Abstract
In this study, we propose an effective algorithm for globally solving the sum of linear ratios problems. Firstly, by introducing new variables, we transform the initial problem into an equivalent nonconvex programming problem. Secondly, by utilizing direct relaxation, the linear relaxation programming problem of the equivalent problem can be constructed. Thirdly, in order to improve the computational efficiency of the algorithm, an out space pruning technique is derived, which offers a possibility of pruning a large part of the out space region which does not contain the optimal solution of the equivalent problem. Fourthly, based on out space partition, by combining bounding technique and pruning technique, a new out space branch-and-bound algorithm for globally solving the sum of linear ratios problems (SLRP) is designed. Finally, numerical experimental results are presented to demonstrate both computational efficiency and solution quality of the proposed algorithm.
1. Introduction
Sum of linear ratios problems (SLRP) have broad wide applications in information technology, control science and engineering, transportation design, government plan, economy and finance [1–5], and so on. In these applications, especially, the number of linear ratios is usually less than four or five. In addition, due to the fact that the sum of linear ratios problems (SLRP) possess generally multiple local optimal solutions that are not globally optimal, the kinds of problems pose significant theoretical difficulty and computational complexity. Therefore, they have attracted interest of many researchers and practitioners.
In this paper, we will investigate the following sum of linear ratios problems: where , , , and are all affine functions, is a nonempty bounded constraint set, and for any , , .
During the past two decades, with the assumption that the denominator and the numerator for any , many optimization algorithms have been developed for globally solving the sum of linear ratios problems (SLRP), for example, simplex algorithms, image space algorithm, branch-and-bound algorithms, and monotonic algorithm (see [6–17]). Recently, Jiao et al. [18–20] presented three branch-and-bound algorithms for sum of linear ratios problem by constructing different linear relaxation technique; Jiao et al. [21] presented a new two-level relaxation algorithm for generalized linear multiplicative programming with generalized polynomial constraints, which can be also used to globally solve the sum of linear ratios problem (SLRP) investigated in this paper. In addition, several new algorithms [22–25] have been also proposed for solving linear sum-of-ratios and nonlinear sum-of-ratios problems.
In this paper, we shall present a new out space branch-and-bound algorithm for the sum of linear ratios problems (SLRP) using out space pruning technique. Firstly, we transform the initial problem into the equivalent nonconvex programming problem by introducing new variables. Next, the linear relaxation programming problem of the equivalent problem can be constructed by utilizing direct relaxation. Secondly, by making full use of the special structure of the equivalent problem and the branch-and-bound algorithm, an out space pruning technique is derived to improve the computational efficiency of the proposed algorithm, which offers a possibility of pruning a large part of the investigated out space region which does not contain the global optimal solution of the equivalent problem. Thirdly, by combining bounding technique and pruning technique, a new out space branch-and-bound algorithm is designed for globally solving the sum of linear ratios problems (SLR). Finally, numerical experimental results are presented to demonstrate both computational efficiency and solution quality of the proposed algorithm.
The remaining sections of this paper are organized as follows. At first, in Section 2, by introducing new variables the equivalent nonconvex programming problem of the initial problem is introduced. Next, by utilizing direct relaxation the linear relaxation programming problem of the equivalent problem is constructed. Second, an out space pruning technique is derived by making full use of the special structure of the equivalent problem and the branch-and-bound algorithm in Section 3. Third, a new out space branch-and-bound algorithm and its global convergence are described in Section 4. Fourth, numerical experiments and their computational results are presented in Section 5. Finally, the concluding remarks of this paper are drawn.
2. Equivalent Problem and Its Linear Relaxation
In this section, to globally solve the problem (SLRP), by the continuity of the linear ratio , we know that or . Here, without loss of generality, we can assume that except for the above assumptions, for each , since where is an appropriate real number such that . Thus, without loss of generality, by some proper transformation we can always guarantee that the numerator of each ratio is greater than or equal to .
Let And construct the initial box then the problem (SLRP) can be converted into the equivalent problem (EQ) as follows. The key equivalence theorem for the problem (SLRP) and the is described as follows.
Theorem 1. If is a global optimum point of the problem , then is a global optimum point of the problem (SLRP). On the contrary, if is a global optimum point of the problem (SLRP), then is a global optimum point of the problem with , , , .
Proof. The conclusion can be easily drawn, so the proof is omitted.
From Theorem 1, to solve the problem (SLRP), we may globally solve its equivalent nonconvex programming problem instead.
For each box , some notations and functions of this paper are listed as follows:
Theorem 2. For the functions , , , , where , we have the following two conclusions:(i).(ii).
Proof. The proof can be easily followed, therefore here it is omitted.
From the above Theorem 2, for any , the corresponding linear relaxation programming problem of the problem can be established as follows, which can offer a reliable upper bound for the global optimum of .
3. Out Space Pruning Technique
For any box , we need to recognize whether or not contains global optimum point of the problem (EQ). Thus, in this section, we shall construct an out space pruning technique for pruning a part of the investigated out space region which does not contain the global optimum point of the problem (SLRP) and use the technique to improve the computational efficiency of the proposed algorithm. The proposed new out space pruning technique aims at replacing the box with a smaller box without pruning any global optimum point of the problem (EQ).
We assume, without loss of generality, that is a currently known lower bound of the global optimum value of the and that denote the maximum value of the problem over and and set
Theorem 3. For any subbox , we have the following two conclusions:(i)Suppose that ; then there does not exist any global optimum point of over .(ii)Suppose that ; then, when , there does not exist any global optimum point of over ; and when , there does not exist any global optimum point of over , where
Proof. (i) Suppose that ; then we have therefore, there does not exist any global optimum point of the problem over .
