Karush-Kuhn-Tucker conditions for interval and fuzzy

0 downloads 0 Views 2MB Size Report
Feb 16, 2018 - to the well know tangency between a differentiable function and its ... The relevant formulations of interval and fuzzy linear programming, ...

Karush-Kuhn-Tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability Luciano Stefaninia , Manuel Arana-Jiménezb,1 a Department

of Economics, Society and Politics, University of Urbino "Carlo Bo", Urbino, Italy of Statistics and Operational Research, University of Cádiz, Spain

b Department

Abstract We extend the interval and fuzzy gH-differentiability to consider interval and fuzzy valued functions of several variables and to include directional gH-differentiability; the proposed setting is more general than the existing definitions in the literature and allows a unified view of total and direction gH-differentiability and for the computation of partial gH-derivatives, directional gH-derivative and level wise gH-differentiability in the fuzzy valued case. A concept of gH-differential is then deduced and its values are used to define an (abstract) local tangency property for a gH-differentiable function, similar to the well know tangency between a differentiable function and its tangent plane. The proposed new setting allows an analysis of conditions for local optimality (dominance with respect to interval and levelwise partial orders well known in the literature) in terms of directional gH-derivatives, including concepts of local convexity, and to formulate KKT-like conditions for non dominated solutions in constrained optimization problems. Keywords: Optimization under uncertainty, fuzzy valued functions of several variables, gH-differentiability, fuzzy directional LgH-derivative, Karush-Kuhn-Tucker optimality conditions.

1. Introduction In this paper we consider nonlinear constrained optimization problems with a single interval or fuzzy valued objective function of several variables. After the influential paper by Bellman and Zadeh [5], the theory of optimization with fuzzy data has considered the general problem where fuzziness enters both the objective functions (linear on nonlinear) and the constraints (usually in the form of inequalities); some milestone papers (e.g. [37], [24], [32]) established the basic ideas about inequalities between fuzzy intervals, the meaning of fuzzy feasibility, the basic arithmetic operations, the

Email addresses: [email protected] (Luciano Stefanini ), [email protected] (Manuel Arana-Jiménez ) 1 Corresponding author

Preprint submitted to Fuzzy Sets and Systems

February 16, 2018

comparison criteria for fuzzy objective values to define optimality. The representation of the fuzzy quantities in terms of the interval-valued level cuts, established the strong connections between fuzzy and interval analysis and motivated a series of common researches to the two areas. Part II of the Handbook edited by Slowinski [26] contains an extended discussion about the interpretation and the implementation of various approaches for fuzzy optimization with linear and nonlinear single and multiple objectives, with general constraints and with continouous or integer/binary decision variables. The relevant formulations of interval and fuzzy linear programming, including some general approaches to duality theory, have been systematized in recent papers (see, e.g., [36], [23] and the references therein). More general types of fuzzy optimization and fuzzy mathematicsl programming problems have been published more recently; a recent collection of papers in fuzzy optimization is [16]. The mathematics of unconstrained and constrained nonlinear interval and fuzzy optimization is the subject of actual research and the more recent papers have taken advantage of the concepts of Hukuhara and generalized-Hukuhara difference (introduced in [27] and [28]) and differentiability (see [3], [29], [30], [4]) for functions of a single variable, and used by some authors to define the partial gH-derivative for functions of several variables. In [33] and [34] the differentiability of an interval-valued function of n variables is defined in terms of the existence of Hukuhara partial derivatives, with the additional requirement that at least n − 1 of them are locally continuous. Similarly, in [21], [7] the differentiability of f at a point is defined by requiring that the partial gH-derivatives have to be locally continuous (analogous to the class C (1) requirement for single-valued functions of several variables). The fuzzy-valued cases have been considered, e.g., in [8], [35] where a simplified definition of level wise differentiability is introduced by requiring that the end-point functions of each level cut [ f ]α be both differentiable in the usual sense as for crisp functions. In this paper, we first propose (section 3) a new concept of generalized (total) differentiability for interval and fuzzy-valued functions of n variables and we discuss the concept of level wise fuzzy differentiability in the general setting introduced in [4] and [30]. A concept of gH-differential is then deduced and its values are used to define an (abstract) local tangency property for a gH-differentiable function, similar to the well know tangency between a differentiable function and its tangent plane. Then (section 4) we define the interval and fuzzy directional gH-derivative and differentiability and we show its connections (dependence) with total gH-differentiability and with partial gH-derivatives. Finally (section 5), we present new results on necessary optimality conditions for optimization problems with a fuzzy objective function of several variables in terms of directional LgH-differentiability. We end with a theorem on Karush-Kuhn-Tucker conditions for non dominated solutions in constrained optimization problems and we illustrate a linearly constrained example.

