Some remarks on optimality conditions for fuzzy optimization problems

REVISTA INVESTIGACIÓN OPERACIONAL VOL. 38, NO. 2, 189-194, 2017 SOME REMARKS ON OPTIMALITY CONDITIONS FOR FUZZY OPTIMIZATION PROBLEMS A. Rufián-Lizana1, R. Osuna-Gómez1, Y. Chalco-Cano2, H. Román-Flores2 1Departamento de Estadística I.O., Universidad de Sevilla, España 2Instituto de Alta Investigación, Universidad de Tarapacá Arica, Chile ABSTRACT In this article we present a new concept of stationary point for gH-differentiable fuzzy functions which generalize previous concepts that exist in the literature. Also, we give a concept of generalized convexity for gH-differentiable fuzzy functions more useful than level-wise generalized convexity (generalized convexity of the endpoint functions). Then we give optimatily conditions for fuzzy optimization problems. KEYWORDS: Fuzzy optimization, stationary point, gH-differentiable function. MSC: 90C70. RESUMEN En este artículo presentamos un nuevo concepto de punto estacionario para funciones difusas gH- diferenciables que generalizan los conceptos previos que existen en la literatura. También damos un concepto de convexidad generalizada para funciones difusas gH-diferenciables más útil que los basados en las funciones extremos. A partir de esos conceptos, damos condiciones de optimalidad para problemas de optimización difusos. 1. PRELIMINARIES Let KCdenote the family of all bounded closed intervals in Rand let FCdenote the family of all fuzzy intervals. So, for any u∈ FCwe denote its α-levels by [u]α= [uα, uα]∈ KC, for all α∈[0,1]. For fuzzy intervals u, v ∈ FC, with α-levels represented by [uα, uα]and [vα, vα], respectively, and for any real number λ, the addition u+vand the scalar multiplication λu are defined as follows (u+v)(x) = sup y+z=x min{u(y), v(z)},(λu)(x) = u(λ−1x),if λ6= 0, 0,if λ= 0. Equivalently, [u+v]α= [(u+v)α,(u+v)α] = [uα+vα, uα+vα], [λu]α=(λu)α,(λu)α= [min{λuα, λuα},max{λuα, λuα}], for every α∈[0,1]. Definition 1 ([17]) Given two fuzzy intervals u,v, the generalized Hukuhara difference (gH-difference, for short) of uand vis 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)are both valid if and only if wis a crisp number. If ugH vexists then, in terms of α-levels, we have [ugH v]α= [u]αgH [v]α= [min{uα−vα, uα−vα},max{uα−vα, uα−vα}], 189 for all α∈[0,1], where [u]αgH [v]αdenotes the gH-difference between two intervals (see [16, 17]). We use the Pompeiu-Hausdorff metric to define the distance between two fuzzy numbers. Given u, v ∈ FCwe 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|} . We recall the usual order for fuzzy intervals [9, 11, 12, 14, 15]. Definition 2 For u, v ∈ FC, it is said that uv, if for every α∈[0,1],uα≤vαand uα≤vα. It is said that uvif uvand ∃α0∈[0,1], such that uα0< vα0or uα0< vα0. It is said that u≺vif uvand ∃α0∈[0,1], such that uα0< vα0and uα0< vα0. Note that is a partial order relation on FC. So vuinstead of uvcan be written. 2. DIFFERENTIABLE FUZZY FUNCTIONS Henceforth, let Kbe an open subset of R, and F:K→ FCbe a fuzzy function. For each α∈[0,1], we define the family of interval-valued functions Fα:K→ KC, associated with F, and given by Fα(x)=[F(x)]α. For any α∈[0,1], we denote Fα(x) = hfα(x), fα(x)i. Here, for each α∈[0,1], the endpoint functions fα, fα:K→Rare called upper and lower functions of F, respectively. Next, we present the concept of gH-differentiable fuzzy functions based on the gH-difference of fuzzy intervals. Definition 3 ([1]) The gH-derivative of a fuzzy function F:K→ FCat x0∈Kis defined as F0(x0) = lim h→0 1 h[F(x0+h)gH F(x0)] .