑈_𝑖. (21)

5. Algorithm and Its Convergence

The branch and bound algorithm of the problem (GFP) is stated as follows:

Step 1. Choose 𝜀≥0, the initial rectangle 𝐻⁰={𝑦∈𝑅^𝑝∣𝑙⁰_𝑖≤𝑦_𝑖≤𝑢⁰_𝑖,𝑖=1,2,…,𝑝}; we can find an optimal solution 𝑥⁰ and the optimal value LB(𝐻⁰) by solving the problem LRP(𝐻⁰). Set LB₀=LB(𝐻⁰), 𝑥^𝑐=𝑥⁰. Set 𝑦^𝑐_𝑖=(∑^𝑛_𝑗=1𝑒_𝑖𝑗𝑥^𝑐_𝑗+𝑟_𝑖)⁻¹, 𝑖∈{1,2,…,𝑝}, UB₀=𝜑₀(𝑥^𝑐,𝑦^𝑐).

  If UB₀−LB₀≤𝜀, stop. (𝑥^𝑐,𝑦^𝑐) and 𝑥^𝑐 are global 𝜀-optimal solutions of problems EP(𝐻⁰) and (GFP), respectively. Otherwise, set 𝑃₀={𝐻⁰}, 𝐹=Ø, 𝑘=1, and go to Step 2.

Step 2. Set UB_𝑘=UB_𝑘−1. Subdivide 𝐻^𝑘−1 into two 𝑝-dimensional rectangles 𝐻^𝑘,1,𝐻^𝑘,2⊆𝑅^𝑝 via the branching rule. Set 𝐹=𝐹∪{𝐻^𝑘−1}.

Step 3. For 𝑗=1,2, compute LB(𝐻^𝑗,𝑘). If LB(𝐻^𝑗,𝑘)≠+∞, find an optimal solution 𝑥^𝑘,𝑗 of problem LRP(𝐻) with 𝐻=𝐻^𝑗,𝑘; set 𝑡=0.

Step 4. Set 𝑡=𝑡+1. If 𝑡>2, go to Step 6. Otherwise, continue.

Step 5. If UB_𝑘≤LB(𝐻^𝑘,𝑡), set 𝐹=𝐹∪{𝐻^𝑘,𝑡}; go to Step 4. Otherwise, set 𝑦_𝑖^𝑘,𝑡=^𝑛_𝑗=1𝑒_𝑖𝑗𝑥_𝑗^𝑘,𝑡+𝑟_𝑖⁻¹,𝑖∈1,2,…,𝑝. (22) Let UB_𝑘=min{UB_𝑘,𝜑₀(𝑥^𝑘,𝑡,𝑦^𝑘,𝑡)}. If UB_𝑘<𝜑₀(𝑥^𝑘,𝑡,𝑦^𝑘,𝑡), go to Step 4. If UB_𝑘=𝜑₀(𝑥^𝑘,𝑡,𝑦^𝑘,𝑡), set 𝑥^𝑐=𝑥^𝑘,𝑡,(𝑥^𝑐,𝑦^𝑐)=(𝑥^𝑘,𝑡,𝑦^𝑘,𝑡). Let 𝐹=𝐹∪𝐻∈𝑃_𝑘−1∣UB_𝑘≤LB(𝐻). (23)

Step 6. Set 𝑃_𝑘={𝐻∣𝐻∈(𝑃_𝑘−1∪{𝐻^𝑘,1,𝐻^𝑘,2}),𝐻∉𝐹}.

Step 7. Set LB_𝑘=min{LB(𝐻)∣𝐻∈𝑃_𝑘}. Let 𝐻^𝑘∈𝑃_𝑘 satisfy LB_𝑘=LB(𝐻^𝑘).

If UB₀−LB₀≤𝜀, stop. (𝑥^𝑐,𝑦^𝑐) and 𝑥^𝑐 are global 𝜀-optimal solutions of the problems EP(𝐻⁰) and (GFP), respectively. Otherwise, set 𝑘=𝑘+1 and go to Step 2.

Next, the convergence of the algorithm is stated in the following theorem.