2

2. Notation on fuzzy intervals We denote by KC the family of all bounded closed intervals in R, i.e., nh i o KC = a, a | a, a ∈ R and a ≤ a . To represent operations for real intervals (and for level sets of fuzzy intervals) the well-known midpoint-radius representation is very useful. For a given interval A = h i a, a , define the midpoint b a and the radius e a, respectively, by b a=

a+a a−a and e a= , 2 2

so that a= b a −e a and a = b a +e a. We will denote an interval by A = [a, a] or, in midpoint notation, by A = (b a;e a). A fuzzy set on Rn is a mapping u : Rn → [0, 1]. For each fuzzy set u, we denote its α-level set as [u]α = {x ∈ Rn | u(x) ≥ α} for any α ∈ (0, 1]. The support of u we denote by supp(u) where supp(u) = {x ∈ Rn | u(x) > 0}. The closure of supp(u) defines the 0-level of u, .i.e. [u]0 = cl(supp(u)) where cl(M) means the closure of the subset M ⊂ Rn . The following definitions and results are well known (see e.g. [2]). Definition 1. A fuzzy set u on R is said to be a fuzzy interval if: 1. 2. 3. 4.

u is normal, i.e. there exists x0 ∈ R such that u(x0 ) = 1; u is an upper semi-continuous function; u(λx + (1 − λ)y) ≥ min{u(x), u(y)}, x, y ∈ R, λ ∈ [0, 1]; [u]0 is compact.

Let [u]α ∈ [u]α =

FC denote the family of all fuzzy intervals. So, for any u ∈ FC we have that K h C fori all α ∈ [0, 1] and thus the α-levels of a fuzzy interval are given by uα , uα , uα , uα ∈ R for all α ∈ [0, 1]. In midpoint notation, we will write u +u

u −u

uα −b uα and uα = b uα +e uα . uα = α 2 α so that uα = e [u]α = (b uα ;e uα ) where b uα = α 2 α and e 1 If [u] is a singleton then we say that u is a fuzzy number. Triangular fuzzy numbers are a special type of fuzzy numbers which are well determined by three real numbers a ≤ b ≤ c, denoted by u = ha, b, ci, with α-levels [u]α = [a + (b − a)α, c − (c − b)α], for all α ∈ [0, 1]. Definition 2. Let u = (u, u) be a fuzzy interval. We say that u is a non-negative fuzzy interval (non-positive fuzzy interval, respectively) if u(0) ≥ 0 (u(0) ≤ 0, respectively). The well-known characterization theorem makes the connection between a fuzzy interval and their endpoint functions (Goestschel and Voxman [13]).

3

Theorem 1. Let u be a fuzzy interval. Then the functions u, u : [0, 1] → R, defining the endpoints of the α-level sets of u (u(α) = uα and u(α) = uα ), satisfy the following conditions: (i) u is a bounded, non-decreasing, left-continuous function in (0, 1] and it is rightcontinuous at 0. (ii) u is a bounded, non-increasing, left-continuous function in (0, 1] and it is rightcontinuous at 0. (iii) u(1) ≤ u(1). Reciprocally, given two functions that satisfy the above conditions they uniquely determine a fuzzy interval. Given A = [a, a], B = [b, b] ∈ KC and τ ∈ R, we have the following classical operations: • A + B = [a + b, a + b], ( • τA = {τa : a ∈ A} =

[τa, τa], [τa, τa],

if τ ≥ 0, if τ ≤ 0

Using midpoint notation, the previous and other standard operations on intervals are the following, given A = [a, a] = (b a;e a), B = [b, b] = (b b; e b) and τ ∈ R: • A + B = (b a +b b;e a +e b), • τA = {τa : a ∈ A} = (τb a; |τ|e a), • −A = (−b a;e a), • A − B = (b a −b b;e a +e b). We refer to Moore [19, 20] and Alefeld and Herzberger [1] for further details on the topic of interval analysis. As a natural extension hof the previous let us consider the fuzzy intervals i h operations, i u, v ∈ FC represented by uα , uα and vα , vα , respectively, and a real number λ. We define the addition u + v and scalar multiplication λu as follows: (u + v)(x) = sup min{u(y), v(z)} y+z=x

( (λu)(x) =

  u λx , 0,

if λ , 0, if λ = 0.

It is well known that in terms of α-levels and taking into account the midpoint notation, for every α ∈ [0, 1], h i h i [u + v]α = (u + v)α , (u + v)α = uα + vα , uα + vα = (b uα + b vα ;e uα + e vα ), (1) and i h i h [λu]α = (λu)α , (λu)α = min{λuα , λuα }, max{λuα , λuα } = (λb uα ; |λ|e uα ). 4

(2)

A crucial concept in obtaining a useful working definition of derivative for fuzzy functions is deriving a suitable difference between two fuzzy intervals. Toward this end we have the following definition. The gH-difference of two intervals A and B, which we recall from [18], [29], [28]), is as follows: ( (a) A = B + C, A gH B = C ⇐⇒ or (b) B = A + (−1)C. Note that the difference of an interval and itself is zero, that is, A gH A = [0, 0]. Furthermore, the gH-difference of two intervals always exists and is equal to a −b b; |e a −e b|) ⊂ A − B. A gH B = [min{a − b, a − b}, max{a − b, a − b}] = (b As an extension of the previous gH-difference of two intervals, we have the gHdifference of two fuzzy intervals (see [27], [28]). Definition 3. Given two fuzzy intervals u, v, the generalized Hukuhara difference (gHdifference for short) is the fuzzy interval w, if it exists, such that ( (i) u = v + w, u gH v = w ⇔ or (ii) v = u + (−1)w. It is easy to show that (i) and (ii) are both valid if and only if w is a crisp number. Note that the case (i) is coincident to Hukuhara difference (see [15]) and so the concept of gH-difference is more general than H-difference. Given u, v ∈ FC , we define the distance between u and v by D(u, v) = =

sup H ([u]α , [v]α ) n o sup max uα − vα , |uα − vα | .

