Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 6174352, 8 pages
https://doi.org/10.1155/2020/6174352

Research Article

A Deterministic Method for Solving the Sum of Linear Ratios Problem

Zhenping Wang and Yonghong Zhang

Department of Mathematics, Henan Normal University, Xinxiang 453007, China

Correspondence should be addressed to Yonghong Zhang; zhangyonghong@htu.edu.cn

Received 6 July 2020; Revised 13 August 2020; Accepted 25 August 2020; Published 28 September 2020

Academic Editor: Paolo Manfredi

Copyright © 2020 Zhenping Wang and Yonghong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Since the sum of linear ratios problem (SLRP) has many applications in real life, for globally solving it, an efficient branch and bound algorithm is presented in this paper. By utilizing the characteristic of the problem (SLRP), we propose a convex separation technique and a two-part linearization technique, which can be used to generate a sequence of linear programming relaxation of the initial nonconvex programming problem. For improving the convergence speed of this algorithm, a deleting rule is presented. The convergence of this algorithm is established, and some experiments are reported to show the feasibility and efficiency of the proposed algorithm.

1. Introduction

This paper considers the sum of linear ratios problem (SLRP) with the following form:

where are the finite functions such that for all , and , and are the real constant coefficients, .

As we know, fractional programming is an important branch of nonlinear optimization. As a special case of fractional programming, the sum of linear ratios problem has been widely concerned. The first reason is that SLRP has many important applications in the real world, such as certain government contracting problems [1], cluster analysis [2], and multiobjective bond portfolio [3]. The second reason is the objective function is neither quasiconvex nor quasiconcave, so it usually has multiple local optima that are not global optima, and it is a very challenging issue to find its global optima. In addition, as pointed by Matsui [4], a special case of SLRP was proved to be NP hard (Theorem 1, [4]), so SLRP is an NP-hard problem.

Up to now, various algorithms have been proposed for solving SLRP. For instance, (i) when , by using the parametric simplex method, an algorithm was developed [5]; (ii) when , through searching iteratively the nonconvex outcome space until a global optimal solution is found, some algorithms were proposed [6, 7]. In addition, to solve special cases of SLRP, based on branch and bound, several other algorithms were presented [2, 3, 8–14].

Although many algorithms have been proposed to find the global optimal solution of fractional programming, as far as we know, the research results of SLRP considered in this paper are still relatively little. Most of these algorithms proposed are intended only for the sum of linear ratios problem without coefficients .

The aim of the present paper is to present a new global optimization algorithm for solving SLRP. To globally solve SLRP, a convex separation technique and a two-part relaxation technique are designed to convert SLRP into a series of linear programming problems. Through a successive refinement process, the solutions of these linear programming problems can be as close as possible to the global optimum of SLRP. The main features of this algorithm are as follows: (1) compared with these methods reviewed above ([3, 6–9, 11, 15], for example), the method given in this paper can solve the general problem SLRP; (2) compared with the method in [2], this method need not introduce new variables; (3) by using a convex separation technique and a two-part relaxation technique, the initial problem SLRP can be converted into a series of linear programming problems, which is more convenient in the computation than the parametric programming (or concave minimization) methods [8]; (4) numerical results show that the proposed method can solve all of the test problems in finding globally optimal solutions with given precision.

The organizational structure of this paper is as follows. To obtain the relaxed linear programming of SLRP, Section 2 introduces a convex separation technique and a two-part relaxation technique. For improving the convergence speed of the proposed algorithm, a deleting rule is designed. In Section 3, the proposed branch and bound algorithm is described based on the relaxed subproblems, and the convergence of the algorithm is proved. Numerical results are given in Section 4, and some concluding remarks are provided in Section 5.

2. Linear Relaxation of SLRP

For solving the problem SLRP, one of the key parts is to construct lower bounds for SLRP and its partitioned subproblems. Towards this end, we propose a novel strategy to generate lower bound by underestimating with a linear function. The detailed construction process is given below.

Let be either the initial box of the problem SLRP or the partitioned box in a branch and bound scheme. For convenience in expression, for , let

For generating the linear relaxation of , a convex separation technique and a two-part relaxation technique are designed.