Theorem 2. (a)  If the algorithm is finite, (𝑥^𝑐,𝑦^𝑐) and 𝑥^𝑐 are global 𝜀-optimal solutions of the problems 𝐸𝑃(𝐻⁰) and (GFP), respectively.
(b)  For 𝑘≥0, let 𝑥^𝑘 denote the incumbent solution 𝑥^𝑐 at the end of step 𝑘. If the algorithm is infinite, then {𝑥^𝑘} is a feasible solution sequence, whose every accumulation point is a global optimal solution of the problem (GFP), and lim_𝑘→∞UB_𝑘=lim_𝑘→∞LB_𝑘=𝜈. (24)

Proof. (a) If the algorithm is finite, without loss of generality, it terminates in step 𝑘(𝑘≥0), since (𝑥^𝑐,𝑦^𝑐) is obtained by solving problem LRP(𝐻), for some 𝐻⊆𝐻⁰ and optimal solution 𝑥^𝑐, set 𝑦^𝑐_𝑖=1∑^𝑛_𝑗=1𝑒_𝑖𝑗𝑥^𝑐_𝑗+𝑟_𝑖,𝑖∈1,2,…,𝑝, (25) where 𝑥^𝑐 is a feasible solution of the problem (GFP) and (𝑥^𝑐,𝑦^𝑐) is a feasible solution of problem EP(𝐻⁰). When UB_𝑘−LB_𝑘≤𝜀, the algorithm terminates. From Steps 1, 2, and 5, it is implied that 𝜑₀(𝑥^𝑐,𝑦^𝑐)−LB_𝑘≤𝜀; by the algorithm, it shows that LB_𝑘≤𝜈. Since (𝑥^𝑐,𝑦^𝑐) is a feasible solution of the problem EP(𝐻⁰), therefore, 𝜑₀(𝑥^𝑐,𝑦^𝑐)≥𝜈.
Taken together, it is implied that 𝜈≤𝜑₀𝑥^𝑐,𝑦^𝑐≤LB_𝑘+𝜀≤𝜈+𝜀. (26) Therefore, 𝜈≤𝜑₀𝑥^𝑐,𝑦^𝑐≤𝜈+𝜀. (27) From the formula 𝑦^𝑐_𝑖=1/(∑^𝑛_𝑗=1𝑒_𝑖𝑗𝑥^𝑐_𝑗+𝑟_𝑖),𝑖=1,2,…,𝑝, we have 𝑓𝑥^𝑐=𝜑₀𝑥^𝑐,𝑦^𝑐. (28) From (27), this implies that 𝜈≤𝑓𝑥^𝑐≤𝜈+𝜀. (29) The proof of (a) is complete.
(b) If the algorithm is infinite, then it generates a sequence of incumbent solutions of the problem EP(𝐻⁰), denoted by {(𝑥^𝑘,𝑦^𝑘)}, for each 𝑘≥1, (𝑥^𝑘,𝑦^𝑘) is obtained by solving the problem LRP(𝐻). For some 𝐻^𝑘⊆𝐻⁰ and optimal solution 𝑥^𝑘∈𝐷, set 𝑦^𝑘_𝑖=1∑^𝑛_𝑗=1𝑒_𝑖𝑗𝑥^𝑘_𝑗+𝑟_𝑖,𝑖∈1,2,…,𝑝. (30) Then the sequence {𝑥^𝑘} consists of feasible solutions of the problem (GFP).
Suppose that 𝑥 is an accumulation point of {𝑥^𝑘}. Assume without loss of generality that lim_𝑘→∞𝑥^𝑘=𝑥. Since 𝐷 is a compact set, 𝑥∈𝐷. Furthermore, because {𝑥^𝑘} is infinite, we assume without loss of generality that, for each 𝑘, 𝐻^𝑘+1⊆𝐻^𝑘, for some point 𝑦∈𝑅^𝑝, lim_𝑘→∞𝐻^𝑘=_𝑘𝐻^𝑘=𝑦. (31) Set 𝐻={𝑦}, for each 𝑘; let 𝐻^𝑘={𝑦∈𝑅^𝑝∣𝐿^𝑘_𝑖≤𝑦_𝑖≤𝑈^𝑘_𝑖,𝑖=1,2,…,𝑝}. Since 𝐻^𝑘+1⊆𝐻^𝑘⊆𝐻⁰, from Step 5, we know that {LB(𝐻^𝑘)} is a nonincreasing sequence, and lim_𝑘→∞LB(𝐻^𝑘) is a finite number and satisfies lim_𝑘→∞LB𝐻^𝑘≤𝜈. (32) For each 𝑘, from Step 3, we know that LB(𝐻^𝑘) is equal to the optimal value of the problem LRP(𝐻^𝑘) and that 𝑥^𝑘 is an optimal solution of this problem. From (31), we have lim_𝑘→∞𝐿^𝑘=lim_𝑘→∞𝑈^𝑘=𝑦=𝐻. (33) Since lim_𝑘→∞𝑥^𝑘=𝑥, 𝐿^𝑘_𝑖≤1/(∑^𝑛_𝑗=1𝑒_𝑖𝑗𝑥^𝑘_𝑗+𝑟_𝑖)≤𝑈^𝑘_𝑖, and the continuity of ∑^𝑝_𝑖=1𝑒_𝑖𝑗𝑥^𝑘_𝑗+𝑟_𝑖, 1∑^𝑛_𝑗=1𝑒_𝑖𝑗𝑥_𝑗+𝑟_𝑖=𝑦_𝑖,𝑖=1,2,…,𝑝. (34) This implies that (𝑥,𝑦) is a feasible solution of the problem EP(𝐻⁰). Therefore, 𝜑₀𝑥,𝑦≥𝜈. (35) Together with (32), we have 𝜑₀𝑥,𝑦≥𝜈≥lim_𝑘→∞LB𝐻^𝑘. (36) Since the branching process is bisection and the branching process of rectangle is exhaustive, we have lim_𝑘→∞LB𝐻^𝑘=𝜈=𝜑₀𝑥,𝑦. (37) Therefore, (𝑥,𝑦) is a global optimal solution of the problem EP(𝐻⁰). By Theorem 1, this implies that 𝑥 is a global optimal solution of the problem (GFP). For each 𝑘, since 𝑥^𝑘 is the incumbent solution of the problem (GFP) at the end of step 𝑘, UB_𝑘=𝑓(𝑥^𝑘); by the continuity of 𝑓, we obtain that lim_𝑘→∞𝑓𝑥^𝑘=𝑓(𝑥). (38) Since 𝑥 is a global optimal solution of the problem (GFP), 𝑓(𝑥)=𝜈. (39) Therefore, lim_𝑘→∞UB_𝑘=𝜈. The proof is complete.