α∈[0,1] α∈[0,1]

where H is the Pompeiu-Hausdorff distance defined by   H(A, B) = max max d(a, B), max d(b, A) a∈A

b∈B

with d(a, B) = minb∈B ||a − b||.

It is known (see [29]) that H(A, B) =

A gH B

where, for C ∈ KC , kCk = max{|c| ; c ∈ C}; then

D(u, v) = sup{

[u]α gH [v]α

; α ∈ [0, 1]}. It is well known that (FC , D) is a complete metric space. We denote by FCC the family of all level-continuous fuzzy intervals [25]. Thus, u ∈ FCC if the application α 7→ [u]α is continuous, i.e given  > 0 there exists δ > 0 such that |α − α0 | < δ implies H([u]α , [u]α0 ) < . It is well known that (FCC , D) is a separable and complete metric space [25]. Moreover, FCC is a closed subspace of FC . Furthermore, given a fuzzy interval u = (u, u), then, u ∈ FCC if and only if u and u are continuous functions (see [11]). 5

3. On a general concept of gH-differentiability In this Section, firstly, we introduce a concept of differentiability for fuzzy valued mappings of several variables. Henceforth, K denotes an open subset of Rn . A function F : K → FC is said to be a fuzzy function. For each α ∈ [0, 1], we associate with F the family of interval-valued functions Fα : K → KC given by Fα (x) = [F(x)]α . For any α ∈ [0, 1], we denote i h bα ; F eα ). Fα (x) = F α (x), F α (x) = (F Here, the endpoint functions F α , F α : K → R are called lower and upper functions of bα and F eα are its midpoint and radius functions, respectively. F, respectively; and, F As an extension of the gH-derivative for an interval-valued function (see [18, 28]), in [4], [30] we find the concept of gH-differentiable fuzzy functions of a single variable (one dimensional case) as follows. Definition 4. Let K ⊂ R with F : K → FC be a fuzzy function and x0 ∈ K and h ∈ R be such that x0 + h ∈ K. Then the generalized Hukuhara derivative (gH-derivative, for short) of F at x0 is defined as F 0 (x0 ) = lim

h→0

F(x0 + h) gH F(x0 ) . h

(3)

If F 0 (x0 ) ∈ FC satisfying (3) exists, we say that F is generalized Hukuhara differentiable (gH-differentiable, for short) at x0 . With respect to the level-wise gH-derivative, Bede and Stefanini [4] introduced the following definition. Definition 5. Let K ⊂ R with F : K → FC a fuzzy function, x0 ∈ K and h be such that x0 + h ∈ K. Given α ∈ [0, 1], the level-wise gH-derivative (LgH-derivative, for short) of the corresponding interval-valued function Fα : K → KC at x0 is defined as 0 F LgH,α (x0 ) = lim

h→0

Fα (x0 + h) gH Fα (x0 ) , h

(4)

0 if it exists. If F LgH,α (x0 ) ∈ KC for all α ∈ [0, 1], we say that F is level-wise generalized differentiable (LgH-differentiable, for short) at x0 and the family of intervals 0 0 {F LgH,α (x0 ) : α ∈ [0, 1]} is the LgH-derivative of F at x0 , denoted as F LgH (x0 ).

As a consequence of the previous definitions, it is derived that LgH-differentiability, and consequently level-wise continuity, is a necessary condition for gH-differentiability, but it is not sufficient (see [4, 7]). Theorem 2. Let F : K → FC be a fuzzy function. If F is gH-differentiable at x0 ∈ K then it is LgH-differentiable and, for each α ∈ [0, 1], h iα   0 (5) F LgH (x0 ) = F 0 (x0 ) α . 6

