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Regional division and reduction algorithm for minimizing the sum of linear fractional functions
- Pei-Ping Shen^{1}Email authorView ORCID ID profile and
- Ting Lu^{1}
https://doi.org/10.1186/s13660-018-1651-9
© The Author(s) 2018
Received: 30 January 2018
Accepted: 7 March 2018
Published: 21 March 2018
Abstract
This paper presents a practicable regional division and cut algorithm for minimizing the sum of linear fractional functions over a polyhedron. In the algorithm, by using an equivalent problem (P) of the original problem, the proposed division operation generalizes the usual standard bisection, and the deleting and reduction operations can cut away a large part of the current investigated region in which the global optimal solution of (P) does not exist. The main computation involves solving a sequence of univariate equations with strict monotonicity. The proposed algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of univariate equations. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.
Keywords
1 Introduction
Problem (FP) is a well-known class among fractional programming problems. Theoretically, it is NP-hard [1, 2]. The primary challenges in solving problem (FP) arise from a lack of useful properties (convexity or otherwise) and from the number of ratios. In general, problem (FP) possesses more local optimal solutions that are not globally optimal [3], and so problem (FP) owns major theoretical and computational difficulties. From an application point view, this problem has a large deal of applications; for instance, traffic and economic domain [4, 5], multistage stochastic shipping problems [6], data envelopment analysis [7], and queueing-location problems [8]. The reader is referred to a survey to find many other applications [4, 5, 9–11].
Many algorithms have been proposed to solve problem (FP) with a limited number of ratios [4, 12–19]. For instance, Wang and Shen [4] give an efficient branch and bound algorithm by using a transformation technique and the linear relaxation programming of the objective function. By applying Lagrangian duality theory, Benson [9] presents a simplicial branch and bound duality-bounds algorithm. Carlsson and Shi [10] propose a linear relaxation algorithm with lower dimension which is performed on a 2N-dimensional domain instead of the original n-dimensional one, the computational time is long with larger N. Ji and Zhang [16] consider a deterministic global optimization algorithm by utilizing a transformation technique and a linearizing method. Jiao et al. [17, 18] present the branch and bound algorithms for globally solving sum of linear and generalized polynomial ratios problems, by solving a sequence of linear relaxation programming problems. In short, most of them (see [4, 9, 16–18] for example) are branch and bound algorithms. The key idea behind such algorithms mentioned above is that the branch and bound operator is performed on an N-dimensional region (or the correspondence relating to N and n) rather than the native n-dimensional feasible set, that is, they all work on a space whose dimension increases with the number N of ratios.
In this article, a new division and reduction algorithm is proposed for globally solving problem (FP). To solve problem (FP), an equivalent optimization problem (P), whose objective function is just a simple univariate, is first presented by exploiting the feature of this problem. Then, in order to design a more efficient algorithm, several basic operations: division, deleting, and reduction, are incorporated into a similar branch and bound framework by utilizing the particular structure of problem (P). Compared with the usual branch and bound (BB) methods (e.g., [4, 5, 9, 10, 16]) mentioned above, the goal of this research is three fold. First, the proposed bounding operation is simple since the lower bound of the subproblem of each node can be achieved easily only by arithmetic computations, distinguishing it from the ones obtained by solving convex/linear programs in the usual BB methods. Second, the reduction operation that does not appear in other BB methods is used to tighten the range of each variable, such that the growth of the branch tree can be suppressed. Moreover, the main computational cost of the algorithm is to implement the reduction operation which solves univariate equations with strict monotonicity. Third, the problem in this paper is more general than the others considered in [10, 11], since we only request \(d_{i}^{\top }y+d_{0i}\neq 0\) for each i. Further, the proposed adaptive division operation both generalizes and is superior to the usual standard bisection in BB methods according to the numerical computational result in Sect. 5. Also, the computational results of the problem with the large number of ratio terms can be obtained to illustrate the feasibility and validity of the proposed algorithm.
