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Solving nonlinear optimization problems with bipolar fuzzy relational equation constraints
- Jian Zhou^{1},
- Ying Yu^{1}Email author,
- Yuhan Liu^{2} and
- Yuanyuan Zhang^{1}
https://doi.org/10.1186/s13660-016-1056-6
© Zhou et al. 2016
- Received: 11 January 2016
- Accepted: 30 March 2016
- Published: 23 April 2016
Abstract
This paper considers the problem of minimizing a nonlinear objective function subject to a system of bipolar fuzzy relational equations with max-\(T_{L}\) composition, where \(T_{L}\) is the Łukasiewicz triangular norm. It shows that the feasible domain, i.e., the solution set of a system of bipolar fuzzy relational equations, can be reformulated as a system of 0-1 mixed integer inequalities. Consequently, such a type of optimization problems can be handled within the framework of 0-1 mixed integer optimization requiring no particular solving techniques.
Keywords
- fuzzy relational equations
- nonlinear optimization
- mixed integer optimization
1 Introduction
For a system of max-T equations \(A\circ\mathbf{x} = \mathbf{b}\), it is well known that the solution set, denoted by \(S(A,\mathbf{b})\), is nonempty if and only if its principal solution \(\hat{\mathbf{x}}\) is indeed a solution which can be constructed and verified in polynomial time. Moreover, when \(S(A,\mathbf{b})\) is nonempty, the principal solution \(\hat{\mathbf{x}}\) becomes the maximum solution and \(S(A,\mathbf{b})\) can be determined by the maximum solution and a finite number of minimal solutions. However, to obtain all the minimal solutions to \(A\circ\mathbf{x} = \mathbf{b}\) is in general a computationally difficult task because the number of minimal solutions could be exponentially large with respect to the input size. For a comprehensive discussion of fuzzy relational equations, the reader may refer to the monograph by Peeva and Kyosev [2] and the surveys by De Baets [3] and Li and Fang [4].
When a particular solution to a system of max-T equations is desired, the associated optimization problem is of concern. The problem of minimizing a linear objective function subject to a system of max-T equations has been intensively investigated with respect to various composition operations. It turns out that such an optimization problem can be transformed into the set covering problem which is known to be NP-hard (see, e.g., [5–14]). The linear fractional optimization constrained by a system of max-T equations was also studied by Wu et al. [15] and Li and Fang [16] with respect to an Archimedean triangular norm.
The problem of minimizing a general nonlinear objective function subject to a system of max-T equations has been tackled by Lu and Fang [17], Khorram and Hassanzadeh [18], and Hassanzadeh et al. [19] using the genetic algorithm. It was pointed out by Li et al. [20] that such an optimization problem can be in general reformulated into a 0-1 mixed integer nonlinear optimization problem so that the traditional solving techniques, e.g., the branch-and-bound method, may apply. However, when the objective function is max-separable and monotone, the associated optimization problem can be solved in polynomial time (see, e.g., [21–23] and references therein).
The bipolar max-\(T_{M}\) equations and the associated linear optimization problem were first investigated in Freson et al. [24]. By resolving each single equation of a system of bipolar max-\(T_{M}\) equations \(A^{+}\circ\mathbf{x} \vee A^{-} \circ\neg\mathbf{x} = \mathbf{b}\), Freson et al. [24] figured out analytically that the whole solution set \(S(A^{+}, A^{-},\mathbf{b})\) is the union of some interval-valued solutions, each of which is determined by a pair of maximal and minimal solutions. As a direct consequence, the bipolar max-\(T_{M}\) equation constrained linear optimization problem can be solved by examining these maximal and minimal solutions. An alternative reformulation for bipolar max-\(T_{M}\) equations was developed by Li and Jin [25] so that the associated linear optimization problem can be treated by integer optimization methods.
Besides, the bipolar max-\(T_{L}\) equation constrained linear optimization problem was studied by Li and Liu [26] using an analogous approach developed by Li and Jin [25, 27]. It turns out that the Archimedean property of \(T_{L}\) leads to a somewhat simpler structure for bipolar max-\(T_{L}\) equations, so that the associated linear optimization problem can be reformulated as a 0-1 integer linear optimization problem. This motivates us to extend this approach to nonlinear optimization scenarios.
The rest of this paper is organized as follows. In Section 2, the reformulation of a system of bipolar max-\(T_{L}\) equations is presented. The bipolar max-\(T_{L}\) equation constrained optimization problem is discussed in Section 3, and the conclusions are presented in Section 4.
2 Bipolar max-\(T_{L}\) equations and their reformulation
In this section, we reveal the critical features of a system of bipolar max-\(T_{L}\) equations and develop its equivalent representation.