On the other hand, the existence of the gH-derivative for a fuzzy function does not necessarily imply that the corresponding endpoint functions are differentiable, such as the following example shows. Example 1. We consider the fuzzy mapping F : R → FC defined by F(x) = C · x, where C is a fuzzy interval where [C]α = [C α , C α ] with C α < C α . Note that in this case F is a generalization of a linear function. Then  h i    hC α x, C α xi if x ≥ 0 Fα (x) =    C α x, C α x if x < 0 We can see that the endpoint functions F α and F α are not differentiable at x = 0. 0 However F is gH-differentiable on R and F (x) = C for all x ∈ R. In general, if F(x) = C · g(x) where g : R → R is a differentiable function and C ∈ FC , then it follows relatively easily that the gH-derivative exists and it is F 0 (x) = C · g0 (x) but the endpoint functions F α and F α are not necessarily differentiable. We have the following connection between LgH-differentiability of F and the differentiability of its endpoint functions F α and F α : Theorem 3. Let F : K → FC be a fuzzy function. F is LgH-differentiable at x0 ∈ K if and only if, for each α ∈ [0, 1], one of the following cases hold: (a) F α and F α are differentiable at x0 and o n oi n h 0 0 0 0 0 [F (x0 )]α = min (F α ) (x0 ), (F α ) (x0 ) , max (F α ) (x0 ), (F α ) (x0 ) ; (b) (F α )0− (x0 ), (F α )0+ (x0 ), (F α )0− (x0 ) and (F α )0+ (x0 ) exist and satisfy (F α )0− (x0 ) = (F α )0+ (x0 ) and (F α )0+ (x0 ) = (F α )0− (x0 ). Moreover o n oi n h 0 0 0 0 0 [F (x0 )]α = min (F α )− (x0 ), (F α )− (x0 ) , max (F α )− (x0 ), (F α )− (x0 ) n o n oi h 0 0 0 0 = min (F α )+ (x0 ), (F α )+ (x0 ) , max (F α )+ (x0 ), (F α )+ (x0 ) Proof. This is a consequence of Theorem 6 in [9] and Definition 5. Note that the gH-differentiability is coincident with the H-differentiability (differentiability in the sense of Hukuhara introduced by Puri and Ralescu [22] as a generalization of the Hukuhara derivative for set-valued functions [15]) only when F α and F α are differentiable and (F α )0 (x) ≤ (F α )0 (x) for all α ∈ [0, 1]. 3.1. gH-differentiability for multiple variable functions As we have seen, the gH-differentiability on R is a concept of differentiability for fuzzy functions more general than the H-differentiability and G-differentiability, but less general than LgH-differentiability (see [4]). Remark that this last is based on the gH-differentiability of every interval-valued function Fα . 7

In order to present an extension of (level-wise) generalized Hukuhara differentiability on Rn , we introduce a concept of differentiability for interval-valued and fuzzyvalued mappings F : K → KC or F : K → FC where K is any open subset of Rn . We will consider first the interval-valued case F : K → KC . b e b − F(x), e b + F(x)] e Definition 6. Let F : K ⊆ Rn → KC , F(x) = (F(x); F(x)) = [F(x) F(x) (0) (0) n and let x ∈ K such that x + h ∈ K, for all h ∈ R with khk < δ for a given δ > 0. We b, w e ∈ Rn , say that F is gH-differentiable at x(0) if and only if there exist two vectors w b = (b bn ), w e = (e en ) and two functions b w w1 , ..., w w1 , ..., w ε(h), e ε(h) with limb ε(h) = lime ε(h) = h→0

h→0

0, such that, for all h , 0, b (0) + h) − F(x b (0) ) = F(x

n X

b j + khkb h jw ε(h),

(6)

j=1

n X e (0) + h) − F(x e (0) ) = h j w F(x e j + khk e ε(h) . j=1

(7)

The interval-valued function DgH F(x(0) ) : Rn → KC defined, for h = (h1 , ..., hn ) ∈ Rn , by   n n X  X b j ; h j w e j  DgH F(x(0) )(h) =  h j w j=1

j=1

is called the gH-differential (or the total gH-derivative) of F at x(0) and DgH F(x(0) )(h) is the interval-valued differential of F at x(0) with respect to h. Note that, in the previous definition, the gH-differential of F at x(0) is well-defined, b, w e ∈ Rn are essentially unique, such as we prove in the next that is, the two vectors w proposition. b e b − F(x), e b + Proposition 1. Let F : K ⊆ Rn → KC , F(x) = (F(x); F(x)) = [F(x) F(x) (0) (0) n e F(x)] and let x ∈ K such that x + h ∈ K, for all h ∈ R with khk < δ. Let us b, w e ∈ Rn , w b = (b bn ), assume that F is gH-differentiable with the two vectors w w1 , ..., w e = (e en ) and the two functions b w w1 , ..., w ε(h), e ε(h) with limb ε(h) = lime ε(h) = 0, verifying h→0

h→0

f∗ , and two functions equations (6) and (7), for all h , 0. If there exist two vectors wb∗ , w b εb∗ (h), εe∗ (h) with lim εb∗ (h) = lim εe∗ (h) = 0, verifying equations (6) and (7), then wb∗ = w h→0

h→0

f∗ = w f∗ = −e e or w and w w. b and wb∗ , and the functions Proof. To prove the result, firstly let us consider the vectors w ∗ b ε(h) and εb (h), which verify equation (6), for all h with 0 < khk < δ. In particular, given j0 ∈ {1, . . . , n}, let us consider h j0 ,k = (h1j0 ,k , h2j0 ,k , . . . , hnj0 ,k ), with h jj00 ,k = 2kδ and h jj0 ,k = 0 for all j , j0 , for k ∈ N. Therefore, from (6), it follows that n X

δ b j +

h j0 ,k

b ε(h j0 ,k ) = (b h jj0 ,k w w j0 + b ε(h j0 ,k )), 2k j=1 8

(8)

n X

δ h jj0 ,k wb∗ j +

h j0 ,k

εb∗ (h j0 ,k ) = (wb∗ j0 + εb∗ (h j0 ,k )). 2k j=1

(9)

Combining (8) and (9), we have that b j0 + b w ε(h j0 ,k ) = wb∗ j0 + εb∗ (h j0 ,k ).

