On the Newton method for solving fuzzy optimization problems

Elsevier Editorial System(tm) for Fuzzy Sets and Systems Manuscript Draft Manuscript Number: FSS-D-14-00513 Title: On the Newton method for solving fuzzy optimization problems Article Type: Full Length Article (FLA) Keywords: Fuzzy optimization; generalized Hukuhara differentiability; Newton method. Corresponding Author: Dr. Y. Chalco Cano, Corresponding Author's Institution: Universidad de Tarapacá First Author: Y. Chalco Cano Order of Authors: Y. Chalco Cano; Geraldo Silva; Antonio Rufián-Lizana 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 In this article we consider optimization problems where the objectivesare fuzzy functions (fuzzy-valued functions). For this class of fuzzy optimization problems we discuss the Newton method to find a non-dominated solution. For this purpose, we use the generalized Hukuhara differentiability notion, which is the most general concept of existing differentiability for fuzzy functions. This work improves and correct the Newton Method previously proposed in the literature. 1 Abstract 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 On the Newton method for solving fuzzy optimization problems 4 Y. Chalco-Cano a,1,G.N.Silva b,2,A.Rufi ´ an-Lizana c,3 aInstituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica Chile bInstituto de Biociˆencias, Letras e Ciˆencias Exatas, UNESP-Univ.EstadualPaulista, Cˆampus de S˜ao Jos´e do Rio Preto, Departamento de Matem´atica Aplicada, S˜ao Jos´e do Rio Preto-SP, Brasil. cDepartamento de Estad´ıstica e I.O., Universidad de Sevilla, Spain. Abstract In this article we consider optimization problems where the objectives are fuzzy functions (fuzzy-valued functions). For this class of fuzzy optimization problems we discuss the Newton method to find a non-dominated solution. For this purpose, we use the generalized Hukuhara differentiability notion, which is the most general concept of existing differentiability for fuzzy functions. This work improves and correct the Newton Method previously proposed in the literature. Key words: Fuzzy optimization, generalized Hukuhara differentiability, Newton method. 1Introduction Fuzzy optimization problems have been studied by many researchers in several directions with a lot of applications. The collection of papers on fuzzy optimization edited by Delgado et al. [11], Lodwick and Kacprzyk [19], Inuiguchi and Ram´ık 1Corresponding author, e-mail: [email protected] 2e-mail: [email protected] 3e-mail: rufi[email protected] 4The research in this paper has been supported by Fondecyt-Chile, project 1120665 and 1120674, by Ministerio de Ciencia y Tecnolog´ıa (Spain) through project MTM 2010- 15383, by the Brazilian National Council for Scientific and Technological Development - CNPq (Brazil) under Grant numbers 309335/2012-4 and 479109/2013-3 and by Sao Paulo State Foundation (FAPESP) under Grant number 2013/07375-0 – through the Center for Mathematical Sciences Applied to Industry – CeMEAI/CEPID. Preprint submitted to Elsevier 27 June 2014 *Manuscript Click here to view linked References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 [15], Rommelfanger and Slowi´nski [26], Slowi´nski and Teghem [27], and the books by Lai and Hwang [16,17] provide reviews about this topic fromaverybroadpoint of view. It is usually difficult to determine the coefficients of an objective function as a real number since, most often, these possess inherent uncertainty and/or inaccuracy. Given that this is the usual state, we consider fuzzy-valued objective function as one approach to tackle uncertainty and inaccuracies in theobjectivefunction coefficients of mathematical programming models (see, e.g., Lodwick [18]). Optimization problems with fuzzy-valued objective