This paper is summarized as follows. In Sect. 2, by using a conversion strategies, the original problem (FP) is transformed into an equivalent optimization problem (P). The adaptive division, deleting, and reduction operations are shown in Sect. 3. In Sect. 4 we give the proposed algorithm and its convergence. Some numerical results demonstrate the feasibility and availability of the algorithm in Sect. 5.
2 Equivalent problem
Proposition 2.1
\(y^{*}\in R^{n}\) is a globally optimal solution for problem (FP) if and only if \((y_{0}^{\ast }, {y}^{*},w ^{*})\in R^{n+N+1}\) with \(y_{0}^{\ast }\in R\) and \(w^{*}\in R^{N}\) is a globally optimal solution for problem (FP2), where \(y_{0}^{\ast }= \sum_{i=1}^{N} w_{i}^{*}(c_{i}^{\top }y^{*}+c_{0i})\) and \(w_{i}^{*}=\frac{1}{d_{i}({y}^{*})+d_{0i}}\) for each \(i=1,\ldots ,N\).
Proof
The proof of this result is obvious. □
From Proposition 2.1, notice that, in order to globally solve problem (FP), we may globally solve problem (FP2) instead. Moreover, it is easy to see that the global optimal values of problems (FP) and (FP2) are equal.
Note that both problem (FP) and problem (P) are equivalent according to Proposition 2.1, hence for globally solving problem (FP), the algorithm to be presented concentrates on how to solve problem (P).
3 Essential operations
For solving problem (P), we first give the following concept about an approximate optimal solution.
Definition 3.1
Remark 3.1
All feasible solutions to problem (P) are ε-feasible. When \(\eta =0\), an \((\varepsilon ,\eta)\)-optimal solution is optimal over all ε-feasible solutions of problem (P). When \(\eta >0\), an \((\varepsilon ,\eta)\)-optimal solution is η-optimal for all ε-feasible solutions to problem (P).
For seeking an \((\varepsilon ,\eta)\)-optimal solution of problem (P), a division and cut algorithm to be developed includes three essential operations: division operation, deleting operation, and reduction operation.
First, the division operation consists in a sequential box division of the original box \(D^{0}\) following in an exhaustive subsection principle, such that any infinite nested sequence of division sets generated through the algorithm reduces to a singleton. This paper takes an adaptive division operation, which extends the standard bisection in the normally used exhaustive subsection principle. Second, by using overestimation of the constraints, the deleting operation consists in eliminating each subbox D generated by the division operation, in which there is no feasible solution. In addition, the reduction operation is used to reduce the size of the current partition set (referred to a node), aiming at tightening each subbox which contains the feasible portion currently still of interest.
At a given stage of the proposed algorithm for problem (P), let V represent the best current objective function value to problem (P). Next, we will show these detailed operations.
3.1 Deleting operation
Theorem 3.1
- (i)
A feasible solution to \(\mathrm{Q}(D)\) satisfying \(\bar{F}(\hat{x})< \varepsilon \) is an ε-feasible solution of \(\mathrm{P}(D)\) with \(\hat{x}_{0}\leq V-\eta \).
- (ii)
If \(V(\mathrm{Q}(D^{0}))>0\), consider the following two cases: (a) problem P(\(D^{0}\)) has no feasible solution if \(V=x_{0}^{u}+\eta \), and (b) an ε-feasible solution \(\tilde{x}=(\tilde{x}_{0},\tilde{x} _{1},\ldots ,\tilde{x}_{n+N})\) of \(\mathrm{Q}(D^{0})\) is an \((\varepsilon , \eta)\)-optimal solution of \(\mathrm{P}(D^{0})\) if \(V=\tilde{x}_{0}\).
Proof
(i) This result is obvious, and here it is omitted.
- (a)
If \(V=x^{u}_{0}+\eta \), by (3.2) we get \(\{x\mid \bar{F}(x) \le 0, x\in D^{0}\}=\emptyset \), which implies that problem \(\mathrm{P}(D)\) has no feasible solution.