2.1. First-Part Relaxation

For each term , since and for all , we can denote

The lower bound and upper bound of can be computed as follows:

By introducing the variable , an equivalent form of can be derived, which is expressed in the following form:

For , it is not difficult to calculate its gradient and Hessian matrix:

Therefore, the following relation holds:

Let ; then, , we have

Thus, the function

is convex on . Based on the above results, we can decompose into the difference between two convex as follows:

where .

Let , and since is a convex function, we have

In addition, , it is not difficult to show

Furthermore, we have

Since , it follows that

Thus, from (10), (11), and (14), we obtain

From (3), it follows that . Hence, the first-part relaxation of about can be obtained, i.e.,

2.2. Second-Part Relaxation

For over the interval , we can derive its linear lower bound and linear upper bound as follows:

where . Therefore, we have

where .

In (16), let . Then, we can obtain the linear lower bound function of as follows:

where

Obviously, .

Consequently, the corresponding approximation relaxation linear programming (RLP) of SLRP in is given as follows:

Theorem 1. Let . Then, for any , we have

Proof. , let and let . Obviously, we only need to prove as .

First, by the definition of , we have

where is a constant vector and satisfies .

Furthermore, according to the definition of , we know that as . Thus, we have as .

Second, consider the difference . By (16), we have

From the definitions of , and , we know that as . Consequently, from [16] (Theorem 1), we have as .

Taken together above, we have

By Theorem 1, it can be seen that will approximate the objective function as .

From the above discussion, it can be seen that the objective value of RLP is smaller than or equal to that of LFP for all feasible points; thus, RLP provides a valid lower bound for the solution of LFP. Thus, for any problem (P), let us denote the optimal value of P by , and then we have

For improving the convergence speed of the proposed algorithm, a deleting technique is proposed, which can be used to eliminate the region that does not contain optimal solutions of SLRP.

Let be the current known upper bound of the optimal objective value of the problem SLRP. Denoting

The following theorem gives the deleting technique.

Theorem 2. For any subrectangle with , let

If there exists such that and , then there is no global optimal solution of SLRP over ; if such that and , then there is no global optimal solution of SLRP over , where

Proof. We first show that , . Consider the -th component of , and it obviously follows that

Note that , and then from the definition of and the above inequality, we have

, since , there does not exist global optimal solution of SLRP over .
Similarly, , if and with some , we can derive that there is no global optimal solution of SLRP over .

3. Algorithm and Its Convergence

Based on the former linear relaxation programming (RLP), this section presents a branch and bound algorithm to solve the problem SLRP. In this method, a sequence of RLP problems over partitioned subsets of needs to be solved.

As the algorithm goes on, the set will be partitioned into some subrectangles. Each subrectangle is associated with a node of the branch and bound tree and a relaxation linear subproblem.

At the -th iteration of the algorithm, let be a collection of active nodes denoted, i.e., each node in is associated with a rectangle . , a lower bound of the optimal value of SLRP needs to be computed. The lower bound of the optimal value of SLRP is computed on the whole initial rectangle at the -th iteration by . Let be the active node with and is partitioned into two subrectangles as described below. Meanwhile, the upper bound for each new node is computed as before. If necessary, the upper bound is updated. After deleting all nodes that cannot be improved, a collection of active nodes for the next stage can be obtained. The above process is repeated until the termination conditions are met.

3.1. Branching Rule

In the proposed algorithm, a simple and standard bisection rule is used. Consider any node subproblem identified by rectangle . The branching rule is briefly introduced as follows:

(a)Let

(b)Let

(c)Let

According to this branching rule, the rectangle is partitioned into two subrectangles and .

Since this branching will derive the intervals to zero along any infinite branch of the branch and bound tree, it is sufficient to ensure convergence. This is very important to ensure the convergence of the algorithm.

3.2. Algorithm Statement

Let be the optimal objective function value of the problem (RLP) over the rectangle . Based on the results above, we give the basic steps of the proposed global optimization algorithm.