(10)

b j0 − wb∗ j0 . Since b ε(h j0 ,k ), εb∗ (h j0 ,k ) → 0 when k → +∞, and from (10), it implies that w In consequence, extending the previous equality for all j0 ∈ {1, . . . , n}, it is derived that f∗ , and the functions b = wb∗ . In a similar way, and reasoing with the vectors w e and w w ∗ e ε(h) and εe (h), it follows that |b w j0 + b ε(h j0 ,k )| = |wb∗ j0 + εb∗ (h j0 ,k )|,

(11)

f∗ j0 |. By its extension to every j0 ∈ {1, . . . , n}, it follows that what implies that |e w j0 | = |w ∗ f j |, for all j ∈ {1, . . . , n}. To finish the proof, let us suppose that there exist two |e w j | = |w f∗ j1 , 0 and w f∗ j2 , 0, since in other e j1 = −w e j2 = w different indexes, j1 , j2 , such that w ( j1 , j2 ),k case the result would be proved. In this case, let us define h = 21 (h j1 ,k + h j2 ,k ), √ f∗ , we e and for w with khk = 2k2δ < δ. By substitution of h( j1 , j2 ),k in equation (7) for w have that √ √   δ  2δ ( j1 , j2 ),k δ  2δ ∗ ( j1 , j2 ),k e j1 + w e j2 + e e j2 + e w ε(h ) = −e w j1 + w ε (h ) , 2k 2k 2k 2k that is, e j2 + |e w j1 + w



e j1 + w e j2 + 2e ε(h( j1 , j2 ),k )| = | − w



2e ε∗ (h( j1 , j2 ),k )|,

for all k ∈ N, what implies that e j2 | = | − w e j1 + w e j2 |, |e w j1 + w e j1 = 0. This last is a contradiction with the previous supposition that and therefore w f∗ j , f∗ j , for all j ∈ {1, . . . , n}, or w e j = −w e j1 , 0. In consequence, it is verified that w ej = w w for all j ∈ {1, . . . , n}. As an immediate consequence of the previous result, we have the following. Corollary 1. Under the conditions of Proposition 1, we have that     n n n n X  X X  X v j w  =  v j wb∗ j ; v j w  v j w f∗ j  , b e ; j j       j=1 j=1 j=1 j=1 for all v = (v1 , . . . vn ). For a gH-differentiable function we define the gH-gradient as a vector of intervals b e Definition 7. Let F : K ⊆ Rn → KC , F(x) = (F(x); F(x)) be gH-differentiable at e j ), j = 1, 2, ..., n, is called x(0) ∈ K. The n-dimensional vector of intervals w j = (b w j ; w (0) the gH-gradient of F at x and denoted by ∇gH F(x(0) ) = (w1 , w2 , ..., wn ) . 9

Note that the gH-gradient is well-defined, since, by Proposition 1, its formulation is unique. Remark 1. Expressed in terms of the lower and the upper functions F(x) and F(x), the gH-differentiability conditions (6)-(7) are the following: (a)

1 2



 P n b j + khkb F(x(0) + h) + F(x(0) + h) − F(x(0) ) − F(x(0) ) = h jw ε(h) j=1

P n (0) (0) (0) (0) e j + khk e F(x + h) − F(x + h) + F(x ) − F(x ) = h j w ε(h) . (b) j=1 b e b Remark that in the form F(x) = (F(x); F(x)) the requirements on the h i two functions F(x) e and F(x) are independent; instead, in the form F(x) = F(x), F(x) the requirements on 1 2

the two functions F(x) and F(x) are interdependent. This is a reason for a preference in the use of the midpoint-radius representation. e is a valid Remark 2. In the above definition of gH-differentiability, we have that if w vector for condition (7), then, by Proposition 1, only −e w is also valid. To have a e we will chose, between w e and −e unique vector w w the one for which the first non-zero ei∗ > 0, where i∗ is the minimum index i for which w ei , 0 component is positive, i.e., w ei = 0 for all i no problem arises). (if w   b j ; w e j , j = 1, 2, ..., n, correRemark 3. We will see in section 4 that the intervals w spond to the partial gH-derivatives of F at x(0) , defined as the directional gH-derivatives in the canonical directions of Rn . b is differentiable at Condition (6) is equivalent to say that the midpoint function F x ; we then have that (0)

b e Proposition 2. For F : K ⊆ Rn → KC , F(x) = (F(x); F(x)) to be gH-differentiable (0) b at x ∈ K, it is necessary that the midpoint function F is differentiable at x(0) ; for its b (0) ) b j , j = 1, ..., n; =w partial derivatives the following is true: ∂F(x ∂x j e is differentiable Condition (7) is more general than to say that the radius function F e (0) ) ∂ F(x (0) (0) e is differentiable at x with partial derivatives at x and, when F ∂x j , then condiej = tion (7) is satisfied with w

e (0) ) ∂F(x ∂x j .