- (b)If \(V=\tilde{x}_{0}\), from (3.2) it is easy to see thatfor any \(x=(x_{0},x_{1},\ldots ,x_{n+N})\in \{x\mid \bar{F}(x)\leq 0 < \varepsilon , x\in D^{0}\}\), which means that$$ x_{0}>V-\eta =\tilde{x}_{0}-\eta , $$$$ \min \bigl\{ x_{0}\mid G_{k}(x)< \varepsilon , k=0,1, \ldots ,m+N, x \in D^{0} \bigr\} \geq \tilde{x}_{0}-\eta . $$
Theorem 3.1 illustrates that by utilizing problem \(\mathrm{Q}(D)\) one can know whether or not there exists a feasible solution \(\hat{x}=(\hat{x}_{0}, \hat{x}_{1},\ldots ,\hat{x}_{n+N})\) of \(\mathrm{P}(D)\) with improving the current objective function value V, i.e., \(\hat{x}_{0}\leq V-\eta \). Further, if \(V(\mathrm{Q}(D^{0}))\ge \varepsilon >0\), then an \((\varepsilon ,\eta)\)-optimal solution of \(\mathrm{P}(D^{0})\) can be obtained, or it can be confirmed that problem \(\mathrm{P}(D^{0})\) has no feasible solution.
Theorem 3.2
Proof
(i) If \(\alpha =0 \), this result is obvious.
3.2 Adaptive division
The division operation repeatedly subdivides an \((n+N+1)\)-dimensional box \(D^{0}\) into \((n+N+1)\)-dimensional subboxes. This operation helps the algorithm confirm the position of a global optimal solution in \(D^{0}\) for problem (P). Throughout this algorithm, we take a new adaptive subdivision principle as follows.
Adaptive subdivision
For given \(\eta >0\), consider any box \(D=[p,q]=\{x\in R^{n+N+1}|p_{i} \le x_{i}\le q_{i},i=0,1,\ldots ,n+N\}\subseteq D^{0}\).
(i) If \(q_{0}>V-\eta \), then let \(\alpha =\frac{q_{0}-V+\eta }{q_{0}-p _{0}}\); otherwise let \(\alpha =0\).
(ii) Denote \(t=\mathrm{{argmax}}\{q_{i}-p_{i}\mid i=0,1,\dots ,n+N\}\). Let \(u_{t}=p_{t}\) and \(v_{t}=q_{t}-\alpha (q_{t}-p_{t})\). Set \(\bar{x} _{t}=(u_{t}+v_{t})/2\).
Based on the above division operation, D is divided into two new boxes \(D_{1}\) and \(D_{2}\). Especially, when \(\alpha =0\), the adaptive subdivision simply reduces to the standard bisection. As we will see from numerical experiments in Sect. 5, the adaptive subdivision is superior to the standard bisection. Moreover, the subdivision can confirm the convergence of the algorithm, and we have the following results.
Theorem 3.3
Suppose that the above adaptive division operation is infinite, then it generates a nested sequence \(\{D^{s _{t}}\}\) of partition sets \(\{D^{s}\}\) generated by the adaptive division operation, so that \(\operatorname{LB}(D^{s_{t}})\rightarrow V(\mathrm{Q}(D^{0}))\) as \(t\rightarrow +\infty \).
Proof
3.3 Reduction operation
For any \(D=[p,q]=\prod_{i=0}^{n+N}[p_{i},q_{i}]\subseteq D^{0}\) generated by the division operation, the box \(D'\) satisfying condition (3.4) is denoted by \(R[p,q]\). To recognize how \(R[p,q]\) is acquired, we first demand to know the parameters γ, \({\alpha }_{k}^{i}\), and \({\beta }_{k}^{i}\) computed for each \(k=0,1,\dots ,m+N\), \(i=0,1, \dots ,n+N\) by utilizing the following rules.
Rule (i): Given the box \([p,q]\subseteq D^{0}\), if \(f_{k}^{i}(1) >\varepsilon \), let \({\alpha }_{k}^{i}\) be the solution to the equation \(f_{k}^{i}({\alpha }_{k}^{i})=\varepsilon \) about the univariate \(\alpha_{k}^{i}\); otherwise let \({\alpha }_{k}^{i}=1\).