6. Numerical Experiment

The proposed algorithm is programmed in MATLAB 7.8 and is run in Pentium(R) 4 CPU 3.20 GHz. In order to compare with the algorithm of the literature [10], we perform three experiments to the literature [10].

Example 1 (see [10]). We choose 𝑝=𝑛=2; for each (𝑥₁,𝑥₂)∈𝑅², the numerator and denominator are 𝑛₁𝑥₁,𝑥₂=37𝑥₁+73𝑥₂𝑛+13,₂𝑥₁,𝑥₂=63𝑥₁−18𝑥₂𝑑+39,₁𝑥₁,𝑥₂=13𝑥₁+13𝑥₂𝑑+13,₂𝑥₁,𝑥₂=13𝑥₁+26𝑥₂+13, (40) and all (𝑥₁,𝑥₂)∈𝐷 satisfy 5𝑥₁−3𝑥₂=3,1.5≤𝑥₁≤3. (41)

From our algorithm, we firstly should solve the following linear programming problems: min𝑑_𝑖(𝑥),s.t.𝐴𝑥≤𝑏,max𝑑_𝑖(𝑥),s.t.𝐴𝑥≤𝑏,𝑖=1,2,…,𝑝, (42) of which the optimal solutions denote by 𝑥¹_𝑖,𝑥²_𝑖(𝑖=1,2); then 𝑊=𝑊∪𝑥¹_𝑖,𝑥²_𝑖∶𝑖=1,2,…,𝑝, (43) where 𝑊 represent the set of the current feasible solution of the problem EP(𝐻⁰), and the optimal value is denoted by 𝑙_𝑖 and 𝑢_𝑖(𝑖=1,2); then the initial rectangle is 𝐻⁰=0.00960.01920.00640.0140. (44) By solving the linear relaxation programming problem LRP(𝐻⁰), we obtain the optimal solution 𝑥⁰=[2.0016;2.3360] and the optimal value LB(𝐻⁰)=3.9743; then a lower bound of the original problem is LB(𝐻⁰)=3.9743. Set 𝑦⁰_𝑖=^𝑛_𝑗=1𝑒_𝑖𝑗𝑥⁰_𝑗+𝑟_𝑖⁻¹. (45) Then (𝑥⁰,𝑦⁰) is a feasible solution of EP(𝐻⁰), min{𝜑₀(𝑥⁰,𝑦⁰),𝑓(𝑥)∶𝑥∈𝑊}=4.9126, then it provides an upper bound for the global optimal value of the problem EP(𝐻⁰). Next, we choose the rectangle 𝐻⁰ corresponding with the lower bound to branch; we obtain the following rectangles via our algorithm: 𝐻^0,1=0.00960.01440.00640.0140,𝐻^0,2=0.01440.01920.00640.0140. (46) We solve the linear relaxation programming problem LRP in rectangles 𝐻^0,1 and 𝐻^0,2, respectively. In LRP(𝐻^0,1), the optimal solution and the optimal value are [2.2524;2.7540] and 𝑣=4.2345; then in rectangle 𝐻^0,1, the lower bound of the original problem is LB(𝐻^0,1)=4.2345, and the upper bound corresponding with the optimal solution is 4.9617 (>4.9126), so the upper bound is unchanged. In LRP(𝐻^0,2), the optimal solution and the optimal value are [1.8019;2.0032]; and 𝑣=4.5548; then in rectangle 𝐻^0,2, the lower bound of the original problem is LB(𝐻^0,2)=4.5548, and the upper bound corresponding with the optimal solution is 4.9323 (>4.9126), so the upper bound is also unchanged. Then we choose the rectangle corresponding with the lower bound to branch until the 55th iteration, and we can obtain that 𝐻^55,1=0.01860.01890.01350.0137, (47) we solve the linear programming problem LRP in 𝐻^55,1; the lower bound is 4.9125; it satisfies the terminated rule. Therefore, the optimal value and the optimal solution of the original problem are 4.9126 and 𝑥=[1.5000;1.5000]; the lower bound of the optimal value is 4.9125, which is approximate optimal value. The accuracy is 𝜀=0.0001.