A first property relates gH-differentiability of F to its continuity and to gH-difference. b e Proposition 3. Let F : K ⊆ Rn → KC , F(x) = (F(x); F(x)), and let x(0) ∈ K such (0) that x + h ∈ K, for all h ∈ R with |h| < δ. If F is gH-differentiable at x(0) then F is continuous at x(0) and limh→0 F(x0 + h) gH F(x(0) ) = 0. b 0 + h) − Proof. Taking the limit for h → 0 in (6) and (7) we obtain that limh→0 F(x b and F e are continuous at x(0) b (0) ) = 0 and limh→0 F(x e 0 + h) − F(x e (0) ) = 0 so that F F(x and limh→0 F(x0 + h) gH F(x(0) ) = 0 10

From the previous concept of gH-differentiability, for the case K ⊆ R, the gHderivative for an interval-valued function (see [18, 28]) can be derived, as follows. b e Proposition 4. Let F : K ⊆ R → KC , F(x) = (F(x); F(x)), and let x(0) ∈ K such that (0) x + h ∈ K, for all h ∈ R with |h| < δ. Then, F is gH-differentiable at x(0) if and only if there exist w(1) , w(2) ∈ R such that F(x(0) + h) gH F(x(0) ) (12) = (w(1) ; w(2) ), h→0 h 0 (0) g (0) (1) (2) 0 0 d where FgH (x(0) ) = (F ) is the gH-derivative of the gH (x ); F gH (x )) = (w ; w interval-valued function F at x(0) . lim

Proof. Firstly, in midpoint notation, let us consider the following equality for any h ∈ R:   e (0) + h) − F(x e (0) )  b (0) + h) − F(x b (0) ) F(x F(x(0) + h) gH F(x(0) )  F(x  . =  ; (13)  h h |h| To say that there exist w(1) , w(2) ∈ R such that (12) is fulfilled, taking into account (13) and the metric space (KC , H), is equivalent to say that there exist b (0) + h) − F(x b (0) ) F(x lim h→0 h e (0) + h) − F(x e (0) ) F(x lim h→0 |h|

=

w(1) ,

=

w(2) ,

what, at the same time, is equivalent to say that there exist two functions ε(1) (h), ε(2) (h), with lim ε(i) (h) = 0, i = 1, 2, such that h→0

b (0) + h) − F(x b (0) ) = hw(1) + |h|ε(1) (h), F(x

(14)

e (0) + h) − F(x e (0) ) = |h|w(2) + |h|η(h) ≥ 0. F(x

(15)

Then, if we define ε(2) (h) = η(h) when h ≥ 0 and ε(2) (h) = −η(h) when h < 0, we get e (0) + h) − F(x e (0) ) = hw(2) + |h|ε(2) (h) , F(x (16) and limε(2) (h) = 0.

h→0

e (0) + h) − F(x e (0) ) = hw(2) + |h|ε(2) (h) , F(x

(17)

i.e., taking int account (14) and (16), by Definition 6, F is gH-differentiable at x(0) . 

According to i Definition 6, a necessary condition for gH-differentiability of F =  h b e b (equivalently the sum F + F = 2F) b F; F = F, F is that the midpoint function F 11

b : K −→ R. An is differentiable at x(0) in the usual sense, as a real-valued function F immediate sufficient condition for the gH-differentiability is obtained if also the radius e (equivalently the difference F − F = 2F), e is differentiable as a real-valued function F function. b e Proposition 5. Let F : K ⊆ Rn → KC , F(x) = (F(x); F(x)), and let x(0) ∈ K such that (0) n b e : K −→ R be both x + h ∈ K for all h ∈ R with |h| < δ. Let F : K −→ R and F (0) (0) differentiable at x . Then F is gH-differentiable at x and   n n e (0) )  b (0) ) X X ∂ F(x ∂ F(x (0)  , h ∈ Rn . DgH F(x )(h) =  h j ; h j  ∂x ∂x j j j=1 j=1 b and Proof. Indeed, in Definition 6, condition (6) follows from the differentiability of F e condition (7) follows from the differentiability of F. Proposition 6. Let F : K ⊆ Rn → KC , F(x) = (F(x); F(x)), and let x(0) ∈ K such that x(0) + h ∈ K for all h ∈ R with |h| < δ. Let F : K −→ R and F : K −→ R be both differentiable at x(0) . Then F is gH-differentiable at x(0) and   n n X G j + G j X G j − G j   , h ∈ Rn DgH F(x(0) )(h) =  h j ; h j  2 2 j=1 j=1 where Gj = b= Proof. Consider F

F+F 2

∂F(x(0) ) ∂F(x(0) ) and G j = , j = 1, ..., n. ∂x j ∂x j F−F b 2 ; from the differentiability of F and F both F G j +G j G j −G j e (0) ) b (0) ) ∂F(x ∂F(x = 2 and ∂x j = 2 and the conclusion ∂x j

e= and F

e are differentiable with and F follows from proposition 5.

It is interesting to observe that our definition of gH-differentiability can be expressed in terms of gH-difference. b e Proposition 7. Let F : K ⊆ Rn → KC , F(x) = (F(x); F(x)), and let x(0) ∈ K such that (0) x + h ∈ K, for all h ∈ R with |h| < δ. Then F is gH-differentiable at x(0) if and only e ∈ Rn such that the following limit condition is b ∈ Rn and w if there exist two vectors w true (0 is here the interval (0; 0))   F(x(0) + h) gH F(x(0) ) gH W(h) lim =0 (18) h→0 khk where W(h) is the interval-valued function defined by   n n  X X b j ; h j w e j  . W(h) =  h j w j=1 j=1 In this case, for the differential function, we have DgH F(x(0) )(h) = W(h). 12