Lemma 1
For any \(a^{+}, b\in[0,1]\), \(T_{L}(a^{+}, x)\le b\) if and only if \(x \le S_{L}(\neg a^{+}, b)\) where \(S_{L}:[0,1]^{2} \rightarrow[0,1]\) is the Łukasiewicz t-conorm defined as \(S_{L}(x,y) = \min(x+y,1)\). Analogously, \(T_{L}(a^{-}, \neg x) \le b\) if and only if \(x\ge T_{L}(a^{-}, \neg b)\) for any \(a^{-}, b\in[0,1]\).
Lemma 1 can be verified directly. Moreover, if \(b\in (0,1]\), \(T_{L}(a^{+}, x) = b\) if and only if \(a^{+} \ge b\) and \(x= S_{L}(\neg a^{+}, b)\) while \(T_{L}(a^{-}, \neg x) = b\) if and only if \(a^{-} \ge b\) and \(x= T_{L}(a^{-}, \neg b)\). The only exception occurs when \(b=0\), in which case, \(T_{L}(a^{+},x) = 0\) implies \(0\le x \le S_{L}(\neg a^{+},0) = \neg a^{+}\) and \(T_{L}(a^{-},\neg x) = 0\) implies \(a^{-}=T_{L}(a^{-},1) \le x \le1\), respectively.
Note that for the elements of \(\check{\mathbf{x}}\) and \(\hat{\mathbf {x}}\) if \(\check{x}_{j} = \hat{x}_{j}\) for some \(j\in N\), the value of \(x_{j}\) is fixed in any possible solution. In such a case, the variable \(x_{j}\) can be removed in further analysis as well as those equations where either \(T_{L}(a_{ij}^{+}, \hat{x}_{j}) = b_{i}\) or \(T_{L}(a_{ij}^{-}, \neg \check{x}_{j}) = b_{i}\) holds. Consequently, \(A^{+}\circ\mathbf{x} \vee A^{-}\circ\neg\mathbf{x} =\mathbf{b}\) can be reduced to a system of bipolar max-\(T_{L}\) equations with fewer variables such that the lower and upper bounds are strictly different. Hereafter, we assume without loss of generality that \(\check{\mathbf {x}}\) and \(\hat{\mathbf{x}}\) are strictly different for a system of bipolar max-\(T_{L}\) equations \(A^{+}\circ\mathbf{x} \vee A^{-}\circ\neg \mathbf{x} = \mathbf{b}\) under consideration, i.e., \(\check {x}_{j} < \hat{x}_{j}\) for all \(j\in N\).
Moreover, as indicated by Li and Liu [26], the equations with a zero right hand side play no role once \(\check{\mathbf{x}}\) and \(\hat{\mathbf{x}}\) have been obtained. Therefore, we may assume as well that the right hand side vector b is strictly positive for a system of bipolar max-\(T_{L}\) equations \(A^{+}\circ\mathbf{x} \vee A^{-}\circ\neg\mathbf{x} = \mathbf{b}\) under consideration.
Theorem 1
Proof
Theorem 1 demonstrates that a system of bipolar max-\(T_{L}\) equations can be equivalently expressed as a system of 0-1 mixed integer linear inequalities involving an additional pair of 0-1 vectors. Although this alternative formulation has a relatively larger size, it eliminates the nonlinear structure in bipolar max-\(T_{L}\) equations and allows us to tackle the associated optimization problem by the usual optimization techniques. Besides, Theorem 1 implies that the solution set of a system of bipolar max-\(T_{L}\) equations is a union of some interval-valued vectors analogous to that of bipolar max-\(T_{M}\) equations. Unfortunately, determining all these interval-valued vectors requires an enumeration of all minimal solutions to a system of 0-1 integer linear inequalities, the number of which could be exponentially large.
Example 1
3 Bipolar max-\(T_{L}\) equation constrained optimization
Besides, because the feasible domain \(S(A^{+}, A^{-}, \mathbf{b})\), when it is nonempty, is a union of several interval-valued solutions as illustrated in Example 1, such an optimization problem can also be viewed as a disjunctive optimization problem once \(S(A^{+}, A^{-}, \mathbf {b})\) has been explicitly determined. Consequently, the bipolar max-\(T_{L}\) equation constrained optimization problem can be theoretically decomposed into a series of box-constrained optimization problems and be solved separately. This strategy works for small size problem instances, but it would inevitably suffer some computational obstacles for large size problem instances.
Example 2
In general, the problem of minimizing a quadratic objective function subject to a system of bipolar max-\(T_{L}\) equations can be reformulated into a 0-1 mixed integer quadratic optimization problem, which has been intensively investigated in the literature and can be numerically solved with the aid of some commercial software packages.
4 Conclusions
Following the ideas in Li and Jin [25, 27] and Li and Liu [26], it is demonstrated that a system of bipolar max-\(T_{L}\) equations can be represented equivalently by a system of 0-1 mixed integer linear inequalities. Consequently, the bipolar max-\(T_{L}\) equation constrained optimization problem can be handled within the framework of 0-1 mixed integer optimization.
Declarations
Acknowledgements
This work was supported by a grant from the National Social Science Foundation of China (No. 13CGL057).
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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