The above example satisfies (𝑛,𝑝)=(2,2), where 𝑛 denote the number of variables; our algorithm can have a good approach within accuracy. In Example 2, (𝑛,𝑝)=(3,3); in Example 3, (𝑛,𝑝)=(3,4) we still get good results. Along with the increase of 𝑛 and 𝑝, the computation complexity is increasing. For example, in Example 3, (𝑛,𝑝)=(3,4), we can quickly obtain the approximate optimal value and the optimal value by using this paper’s algorithm, but its effect is poorer than the former example. The result of Example 1 is shown in Table 1.

Table 1 

Example 2 (see [10]). min3𝑥₁+5𝑥₂+3𝑥₃+503𝑥₁+4𝑥₂+5𝑥₃++503𝑥₁+4𝑥₂+504𝑥₁+3𝑥₂+2𝑥₃++504𝑥₁+2𝑥₂+4𝑥₃+505𝑥₁+4𝑥₂+3𝑥₃,+50s.t.2𝑥₁+𝑥₂+5𝑥₃𝑥≤10,₁+6𝑥₂+2𝑥₃≤10,9𝑥₁+7𝑥₂+3𝑥₃𝑥≥10,₁,𝑥₂,𝑥₃≥0. (48) The optimal value is 2.8619.

Example 3 (see [10]). min4𝑥₁+3𝑥₂+3𝑥₃+503𝑥₂+3𝑥₃++503𝑥₁+4𝑥₃+504𝑥₁+4𝑥₂+5𝑥₃+𝑥+50₁+2𝑥₂+4𝑥₃+50𝑥₁+5𝑥₂+5𝑥₃+𝑥+50¹+2𝑥₂+4𝑥₃+505𝑥₂+4𝑥₃,+50s.t.2𝑥₁+𝑥₂+5𝑥₃𝑥≤10,₁+6𝑥₂+3𝑥₃≤10,9𝑥₁+7𝑥₂+3𝑥₃𝑥≥10,₁,𝑥₂,𝑥₃≥0. (49) The optimal value is 3.7109.

We choose 𝜀=1.0𝑒−4; then the approximate optimal solution satisfying accuracy and the iteration times and CPU running time are obtained. The results of our algorithm are shown in Table 2. But the results of the literature [10] are shown in Table 3.

Table 2 

Table 3 

According to Tables 2 and 3, in Example 1, although the optimal solution (3,4)^𝑇 of the literature [10] is feasible, its optimal value 5 is bigger than 4.9126 of our algorithm; in Example 2, the optimal solution (0,3.3333,0)^𝑇 of the literature [10] turns out to be infeasible; in Example 2, the optimal value 4.0000 which corresponds to the optimal solution (0,0.625,1.875)^𝑇 of the literature [10] is actually 3.8384, but it is still bigger than 3.7109 of our algorithm.

From the above comparison we know that the optimal values of our algorithm are much lesser than in the literature [10], and except for Example 1, the iterations of Examples 2 and 3 are much lesser than in the literature [10]. Although our running time is longer than the literature [10], if we can solve the more accurate optimal solution, the price we pay is acceptable.

In conclusion, our algorithm is feasible and effective, and to some degree, it is better than in the literature [10].

7. Conclusion

In this paper, the solving of the sum of linear ratios programming problem is discussed. The problem is equivalently transformed into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product, the linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is proposed and the convergence of the algorithm is proved. Numerical results show the effectiveness of the algorithm, and our algorithm is better than the calculation results of the literature [10].

Acknowledgments

The work is supported by the Foundation of National Natural Science, China (11161001), and by the research project of Beifang University of Nationalities (2013XYZ025).