Elsevier

Fuzzy Sets and Systems

Volume 417, 1 August 2021, Pages 71-92
Fuzzy Sets and Systems

Trapezoidal approximations of fuzzy numbers using quadratic programs

https://doi.org/10.1016/j.fss.2020.05.016Get rights and content

Abstract

In this paper we will prove that the nearest trapezoidal approximation of fuzzy numbers with respect to weighted L2-type metrics with or without additional constraints can be obtained via quadratic programs. Actually, the approach is even more general based on so called finite polyhedral subsets of fuzzy numbers which include most of the important special classes of fuzzy numbers available in the literature. In particular, we will recapture the algorithm to compute the nearest weighted trapezoidal approximation of a fuzzy number by a method which we believe that has the potential to be extended to more complex approximation problems. Then, we will improve the Lipschitz constant of the trapezoidal approximation operator preserving the ambiguity. To achieve this improved result we will exploit the fact that we have an analytical expression for this operator. However, note that the same result is obtained if this solution function is described by quadratic programs. Therefore, for similar problems we still can obtain Lipschitz constants for the approximation operator even if an analytical expression of this operator is not available.

Introduction

In the last decades, numerous researchers investigated on the constrained or unconstrained approximation of fuzzy numbers by fuzzy numbers with simpler form with respect to so called weighted L2-type metrics. First, the goal was to approximate fuzzy numbers by interval, triangular or trapezoidal fuzzy numbers (see, e.g. [1], [2], [5], [9] [13], [35], [38]) but later the studies were extended to other classes of fuzzy numbers (or other specific classes of fuzzy quantities as for example in [3]) such as so called semi-trapezoidal fuzzy numbers, L-R fuzzy numbers or fuzzy numbers with continuous piecewise linear functions in the parametric representation (see, e.g., [4], [11], [17], [18], [26], [36], [34]). One of the primal objectives in these papers is to find an analytical expression (or an effective algorithm) for the approximation operator. Another important objective is to study the properties of these operators, most of all the continuity (or Lipschitz continuity) being investigated in numerous papers (see, e.g., [6], [16], [7], [33], [32]). The main goal of this research is to prove that all these operators are actually solution functions of some parameterized quadratic programs. This can be done because the trapezoidal fuzzy numbers as well as all the other classes mentioned earlier are analogues of so called polyhedral subsets in the Euclidean space. This is why we will call them finite polyhedral subsets of fuzzy numbers. We believe that this approach based on quadratic programs has certain advantages in computing fuzzy number approximations and in addition, sometimes it is more effective in estimating Lipschitz constants of the approximation operators. We will have two concrete examples to support this claim. First, we will propose an algorithm to compute the weighted trapezoidal approximation of a fuzzy number by using a quadratic program that depends on the given fuzzy number in the objective function and also on the linear constraints. We will obtain an algorithm in three steps noting that at each iteration we will obtain a candidate solution obtained as a subvector of the unique solution of a Cramer system. Let us note that this algorithm will lead to a similar discussion as in Algorithm 1 in [35] where the author uses an approach based on projections onto closed convex sets. We believe that using quadratic programs we can try to solve even more complex approximation problems in terms of the variables and constraints that are involved. In the second application, we will improve the Lipschitz constant of the trapezoidal approximation operator preserving the ambiguity with respect to the so called Euclidean metric (that is, the weights are constantly equal to 1) introduced in [9]. We will do that using some auxiliary results, namely, Theorem 9 in [16] (see also Lemma 2.5 in [14]) and the fact that the so called extended trapezoidal approximation (see [31]) is nonexpansive. Although we can ease on the computation because we have an explicit solution function, note that we get the same conclusion if we express the solution using quadratic programs. Therefore, this method can be used also in situations when the solution is obtained via quadratic programs, which could be the case for more complex problems where an explicit expression of the solution is difficult to achieve.