(1) If F0(x0)∈ FCsatisfying (1) exists, we say that Fis generalized Hukuhara differentiable (gH- differentiable, for short) at x0. Next we characterize the gH-differentiability of a fuzzy function in terms of the differentiability of its endpoint functions. Theorem 1 ([2]) Let F:K→ FCbe a fuzzy function and x∈K. Then Fis gH-differentiable at xif and only if one of the following four cases holds: (a)fαand fαare differentiable at x, uniformly in α∈[0,1],(fα)0(x)is monotonic increasing and (fα)0(x)is monotonic decreasing as functions of αand (f1)0(x)≤(f1)0(x). In this case, F0 α(x) = h(fα)0(x),(fα)0(x)i, for all α∈[0,1]. (b)fαand fαare differentiable at x, uniformly in α∈[0,1],(fα)0(x)is monotonic decreasing and (fα)0(x)is monotonic increasing as functions of αand (f1)0(x)≤(f1)0(x). In this case, F0 α(x) = h(fα)0(x),(fα)0(x)i, for all α∈[0,1]. 190 (c) (fα)0 +/−(x)and (fα)0 +/−(x)exist uniformly in α∈[0,1],(fα)0 +(x) = (fα)0 −(x)is monotonic increasing and (fα)0 +(x)=(fα)0 −(x)is monotonic decreasing as functions of αand (f1)0 +(x)≤ (f1)0 +(x). In this case, F0 α(x) = h(fα)0 +(x),(fα)0 +(x)i=h(fα)0 −(x),(fα)0 −(x)i, for all α∈[0,1]. (d) (fα)0 +/−(x)and (fα)0 +/−(x)exist uniformly in α∈[0,1],(fα)0 +(x) = (fα)0 −(x)is monotonic decreasing and (fα)0 +(x)=(fα)0 −(x)is monotonic increasing as functions of αand (f1)0 +(x)≤ (f1)0 +(x). In this case, F0 α(x) = h(fα)0 +(x),(fα)0 +(x)i=h(fα)0 −(x),(fα)0 −(x)i, for all α∈[0,1]. There is another one definition of differentiability in terms of the differentiability of its endpoint functions. Definition 4 Given F:K→ FCa fuzzy function, it is said to be a level-wise differentiable at x0∈Kif both endpoint functions fαand fαare differentiable at x0for each α∈[0,1]. We can see that the concept of level-wise differentiable fuzzy functions is very restrictive due to fuzzy arithmetic. The gH-differentiability is more general. For more details see [1, 2, 4]. 3. NECESSARY OPTIMALITY CONDITIONS FOR FUZZY OPTIMIZATION PROBLEMS Let us consider F:K⊆R→ FCa fuzzy function and let us suppose that Fis gH-differentiable on K, an open subset of R. We introduce the following minimum definition for F. Definition 5 x∗∈Kis said to be a minimum for Fif there does not exist x∈Ksuch that F(x)F(x∗). Note that in other articles minimum is also called non-dominated solution or efficient solution [3, 5, 9, 14, 15]. In the literature there are different definitions of stationary point for fuzzy functions which generalize the stationary point for classical mathematical programming. Some of these definitions use the defuzzification techniques and others that are very restrictive. We propose a new fuzzy stationary point definition that generalizes the existing ones. Definition 6 x∗∈Kis said to be a fuzzy stationary point for Fif 0∈[F0(x∗)]αfor some α∈[0,1]. Using different concepts of differentiability, definitions of stationary point have been introduced. For instance, using level-wise differentiability [5, 9, 14, 15] and GH-differentibility [3, 8]. Since gH- differentiability is more general than level-wise and GH-differentiability, then Definition 6 is more general. Theorem 2 Let Fbe a gH-differentiable fuzzy function. If x∗is a minimum of Fthen x∗is a fuzzy stationary point for F. Proof Since Fis gH-differentiable then taking into account Theorem 1 we have the cases (a), (b), (c) and (d). So, by the same arguments in the proof of Theorem 3 in [8] we obtain the result. Corollary 1 x∗∈Kis a fuzzy stationary point if and only if 0∈[F0(x∗)]0 Of course, a stationary point is not always a minimum. 