Rule (ii): For given boxes \(D=[p,q]\) and \(D'=[\bar{p},q]\) with \(D'\subseteq D\subseteq D^{0}\), if \(g_{k}^{i}(1) >\varepsilon \), one can solve the univariate equation \(g_{k}^{i}({\beta }_{k}^{i})= \varepsilon \) to obtain \({\beta }_{k}^{i}\); otherwise let \({\beta } _{k}^{i}=1\). If \(\bar{p}_{0}< V-\eta < q_{0}\), then set \(\gamma =\frac{V- \eta -\bar{p}_{0}}{q_{0}-\bar{p}_{0}}\); otherwise let \(\gamma =1\).
Notice that it is easy to get \({\alpha }_{k}^{i}\) and \(\beta_{k}^{i}\), since the univariate functions \(f_{k}^{i}(\lambda)\) and \(g_{k}^{i}( \mu)\) are strictly monotonic in Rules (i) and (ii).
Theorem 3.4
- (i)
If \(p_{0}\leq V-\eta \) and \(\bar{F}(p)<\varepsilon \), then \(R[p,q]=[p,p]\), and
- (ii)
If \(p_{0}>V-\eta \) or \(\max \{f_{k}^{i}(0) | k=0,1,\dots ,m+N \}> \varepsilon \) holds for some \(i\in \{0,1,\ldots ,n+N\}\), then \(R[p,q]=\emptyset \).
Proof
(i) The proof of this result is easy, here it is omitted.
(ii) The former of the conclusion is apparent, we only need to give the proof of the latter.
Theorem 3.5
Proof
For any given \(x=(x_{0},\ldots ,x_{n+N})^{T}\in [p,q]\), we first confirm that \(x\geq \hat{p}\).
If \(\hat{\alpha }^{i}=1\), we can obtain \(x_{i}<\hat{p}_{i}=q_{i}- \hat{\alpha }^{i}(q_{i}-p_{i})=p_{i}\) from (3.6), contradicting \(x\in [p,q]\), then \(x\geq \hat{p}\).
If \(\hat{\beta }^{i}=1\), from (3.8) we can acquire \(x_{i}>\hat{q} _{i}=\hat{p}_{i}+\hat{\alpha }^{i}(q_{i}-\hat{p}_{i})=q_{i}\), contradicting \(x\in [p,q]\).
From Theorem 3.5, by Rules (i) and (ii) the main computational effort for deriving \(R[p,q]\) is to solve some univariate equations about the variables \({\hat{\alpha }}_{k}^{i}\) and \(\hat{\beta }_{k}^{i}\), which is easy to solve, for example, by using the bisection approach. What is more, as seen below, the cost of main computation in the proposed algorithm is also to form \(R[p,q]\).
4 Algorithm and its convergence
According to the above discussions, the proposed algorithm is shown as follows.
Algorithm statement
Step (0) Given tolerances \(\varepsilon ,\eta >0\), if there is no known feasible solution at present, set \(V=x^{u}_{0}\) with \(D^{0}=[x^{l},x^{u}]\); otherwise let \(x^{\ast }\) be the best feasible solution of problem (P), and set \(V=x_{0}^{\ast }\). Let \(M_{0}=\{D ^{0}\}\), \(N_{0}=\emptyset , t=0\).
- (a)
If \(R[p,q]=\emptyset \), eliminate D.
- (b)
If \(R[p,q]=[p,p]\), discard D and update \(x^{\ast }=p\), \(V=p_{0}\).
- (c)
If \(R[p,q]\neq \emptyset \), set \(D=R[p,q]\) and compute the lower bound \(\operatorname{LB}(D)\) by Theorem 3.5. If \(\operatorname{LB}(D)>\varepsilon \), delete D.
- (a)
If \(\bar{N_{t}}=\emptyset \), stop: (i) if \(V=x^{u}_{0}\), problem (P) is infeasible; (ii) if \(V=x_{0}^{\ast }\), \(x^{\ast }\) is an \((\varepsilon ,\eta)\)-optimal solution of problem (P).