 Step 1. Choosing . An optimal solution and the optimal value are found for the problem (RLP) with , and set

 If , then stop. is a global -optimal solution for the problem SLRP. Otherwise, set

 Step 2. Setting . is subdivided into two rectangles via the branching rule. Let . Step 3. For each node , the lower bound for each linear constraint function is computed over the present considered rectangle, i.e., computing lower bound:

 where and denote the lower bound and the upper bound of the present considered rectangle, respectively. If there exists such that

 then we will put the corresponding node into . If are both put into , i.e.,

 then go to Step 7. Step 4. For the undeleted subrectangle and/or , parameters are updated. is computed, and an optimal solution is found for the problem (RLP) with , where . If necessary, is updated as follows:

 and let denote the point which satisfies . Step 5. If , then set

 Step 6. Setting

 Step 7. Setting

 Step 8. Setting , and let satisfy . If , then stop. is a global -optimal solution for the problem SLRP. Otherwise, set and go to Step 2.

3.3. Convergence of the Algorithm

The following theorem gives some convergence properties of the algorithm.

Theorem 3.

(i)If the algorithm is finite, then upon termination, is a global -optimal solution of SLRP.(ii)If the algorithm is infinite, then any accumulation point of the sequence , which is generated along any infinite branch of the branch and bound tree, will be the global solution of SLRP.

Proof.

(i)If the algorithm is finite, then it will terminate at some stage , . Upon termination, by the algorithm, we have

 According to Steps 1 and 4, it follows that

 Let be the optimal value of the problem SLRP. By Section 2, it can be known that

 Since is a feasible solution of SLRP, we have

 Combined with the above discussion, this implies that

 Thus, we have

 and the proof of part (i) is complete.(ii)When the algorithm is infinite, by the algorithm, we know that is a nondecreasing sequence and bounded above by . So, we have . Since is a compact set, there must be one convergent subsequence . Suppose that . By the proposed algorithm, there exists a decreasing subsequence where with , and . From Theorem 1, we have

By the definitions of RLP and SLRP, is a feasible solution of SLRP obviously. Combining (50), it follows that is a global solution of SLRP, and the proof of part (ii) is complete.

4. Numerical Experiment

To test the performance of the proposed global optimization algorithm, it is compared with some other algorithms based on five test problems. The convergence tolerance is set to  = 1.0e − 2 in our experiment. The results are summarized in Table 1.

Table 1: Computational results of test problems.

Example
Methods
Optimal solution
Optimal value
Iter
Time

1
[13]
(0.0, 3.3333, 0.0)
1.9
8
0

(0.0, 3.3333, 0.0)
1.9
1
0.015

2
[17]
(1.0, 0.0, 0.0)
−4.081481
38
0.02

(1.0, 0.0, 0.0)
−4.0815
5
0.25

3
[5]
(0.0, 3.3333, 0.0)
−3.002923975
119

[16]
(0.0, 3.3333, 0.0)
−3.002923972
15
0

(0.0, 3.3333, 0.0)
−3.0029
14
0.812

4
[13]
(3.0, 4.0)
−3.29167
9
0

(3.0, 4.0)
−3.2917
7
0.25

5
[5]
(0.0, 0.283935547)
1.623183358
71

(0.0, 0.2812)
1.6232
45
2.468

Notations have been used for column headers: Iter: number of algorithm iteration; time: execution time in seconds.

Example 1 (see ref [13]).

Example 2 (see ref [17]).

Example 3 (see ref [5, 16]).

Example 4 (see ref [13]).

Example 5 (see ref [5]).

From Table 1, it is seen that our algorithm can globally solve the problem SLRP successfully.

5. Concluding Remarks

In this paper, for solving the problem SLRP, we present a branch and bound algorithm. In this algorithm, we utilize a convex separation technique and a two-part relaxation technique to obtain a sequence of linear programming relaxation of the initial nonconvex programming problem SLRP, which is embedded in a branch and bound frame. Furthermore, for improving the convergence speed of the proposed algorithm, a deleting rule is designed. Numerical results show that it can solve the problem SLRP effectively.

Data Availability

No data were used to support this study.

Conflicts of Interest

All authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the National Natural Science Foundation NSFC (11671122); the Key Project of Henan Educational Committee (19A110021).

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