Proof. We have that the gH-differences for the intervals appearing in the limit (18) b e always exist; for simplicity, denote by Φ(h) = (Φ(h), Φ(h)) the interval appearing in the numerator   Φ(h) = F(x(0) + h) gH F(x(0) ) gH W(h) so that b Φ(h) =

b (0) + h) − F(x b (0) ) − F(x

n X

bj h jw

j=1

and e Φ(h) =

n X F(x (0) (0) e e j . + h) − F(x ) − h j w e j=1

b is clearly equivalent to First of all, differentiability of the midpoint function F e Φ(h) = 0; on the other hand, lim khk = 0 if and only if

b lim Φ(h) h→0 khk

h→0

lim

P n (0) (0) e e F(x e j + h) − F(x ) − h j w j=1 khk

h→0

= 0,

η(h) = 0 such that i.e., if and only if there exists a function e η(h) with lime h→0

n X e (0) + h) − F(x e (0) ) = khk e F(x e j ≥ 0. η(h) + h j w j=1 Then, if we define e ε(h) = e η(h) when

n P j=1

e j ≥ 0 and e h jw ε(h) = −e η(h) when

n P j=1

e j < 0, h jw

we get n X e (0) + h) − F(x e (0) ) = h j w F(x e j + khk e ε(h) j=1 and lime ε(h) = 0.

h→0

We then have that condition (7) is equivalent to lim Φ(h) khk = 0 and the conclusion follows from the definition of the limit for an interval-valued function in terms of the limits of its midpoint and radius components. The final expression for the differential is obvious. e

The last proposition above allows an interesting interpretation of the total gHderivative DgH F(x(0) ) in terms of a local tangency condition to F(x) at x(0) : Definition 8. We say that the pair (F(x(0) ), DgH F(x(0) )) defines the (abstract) intervalvalued gH-tangent plane to F at x(0) if and only if F is gH-differentiable at x(0) and DgH F(x(0) ) is the corresponding total gH-derivative (or gH-differential function). The 13

value of the gH-tangent plane to F in x(0) , valuated at a point x, is identified with the pair (F(x(0) ), DgH F(x(0) )(x − x(0) )) and has the following limit property:   F(x) gH F(x(0) ) gH DgH F(x(0) )(x − x(0) )

= 0. (19) lim x→x(0)

x − x(0)

At the end of section 5 we will give an example and we will show the gH-tangent plane in terms of the (feasible) directional derivatives at a minimum (non dominated) point of a linearly constrained minimization problem. 3.2. The case of fuzzy-valued functions For a fuzzy-interval-valued function, the definition below gives the concept of LgHdifferentiability, as follows. cα (x); F fα (x)) = [F cα (x) − F fα (x), F cα (x) + Definition 9. Let F : K ⊆ Rn → FC , Fα (x) = (F (0) (0) f Fα (x)] for each α ∈ [0, 1], and let x ∈ K such that x + h ∈ K, for all h ∈ Rn with khk < δ. (A) We say that F is level-wise generalized Hukuhara differentiable (LgH-differentiable, bα , w eα ∈ for short) at x(0) if and only if for each α ∈ [0, 1] there exist two vectors w bα = (b bα,n ), w eα = (e eα,n ), and two functions b Rn , w wα,1 , ..., w wα,1 , ..., w εα (h), e εα (h) with limb εα (h) = lime εα (h) = 0, such that h→0

h→0

cα (x(0) + h) − F cα (x(0) ) = F

n X

bα, j + khkb h jw εα (h),

(20)

j=1

n X (0) (0) fα (x + h) − F fα (x ) = h j w F eα, j + khk e εα (h) ; j=1

(21)

the family of interval-valued functions DgH Fα (x(0) ) for all α ∈ [0, 1] is called the LgHtotal derivative (or LgH-differential) of F at x(0) . (B) We say that F is (fuzzy) gH-differentiable at x(0) if and only if (B.1) it is LgH-differentiable at x(0) according to point (A), and (B.2) for all h ∈ Rn the LgH-differentials DgH Fα (x(0) )(h) for α ∈ [0, 1] form the α-cuts of a fuzzy interval. In this fuzzy situation, condition (20) is equivalent to the differentiability of the (0) cα at x(0) with partial derivatives ∂Fcα (x ) = w bα, j , j = 1, ..., n; condimidpoint function F ∂x j e tion (21) is more general than to say that the radius function F is differentiable at x(0) (0) fα is differentiable at x(0) with partial derivatives ∂Ffα (x ) , then condition and, when F ∂x j

eα, j = (21) is satisfied with w

fα (x(0) ) ∂F ∂x j .

In the particular case of K ⊆ R, LgH-differentiability coincides with that given by Bede and Stefanini [4], and we get their level-wise gH-derivative, as follows.