functions were studied by many researchers. For instance, see [5,8,23,20,30–37,39]. In particular, in [5,9,33,37] Karush-Kuhn-Tucker type optimality conditions for this class of fuzzy optimization problems were obtained. More recently, Pirzada and Pathak in [21] proposed a Newton method to find a non-dominated solution of a fuzzy optimization problem using the Hukuhara differentiability of fuzzy-valued functions. The concept of Hukuhara differentibility (H-differentiability, for short) for fuzzy functions is very restrictive. For instance, F(x)=C·x,whereCis any fuzzy interval and xis a real number, is not H-differentiable being that Fis a generalization of alinearfunction.Ingeneral,afuzzyfunctiondefinedbyF(x)=C·g(x), where gis a differentiable real function and Cis a fuzzy interval, is not always H-differentiable. However, it is always gH-differentiable. It is well-known that the concept of gH- differentiable fuzzy function (generalized Hukuhara differentiable fuzzy function) is a more general concept than level-wise differentiability [37,38], Hukuhara differentiability [14], and G-differentiability [1–3,6,7]. Thus, the more useful concept of differentiability for fuzzy functions is gH-differentiability. The conditions imposed to implement the Newton method introduced by Pirzada and Pathak in [21] in Theorem 4.1 are very restrictive becausetheyaskthatinthe neighborhood of a nondominated solution, all points must be comparable, but the order relation used is only partial. Moreover, it is also required that the nondominated solution is an ideal point of the endpoint functions oftheobjectivefuzzy functions. The examples presented in the same paper do not obey the conditions required by Theorem 4.1. In addition, the objective functions of all examples they consider are not H-differentiable. Even so, they apply the Newton method to the examples. Not surprisingly, when they apply to the Example 4.1,theyobtainapoint that is not a non dominated solution, although the authors claim it is. In this paper we formulate the Newton method to find a non-dominated solution of fuzzy optimization problems without the requirementthatallfeasiblesolutions in the neighborhood of a nondominated solution be comparable and use gH-differentibility instead of H-differentiability. Finally, we correct the examples considered in [21]. 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2Notationandthespaceoffuzzyintervals AfuzzysetonRnis a mapping u:Rn→[0,1]. For each fuzzy set u,wedenote its α-level set as [u]α={x∈Rn|u(x)≥α}for any α∈(0,1]. The support of uis denoted by supp(u), where supp(u)={x∈Rn|u(x)>0}.Theclosureofsupp(u) defines the 0-level of u,.i.e.[u]0=cl(supp(u)), where cl(M)meanstheclosureof the subset M⊂Rn. Definition 1 AfuzzysetuonRis said to be a fuzzy interval if: (1) u is normal, i.e. there exists x0∈Rsuch that u(x0)=1; (2) u is an upper semi-continuous function; (3) u(λx+(1 −λ)y)≥min{u(x),u(y)},x,y∈R,λ∈[0,1]; (4) [u]0is compact. Let FCdenote the family of all fuzzy intervals. So, for any u∈F Cwe have that [u]α∈K Cfor all α∈[0,1], where KCdenotes the space of all compact intervals in R,andthustheα-levels of a fuzzy interval are given by [u]α=!uα,uα",uα,uα∈R for all α∈[0,1]. If [u]1is a singleton then we say that uis a fuzzy number. Triangular fuzzy numbers are a special type of fuzzy numbers which are well determined by three real numbers a≤b≤cand we write u=(a,b,c)and [u]α=[a+(b−a)α,c−(c−b)α], for all α∈[0,1]. For fuzzy intervals u,v∈F Crepresented by !uα,uα"and !vα,vα",respectively,and for any real number λ,wedefinetheadditionu+vand scalar multiplication λuas follows: (u+v)(x)=sup y+z=x min{u(y),v(z)} (λu)(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u'x λ(,if λ!0, 0,if λ=0. It is well known that, for every α∈[0,1], [u+v]α=!