- (b)
If \(\bar{N_{t}} \neq \emptyset \), Theorem 3.4 is applied to each \(D\in \bar{N_{t}}\), then eliminate D and update \(x^{\ast }=p\), \(V=p_{0}\) if necessary.
Step (3) Denote \(\bar{N_{t}}\) to be the collection of boxes after accomplishment of Step (2). Choose \(D^{t}=[p^{t},q^{t}] \in {\mathrm{{argmin}}}\{\operatorname{LB}(D)\mid D\in \bar{N_{t}}\}\), and denote \(\operatorname{LB}_{t}=\operatorname{LB}(\bar{N _{t}})\). If \(\operatorname{LB}_{t}>0\), then terminate: the conclusion is the same as situation (a) of Step (2); otherwise go to Step (4).
Step (4) If \(q_{0}^{t}>V-\eta \), let \(s^{t}=p^{t}+\alpha_{t}(q ^{t}-p^{t})\) with \(\alpha_{t}=\frac{q_{0}-V+\eta }{q_{0}-p_{0}}\); otherwise set \(s^{t}=p^{t}\) or \(s^{t}=(p^{t}+q^{t})/2\). If \(\bar{F}(s ^{t})<\varepsilon \), update \(x^{\ast }=s^{t}\), \(V=s_{0}^{t}\).
Step (5) Divide \(D^{t}\) into two subboxes by the adaptive division, and set \(M_{t+1}\) to be the collection of these two subboxes of \(D^{t}\). Let \(N_{t+1}=\bar{N_{t}}\backslash \{D^{t}\}\). Set \(t=t+1\), and return to Step (1).
Remark
By utilizing a local solver in Step (4) of the proposed algorithm, instead of evaluating one point, we may acquire a point with the better objective function value V, and the iteration count of the algorithm may be reduced. However, because the computational cost increases rapidly with the quality of the objective value V, the running time is not always decreasing. So a trade-off must be made, practically just one evaluating point as in the above algorithm is used. Moreover, to implement the algorithm, all that is required is the ability to solve univariate equations with monotonicity and to execute simple algebraic steps.
The convergence of the algorithm is shown by the following theorem.
Theorem 4.1
For given tolerances \(\varepsilon ,\eta >0\), the above algorithm always terminates after finitely many iterations, obtaining an \((\varepsilon ,\eta)\)-optimal solution of problem (P), or a demonstration that the problem is infeasible.
Proof
5 Numerical experiments
Ex. | Ref. | Solution | ε | η | Optimum | Iter. | \(L_{\max}\) | Time(s) |
---|---|---|---|---|---|---|---|---|
1 | [ours] | (0.0000, 1.66666667, 0.0000) | 10^{−3} | 10^{−6} | 3.710919 | 8 | 4 | 0.1830 |
[4] | (0.0000, 0.625, 1.875) | 10^{−4} | 4.0000 | 58 | 18 | 2.968694 | ||
[17] | (1.1111, 1.36577e−05, 1.35168e−05) | 10^{−9} | ||||||
2 | [ours] | (5.0000, 0.0000,0.0000) | 10^{−3} | 10^{−6} | 2.861905 | 16 | 8 | 0.1250 |
[4] | (0, 3.3333, 0) | 10^{−4} | 3.0029 | 80 | 64 | 8.566259 | ||
3 | [ours] | (1.5000, 1.5000) | 10^{−3} | 10^{−6} | 4.9125874 | 56 | 14 | 1.0870 |
[4] | (3.0000, 4.0000) | 10^{−4} | 5 | 32 | 32 | 1.089285 | ||
[17] | (3.0000, 4.0000) | 10^{−6} | ||||||
4 | [ours] | (1.1111, 0.0000, 0.0000) | 10^{−2} | 10^{−5} | −4.090703 | 185 | 55 | 3.2510 |
[16] | (1.0715, 0, 0) | 10^{−6} | −4.087412 | 17 | ||||
5 | [ours] | (0.0000, 0.3333, 0.0000) | 10^{−3} | 10^{−6} | −3.0029 | 17 | 3 | 0.1290 |
[16] | (0, 0.33329, 0) | 10^{−6} | −3.000042 | 30 | ||||
6 | [ours] | (1.0000, 0.0000) | 10^{−3} | 10^{−6} | 1.428571 | 6 | 2 | 0.0470 |
[16] | (1.0000, 0.0000) | 10^{−6} | 1.428571 | 10 |
Average performances of adaptive division and bisection when \(n=3\) and \(\varepsilon =0.05\)
N | 600 | 800 | 1000 | 1200 | 1500 | |
---|---|---|---|---|---|---|
Adaptive division | CPU time (s) | 305.709 | 479.106 | 563.286 | 754.851 | 1009.027 |
Bisection | CPU time (s) | 386.527 | 573.421 | 769.683 | 971.858 | 1227.389 |
The notation in Table 1 is as follows:
• Ref.: reference;
• Iter.: the number of the algorithm iterations;
• Time (s): CPU seconds required for solving a problem;
• \(L_{\max}\): the maximal number of the active node necessary.