191 4. SUFFICIENT OPTIMALITY CONDITIONS FOR FUZZY OPTIMIZATION PROBLEMS In order to ensure that all stationary points are minima, we need to introduce convexity or generalized convexity notions for fuzzy functions, similar to classical optimization problems. The following convexity definition for fuzzy function is well-known in the literature, for instance see [3, 6, 7, 9, 11, 13, 14, 15]. Definition 7 A fuzzy function F:K→ FCis said to be convex on a convex set K⊂Rif for all x, y ∈K,λ∈[0,1], F(λx + (1 −λ)y)λF (x) + (1 −λ)F(y). A convex fuzzy function Fis closely relate to its endpoint functions fαand fαconvexity. More precisely, we have the following result. Proposition 1 Let Kbe a nonempty convex subset of Rand let F:K→ FCbe a fuzzy function. Then, Fis convex on Kif and only if, for each α∈[0,1],fα, fαare convex functions on K. We remark that the concept of convexity is very restrictive by two reasons: due to fuzzy arithmetic, since the space of fuzzy interval is not a linear space, we have that usual fuzzy functions such as F(x) = u·x, which is the extension of a linear function, is not convex. On the other hand, convex and gH-differentiable fuzzy function cannot be characterized by inequality (2) F(y)F0(x)·(y−x) + F(x),(2) for all x, y ∈K[8]. This is the main difference with the characterization of convex and differentiable functions in classical optimization. Consequently, we can not extend the well-known results of classic convex optimization problems in a natural form to fuzzy optimization problems. Recently, concept of invex fuzzy functions as a generalization of convex fuzzy functions have been introduced [5, 10]. By same reasons to convexity these concepts are very restrictive. Also, In the literature we can find articles where convex (invex, pseudoinvex, preinvex) level-wise fuzzy functions are introduced. This concepts are based on the convexity (invex, pseudoinvex, preinvex) of the endpoint functions [5, 6, 7, 9, 11, 12, 13, 14, 15]. These concepts are also very restrictive due to fuzzy arithmetic such as we have said above. In addition, as it was also said above, level-wise differentiability is not good tool to differentibility fuzzy. In the article [8] was considered a concept more general than previous concepts of generalized convexity and, such as we can see next, this tool is more useful. Definition 8 Let Fbe a gH-differentiable fuzzy mapping. Fis said to be pseudoinvex on Kif for all x, y ∈K, there exists η:K×K→Rsuch that F(y)F(x)⇒F0(x)η(x, y)≺0. Note that in the previous Definition 8 we consider the gH-differentiability instead of GH-differentiability as was used in [8]. Theorem 3 If Fis pseudoinvex on Kthen every fuzzy stationary point is a minimum. Proof The demonstration follows by the same arguments of Theorem 4 in [8]. In the next example we present a fuzzy function that is not convex and is not level-wise differentiable, but it is gH-differentiable and pseudoinvex. 192 Example 1 Let us consider F:R→ FCdefined by its level sets [F(x)]α= [−(1 −α),(1 −α)]x Its endpoint functions are: fα(x)(1 −α)xif x≤0 −(1 −α)xif x > 0and fα(x)−(1 −α)xif x≤0 (1 −α)xif x > 0 In this case, Fis not level-wise differentiable, but it is gH-differentiable. Also Fis not convex, since fαis not convex, but Fis pseudoinvex and all the points are stationary points and by Theorem 3 these are also minima. ACKNOWLEDGEMENTS The research in this paper has been partially supported by grants 1151154, 1151159 (Fondecyt, Chile) and MTM2015-66185-P (MINECO, FEDER,UE). RECEIVED: JULY, 2016 REVISED: NOVEMBER, 2016 REFERENCIAS [1] BEDE, B. and STEFANINI, L. 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