(u+v)α,(u+v)α"=!uα+vα,uα+vα"(1) and [λu]α=!(λu)α,(λu)α"=!min{λuα,λ uα},max{λuα,λ uα}".(2) Acrucialconceptinobtainingausefulworkingdefinitionofderivative for fuzzy functions is deriving a suitable difference between two fuzzy intervals. Toward this end we have the following definition. 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Definition 2 ([29]) Given two fuzzy intervals u, v,the generalized Hukuhara difference (gH-difference for short) is the fuzzy interval w, if it exists, such that u⊖gH v=w⇔⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (i)u=v+w, or (ii)v=u+(−1)w. It is easy to show that (i)and(ii)arebothvalidifandonlyifwis a crisp number. Note that the case (i) is coincident to Hukuhara difference (see [14]) and so the concept of gH-difference is more general than H-difference. If u⊖gH vexists then, in terms of α-levels, we have [u⊖gH v]α=[u]α⊖gH [u]α=!min{uα−vα},max{uα−vα}", for all α∈[0,1], where [u]α⊖gH [u]αdenotes the gH-difference between two intervals (see [28,29]). Given u,v∈F C,we define the distance between uand vby D(u,v)=sup α∈[0,1] H([u]α,[v]α) =sup α∈[0,1] max )***uα−vα***,|uα−vα|+. So, (FC,D)isacompletemetricspace. 3Differentiable fuzzy functions Henceforth, Kdenotes an open subset of Rn.AfunctionF:K→F Cis said to be afuzzyfunction.Foreachα∈[0,1], we associate with Fthe family of intervalvalued functions Fα:K→K Cgiven by Fα(x)=[F(x)]α.Foranyα∈[0,1], we denote Fα(x)=!fα(x),fα(x)". Here, the endpoint functions fα,fα:K→Rare called upper and lower functions of F,respectively. Next we present the concept of gH-differentiability of fuzzy functions in the one dimensional case. Definition 3 ([3]) Let K ⊂Rwith F :K→F Cafuzzyfunctionandx 0∈Kandh be such that x0+h∈K. Then the generalized Hukuhara derivative (gH-derivative, 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 for short) of F at x0is defined as F′(x0)=lim h→0 F(x0+h)⊖gH F(x0) h.(3) If F′(x0)∈F Csatisfying (3) exists, we say that F is generalized Hukuhara differentiable (gH-differentiable, for short) at x0. The gH-derivative for an interval-valued function [28] is similartoDefinition3. More precisely, an interval-valued function F:K→K Cis gH-differentiable at x0∈K,withgH-derivative F′(x0)∈K C,if(3)existswithrespecttothelimitinthe metric space (KC,H), where the difference is given by the gH-difference between intervals (see [28]). Theorem 1 Let F :K→F Cbe a fuzzy function. If F is gH-differentiable then the interval-valued function Fα:K→K Cis gH-differentiable for each α∈[0,1]. Moreover ,F′(x)-α=F′ α(x).(4) Proof. The proof is a consequence of the definition of gH-differentiability. Example 1 Consider the fuzzy mapping F :R→F Cdefined by F(x)=C·x, where C is a fuzzy interval and [C]α=[Cα,Cα]with Cα<Cα.NotethatFisa generalization of a linear function and for each α∈[0,1] we have Fα(x)=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ !Cαx,Cαx"if x ≥0; !Cαx,Cαx"if x <0. Thus the endpoint functions f αand fαare not differentiable at x =0.HoweverF is gH-differentiable on Rand F′(x)=Cforallx∈R.Ingeneral,ifF(x)=C·g(x), where g :R→Ris a differentiable function and C ∈F C,thenitfollowsrelatively easy that the gH-derivative exists and it is F′(x)=C·g′(x),buttheendpoint functions f αand fαare not necessarily differentiable. In general we have the following result which connects gH-differentiability of F and the differentiability of its endpoint functions fαand fα. Theorem 2 Let F :K→F Cbe a fuzzy function. If F is gH-differentiable at x0∈K then, for each α∈[0,1],oneofthefollowingcaseshold: (a)fαand fαare differentiable at x0and [F′(x0)]α=!min )(fα)′(x0),(fα)′(x0)+,max )(fα)′(x0),(fα)′(x0)+"; 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 (b)(fα)′ −(x0),(fα)′ +(x0),(fα)′ −(x0)and (fα)′ +(x0)exist and satisfy (fα)′ −(x0)=(fα)′ +(x0) and (fα)′ +(x0)=(fα)′ −(x0).Moreover [F′(t0)]α=!min )(fα)′ −(x0),(fα)′ −(x0)+,max )(fα)′ −(x0),(fα)′ −(x0)+" =!min )(fα)′ +(x0),(fα)′ +(x0)+,max )(fα)′ +(x0),(fα)′ +(x0)+" Proof. The proof is a consequence of Theorem 9 in [6] and Theorem 1. Remark 1 Note that the gH-differentiability is coincident with the H-differentia- bility (differentiability in the sense of Hukuhara introduced by Puri andRalescu [22] as a generalization of the Hukuhara derivative for set-valued functions [14]) only when f αand fαare differentiable and (fα)′(x)≤(fα)′(x)for all α∈[0,1]. Thus, the gH-differentiability is a more general concept of differentiability for fuzzy functions than the H-differentiability. The gH-differentiability concept is also more general than G-differentiability, see [1]. We are now going to define the partial derivative for a fuzzy function Fdefined on K⊂Rn,i.e.,F(x)=F(x1, ..., xn)∈F Cfor each x=(x1, ..., xn)∈K.For this, given a fuzzy function F:K→F C,wedenotethefuzzyintervalF(x)by F(x)=!f(x),f(x)"and, for each α∈[0,1], Fα(x)=!fα(x),fα(x)". Definition 4 Let F be a fuzzy function defined on K ⊂Rnand let x0='x(0) 1, ..., x(0) n( be a fixed element of K. We consider the fuzzy function hi(xi)=F(x(0) 1, ..., x(0) i−1,xi,x(0) i+1, ..., x(0) n). If hiis gH-differentiable at x(0) i,thenwesaythatFhastheithpartialgH-derivative at x0(denoted by (∂F/∂xi)(x0))and(∂F/∂xi)(x0)=(hi)′(x(0) i). Definition 5 Let F be a fuzzy function defined on K and let x0='x(0) 1, ..., x(0) n(∈Kbefixed.WesaythatFisgH-differentiable at x0if all the partial gH-derivatives (∂F/∂x1)(x0),..., (∂F/∂xn)(x0)exist on some neighborhood of x0and are continuous at x0. Note that if Fis gH-differentiable at x0,then(∂F/∂xi)(x0)isafuzzyinterval.So, for each α∈[0,1], we denote .∂F ∂xi (x0)/α =∂Fα ∂xi (x0)=⎡⎢⎢⎢⎢⎣ ∂Fα ∂xi (x0),∂Fα ∂xi (x0)⎤⎥⎥⎥⎥⎦. We obtain '∂Fα/∂xi((x0)and'∂Fα/∂xi((x0)fromTheorem2. Next we present an interesting proposition which will be usedtoobtainourmain results. 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Proposition 1 Let F :K→F Cbe a fuzzy function. If F is gH-differentiable at x0∈Kthen,foreachα∈[0,1],thereal-valuedfunction f α+fα:K→Ris differentiable at x0.Moreover, ∂Fα ∂xi (x0)+∂Fα ∂xi (x0)= ∂'fα+fα( ∂xi (x0).(5) Proof. The proof is a consequence of Theorem 2. From previous definition we can define the gradient of a fuzzy function as follows. Definition 6 Given the fuzzy function F :K→F C,thegradientofFatx 0,denoted by ˜ ∇F(x0),isdefinedby ˜ ∇F(x0)=66∂F ∂x17(x0), ..., 6∂F ∂xn7(x0)7,(6) where (∂F/∂xj)(x0)is the jth partial G-derivative of F at x0. Note that ˜ ∇F(x)isan-dimensional fuzzy vector. For the gradient of a fuzzy function we use the symbol ˜ ∇,whereasforthegradientofareal-valuedfunctionweuse the symbol ∇. Definition 7 Let F :K⊂Rn→F Cbe a fuzzy function, where K ⊂Rnis an open set. Suppose now that there is x0∈KsuchthatgradientofF,˜ ∇F, is itself gH- differentiable at x0,thatis,foreachi,thefunction∂F ∂xi:K→F Cis gH-differentiable at x0.DenotethegH-partialderivativeof∂F ∂xiby D2 ijF(x0)or ∂2F ∂xixj (x0),if i !j, and D2 iiF(x0)or ∂2F ∂x2 i (x0),if i =j. If F is twice gH-differentiable at each x0in K, we say that F is twice gH-differentiable on K, and if for each i,j=1,2, ..., n, the cross-partial derivative ∂2F ∂xixjis continuous function from K to FC,wesaythatFistwicecontinuouslygH-differentiable on K. We define a m-times continuously gH-differentiable fuzzy function in way similar to Definition 7, that is, F:K→F Cis m-times continuously gH-differentiable on Kif and only if all of the partial gH-derivatives of order m∈Nexist and are continuous (in the sense of fuzzy function). If Fis gH-differentiable we have that the endpoint function fαand fαare not necessarily differentiable. 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