Example 1
Example 2
(see [4])
Example 3
Example 4
(see [16])
Example 5
(see [16])
Example 6
(see [16])
From Table 1, we can obtain that solving all of the examples by the proposed algorithm yields the \((\varepsilon ,\eta)\)-optimal solutions with much better objective function values and being feasible. In addition, for Example 2, it is observed that the computational solution \(x^{*}=(0,3.3333,0)\) of Ref. [4] does not satisfy the constraint \(x_{1}+6x_{2}+2x_{3} \leq 10\), i.e., \(x^{*}\) is infeasible.
For the adaptive division and bisection the computational time of solving examples with larger N ranging from 600 to 1500 are listed in Table 2. We can see that even if it takes about seventeen minutes to solve problem (P) of size \((n,N)=(3,1500)\), the adaptive division takes less time required by bisection for each N, which confirms the feasibility and availability of the proposed algorithm. We can draw a conclusion that the algorithm has more than enough performance, at least for three dimensions.
Based on the above results, it is easy to see that the adaptive division has many advantages over the usual bisection in the computation. Additionally, when n is not larger than 3, the algorithm can rapidly solve problems, even if N takes on values as high as 1500. On the other hand, when the number of variables is as high as 100, with increasing n, we need to solve more equations in reduction operation, therefore, it is reasonable that the computational time is larger. Moreover, the lower bound is acquired only by simple arithmetic computations, that is, by executing simple algebraic steps, which is different from the ones used in solving convex or linear programs in common branch and bound methods. Finally, the computation for obtaining \({\hat{\alpha }}_{k}^{i}\) and \(\hat{\beta }_{k}^{i}\) is easy by solving the equations with univariate and monotonicity in reduction operation, which is also the main computational cost of the algorithm.
6 Results and discussion
In this article, a new division and reduction algorithm is proposed for globally solving problem (FP). First, the original problem (FP) is converted into an equivalent optimization problem (P), in which the objective function is a single variable and the constraint functions are the difference of two increasing functions. Second, several basic operations (i.e., division, deleting, and reduction) are presented for designing a more efficient algorithm to problem (P). Finally, the numerical computational results show the feasibility and efficiency of the proposed basic operations, compared with the usual branch and bound (BB) methods (e.g., [4, 5, 9, 10, 16, 17]). Additionally, as further work, we think the ideas in this article can be extended to the sum of nonlinear ratios optimization problems; for example, the numerator and denominator of each ratio in the objective function to problem (FP) are replaced with a generalized polynomial function, respectively.
Declarations
Acknowledgements
The authors are grateful to the responsible editor and the anonymous referees for their valuable comments and suggestions, which have greatly improved the earlier version of this paper. This paper is supported by the National Natural Science Foundation of China (11671122; 11171094), the Program for Innovative Research Team (in Science and Technology) in University of Henan Province (14IRTSTHN023).
Authors’ contributions
PPS carried out the idea of this paper, the description of the division and reduction algorithm, and drafted the manuscript. TL carried out the numerical experiments of the algorithm. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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