(ii) Suppose that ; then we can get the following several conclusions.
When , for any and , since we can get Therefore, there does not exist any global optimum point of over
Similarly, when , and for any and , since we haveTherefore, there does not exist any global optimum point of over .
By the above Theorem 3, we can construct an out space pruning technique to prune a part of the investigated out space region which does not contain the global optimum point of the problem (EQ). Assume that a subboxwill be pruned; then from the Theorem 3, the investigated box can be renewed by a subbox
4. Algorithm and Its Convergence
In this section, by utilizing the above new pruning technique, we will present a new out space branch-and-bound algorithm for globally solving the problem (SLRP). In the algorithm, the branching operation is performed in out space . Assume that is or a subbox of it, which will be partitioned, the branching rule is selected as follows.
Set
Obviously, from [26] we can get that this branching rule is exhaustive; that is, if represents a nested subsequence of boxes (i.e., , for all ) formed by partitioning process, then there exists a unique point satisfying .
4.1. Out Space Branch-and-Bound Algorithm
Based upon the above linear relaxation bounding problem, the new out space pruning technique, and bisection method, an out space branch-and-bound algorithm is proposed for globally solving the (SLRP) as follows.
Step 1. Let , ,, , and , And let and construct the initial box Using simplex method to solve LRP, set its optimum solution and optimum value , respectively. Let , , . If , then and are global optimum solutions of and the (SLRP), respectively. Otherwise, let , and continue to the following Step 2.
Step 2. Set . For each investigated subbox , use the former out space pruning technique to prune the investigated subbox , let the remaining box be , and set , .
Step 3. Partition into two subboxes using the bisection technique. Let the new partitioned subboxes set be . Set . For each , solve the (LRP) using simplex method to get and , and set , . If , then let , ; otherwise, update , if necessary.
Step 4. Let the remaining partitioned set , and let the new upper bound Let and . If , then we get that and are global optimum points of the and the (SLRP), respectively. Otherwise, let and return to Step 2.
4.2. Convergence Analysis
In the following, we describe the global convergence of the above algorithm.
Theorem 4. The above algorithm either stops finitely to obtain the global optimum point of the (SLRP) or produces an infinite sequence whose accumulation point must be the global optimum point of the (SLRP).
Proof. If the proposed algorithm stops finitely at iteration, where , obviously, when the algorithm stops, we can get and by solving the LRP(), which are the feasible points of the (SLRP) and the (EQ), where and by terminating step of the algorithm, the updating operations of the lower bound and upper bound, and Theorems 1 and 2, we can easily follow that By combining the above all inequalities and equality, we can easily get that Therefore, if the algorithm stops finitely at iteration, then is the global –optimal solution of the (SLRP).
If the above algorithm generates an infinite sequence of incumbent solutions by solving the LRP, let then we can get an infinite sequence of incumbent solutions for . By the continuity character of the function and and , and the exhaustiveness of the bisection rule, we can get the following conclusions, for each :for each ,Therefore, is a feasible point of the (), also since is a decreasing real number sequence bounded by , we can follow that Thus, from the updating method of the lower bound and the continuity character of the , we can get that Hence, is the global optimum point of the (SLRP); the conclusion is proved.
4.3. Numerical Example
In this subsection, some well-known test problems are implemented on Intel(R) Core(TM)2 Duo CPU microcomputer to verify the performance of the proposed out space branch-and-bound algorithm. The proposed out space branch-and-bound algorithm is coded in C++ program, and the simplex method is used to solve each linear relaxation programming problem. Numerical problems and their computational results are summarized in Table 1. Some notations have been also used for column headers in Table 1 as follows. Ave. Iter. represents the average number of iterations of the algorithm; Ave. L. represents the average number of the necessary maximum nodes of the algorithm; Ave. Time (s) stands for the execution time of algorithm in seconds.
where , , , are randomly generated in ; are generated in ; , . In the investigated problem, denotes the number of the constraints, denotes the dimension of the problem, and denotes the number of ratios.
From Table 1, compared with the known algorithms, numerical results indicate that the proposed algorithm can be used to globally solve the problems (SLRP) with the higher computational efficiency.
5. Concluding Remarks
In this paper, we present an effective algorithm for globally solving the sum of linear ratios problem (SLRP). By introducing new variables, we transform the initial sum of linear ratios problems (SLRP) into the equivalent nonconvex programming problem. Next, the linear relaxation programming problem of the equivalent problem is established by utilizing direct relaxation. To improve the computational efficiency of the algorithm, an out space pruning technique is derived, which offers a possibility of pruning a large part of the investigated out space region which does not contain the optimal solution of the equivalent problem. Based on out space partition, by combining bounding technique and reduction technique, a new outcome space branch-and-bound algorithm for globally solving the sum of linear ratios problem (SLRP) is constructed. Finally, numerical problems and their computational results are presented to demonstrate both computational efficiency and solution quality of the proposed algorithm.
In future research, we can extend the proposed algorithm and reduction technique to solve nonlinear fractional problems, like concave-convex sum-of-ratios problem, convex-convex sum-of-ratios problem, linear fractional multiplicative problem, generalized linear fractional programming problem, and so on.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper is supported by the Cultivation Plan of Young Key Teachers in Colleges and Universities of Henan Province (2016GGJS-107), the Basic and Advanced Technology Research Project of Henan Province (152300410097), the Key Scientific Research Project of Universities in Henan Province (17B413002, 16A110014, and 17A110021), the Major Scientific Research Projects of Henan Institute of Science and Technology (2015ZD07), the National Natural Science Foundation of Henan Province (152300410097), and Science and Technology Project of Henan Province (162102210294).