In Section 2 we present some basic theory on fuzzy numbers. In Section 3 we recall the weighted L2-type metrics defined on the set of fuzzy numbers as they were given by Yeh in [35]. The set of fuzzy numbers can be embedded in a Hilbert space with respect to these metrics and this is why we refer to them as being inner product type metrics. Note that besides these weighted L2-type metrics there are other inner product type metrics such as those introduced in [30]. We mention these types of metrics too. In Section 4 we present a general approach on the existence and uniqueness of the best approximation with respect to a weighted L2-type metric. We can do that noticing that the target set where the best approximation is searched is actually an analogue of the well-known polyhedral subsets of the Euclidean space. This is why we refer to these sets as finite polyhedral subsets of fuzzy numbers. In particular, the sets of triangular fuzzy numbers, trapezoidal fuzzy numbers, L-R fuzzy numbers, semi-trapezoidal fuzzy numbers (introduced in [26]), polygonal fuzzy numbers (introduced in [4] and called piecewise linear fuzzy numbers in [18]), all of them are finite polyhedral subsets of fuzzy numbers. It means that we can easily prove the existence and uniqueness of the best approximation in these sets with respect to weighted L2-type metrics. The same is true when we consider subsets obtained via linear (in the embedded Hilbert space) inequality constraints. In this way, we can give simpler unified results on the existence and uniqueness of various approximations from the papers mentioned earlier. In Section 5 we present some generalities of quadratic programming including a characterization of the solution using a Karush-Kuhn-Tucker type theorem. In particular, an approximation problem in an inner product space will be transformed into a quadratic program. This is important because the approximation problems in the space of fuzzy numbers can be modelled in similar manner since we can use embeddings in Hilbert spaces. Consequently, in Section 6 we will propose an algorithm to compute the weighted trapezoidal approximation of a fuzzy number, that is, the nearest trapezoidal fuzzy number to a given fuzzy number with respect to a weighted L2-type metric. In paper [35] (see Algorithm 1 at page 3074) the author proposes an algorithm by using projections onto linear subspaces. We recapture this algorithm by transforming the problem into a quadratic program. As we will have three constraints, in general we would have 8 cases to get the solution. But we will actually need only three cases because at each iteration we know at least the value of one variable (thanks to Lemma 3.4 in [35] among other properties). We believe that this approach has the potential to be further developed considering approximation problems with a higher number of variables. We test the algorithm on concrete examples, one of them discussed in [35] too, as a proof of correctness. In Section 7, we will find a new, sharper Lipschitz constant for the trapezoidal approximation operator preserving the ambiguity introduced in [9] comparing to the one obtained in [9] and later improved in [15]. Still, it remains an open problem the finding of the sharpest Lipschitz constant for this operator. More details about this mater including comparison with other operators where the sharpest constant could be obtained (by a direct computational approach) are given inside the section. Although we will exploit the fact that we have an analytical representation of the approximation operator, the same Lipschitz constant is obtained if the approximation operator is given as the solution function of a parameterized quadratic program. The paper ends with conclusions where the main results are briefly recalled and some directions of research are proposed.

Section snippets

Preliminaries on fuzzy numbers

A fuzzy number A is a real fuzzy set having a membership function A:R[0,1], which is upper semicontinuous, quasi-concave, with bounded support and which attaints at least once the value 1. This means that there exist the real numbers abcd such that A(x)=0, if x[a,d], A is nondecreasing on [a,b] and nonincreasing on [c,d] and A(x)=1 for all x[b,c]. For some α(0,1], the α−cut (or α level set) of A is given byAα={xR:A(x)α}.The 0−cut has a slightly different definition, namelyA0=cl({xR:A(x

Metrical structures on fuzzy numbers

There are numerous metrics defined on the space of fuzzy numbers (see [20] for more details). In this paper we focus on inner product type metrics. Consider the functions λL,λU:[0,1]R that are strictly positive almost everywhere on [0,1] and integrable on [0,1]. We call these functions weight functions and we use the notation λ=(λL,λU). Then consider the set LλL2[0,1]×LλU2[0,1], where f=(f1,f2)LλL2[0,1]×LλU2[0,1] if f1,f2 are real functions on [0,1] such that the integrals 01[f1(α)]2λL(α)dα

Projections onto finite polyhedral subsets of fuzzy numbers

In this section we will present some general results with respect to so called finite polyhedral subsets of fuzzy numbers. Although we will employ only the metrics dλ introduced in the previous section, the results are valid for any inner product type metric. By inner product type metric d on F(R), generically we understand that (F(R),+,,d) can be embedded in an inner product space. In all that follows, if otherwise not specified, addition and scalar multiplication are considered in LλL2[0,1]×L

Quadratic programs

The major goal of this study is to prove that using the setting of quadratic programming we can contribute to the approximation of trapezoidal fuzzy numbers concerning alternative algorithms in finding the solution, as well as improving estimations concerning Lipschitz continuity of the solution function. Therefore, in this section we will make a resume of some important results concerning the solving of quadratic programs. We start with preliminaries on notations and with a description of the

An algorithm to compute weighted trapezoidal approximations of fuzzy numbers

In paper [35] (see Theorem 5.2 there), the author found an algorithm to compute the trapezoidal approximation of a fuzzy number with respect to metric dλ given in (1). It is a linear algorithm since the solution is obtained after at most four iterations as the number of the variables of the trapezoid. We will propose an alternate algorithm by rewriting the problem as a quadratic program. The final solution is obtained by examining a candidate solution at each step. In Theorem 5.2 from [35], the

Sharper Lipschitz constant of the trapezoidal approximation operator preserving the ambiguity

In this section, inspired by the considerations in the previous section, we will improve the Lipschitz constant of the trapezoidal approximation operator preserving the ambiguity introduced in [9]. In that paper we proved that this operator is Lipschitz continuous, then in [15] the result was improved by providing as Lipschitz constant the value 6. We will further improve this constant by using matrix representations for the solution function. It is worth mentioning that for other operators,

Conclusions

Numerous papers investigate on the approximations of fuzzy numbers by fuzzy numbers with simpler form using as approximation tools various L2-type metrics. In this paper we proved that all these problems are effectively approached using the quadratic programming setting. This approach enables us to propose effective algorithms to obtain the solution but it also helps us to find (sometimes improve) the estimations in terms of Lipschitz continuity. For the future, it is worth investigating for

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This contribution was supported by a grant of Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P1-1.1-PD-2016-1416, within PNCDI III.

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