14

Proposition 8. Let F : K ⊆ R → FC be a fuzzy function whose α-levels are defined cα (x); F fα (x)) for all α ∈ [0, 1], and let x(0) ∈ K such that x(0) + h ∈ K, as Fα (x) = (F for all h ∈ R with |h| < δ. Then, F is LgH-differentiable at x(0) if and only if for each (2) α ∈ [0, 1] there exist w(1) α , wα ∈ R such that there exists (2) Fα (x0 + h) gH Fα (x(0) ) = (w(1) (22) α ; wα ), h→0 h (1) (2) 0 (0) ] (0) 0 0 where FgH,α (x(0) ) = (F[ gH,α (x ); F gH,α (x )) = (wα ; wα ) is the LgH-derivative of F α (0) at x . lim

Proof. The proof is similar to that given in Proposition 4. Thus, notice that Definition 9 provides a natural extension of LgH-differentiability to the case a fuzzy-valued function of several variables, defined on a subset in Rn .   ∂ F (x(0) ) bα, j ; w eα, j , j = 1, 2, ..., n, define the If the partial LgH-derivatives gH ∂xα j = w level-cuts of a fuzzy interval, we say that the fuzzy-valued function F is gH-differentiable ∂ F(x(0) ) its partial gH-derivative (as a fuzzy interval) at x(0) and we denote again by gH∂x j (0) with respect to x j at x . 4. Directional gH-derivative In this section and in order to obtain necessary optimality conditions later, we introduce directional derivatives. To this purpose, firstly we introduce a definition of directional derivative for interval-valued functions on a subset K ⊆ Rn , which will be later extended to fuzzy-valued functions. Definition 10. Consider an interval-valued function F : K → KC , where K is any open subset of Rn . If d ∈ Rn is any admissible direction at x(0) ∈ K, we say that F has the one-sided directional gH-derivative at x(0) in direction d if the following right limit exists and is an interval: F(x(0) + td) gH F(x(0) ) . t→0+ t   (0) (0) 0 0 d g In midpoint notation, we will write this interval as F gH (x ; d); F gH (x ; d) . If also the left limit for t → 0− of the function above exists and the two are equal, we say that F has the two-sided directional gH-derivative in direction d at x(0) . 0 FgH (x(0) ; d) = lim

If d = e j or d = −e j , where e j is the j-th canonical direction in Rn , then, if they exist, we say that 0 • FgH (x(0) ; e j ) is the right-partial (interval-valued) gH-derivative of F at x(0) with

respect to x j , denoted

∂gH F(x(0) ) ; ∂x+j

15

0 b A) e is the interval −A = (−A; b A)) e • −FgH (x(0) ; −e j ) (the opposite interval of A = (A; (0) is the left-partial (interval-valued) gH-derivative of F at x with respect to x j ,

denoted

∂gH F(x(0) ) . ∂x−j

0 0 If FgH (x(0) ; e j ) and −FgH (x(0) ; −e j ) both exist and are equal, we obtain the partial gHderivative of F at x(0) ∈ K with respect to x j :

F(x(0) + te j ) gH F(x(0) ) ∂gH F(x(0) ) = lim . t→0 ∂x j t Proposition 9. Let F : K → KC , K an open subset of Rn . If F is gH-differentiable at x(0) ∈ K, then all partial gH-derivatives of F exist and, with reference to (6)-(7), it is ∂gH F(x(0) )   b j ; w e j , j = 1, 2, ..., n. = w ∂x j Proof. From the definition of gH-differentiability of F at x(0) , taking h = te j , t , 0, we have that for any j = 1, ..., n,   e (0) + te j ) − F(x e (0) ) b (0) + te j ) − F(x b (0) ); F(x F(x(0) + te j ) gH F(x(0) ) = F(x   = tb w + |t|b ε(te ); te w + |t| e ε(te ) j

j

j

j

with lim b ε(te j ) = lim e ε(te j ) = 0; then the following limit exists t−→0

t−→0

F(x(0) + te j ) gH F(x(0) ) lim t−→0 t

! |t| |t| ej + e bj + b lim w ε(te j ); w ε(te j ) t−→0 t t   e j . b j ; w w

= =

In terms of midpoint notation, the directional gH-derivative is given as follows; we have (here t > 0)   e (0) + td) − F(x e (0) )  b (0) + td) − F(x b (0) ) F(x F(x(0) + td) gH F(x(0) )  F(x  =  ;  t t t 0 (0) (0) 0 0 d g so that FgH (x(0) ; d) = (F gH (x ; d); F gH (x ; d)), with (0) 0 d F gH (x ; d) = (0) 0 Fg gH (x ; d) =

b (0) + td) − F(x b (0) ) F(x , t→0+ t e (0) + td) − F(x e (0) ) F(x lim . t→0+ t lim

b has (standard) directional derivative (F) b 0 (x(0) ; d) It follows that, if the midpoint function F(x) 0 (0) e e and the radius function F(x) has (standard) directional derivative (F) (x ; d), then F(x) has directional gH-derivative and  0 (0)  0 b 0 (x(0) ; d); (F) e (x ; d) . FgH (x(0) ; d) = (F) 16

It is clear that partial gH-derivatives of F, defined in terms of directional gHderivatives with canonical directions, may exist even if F is not gH-differentiable. e is not differentiable in On the other hand, F may be gH-differentiable even if F e and in the standard sense; in this case it is interesting to investigate conditions on F particular on the existence of its partial derivatives. b F) e is gHProposition 10. Let F : K → KC , K an open subset of Rn . If F = (F; differentiable at x(0) ∈ K then e (0) ) exists, then a) if, for a given j, the partial derivative ∂F(x ∂x j   e (0) )  b (0) ) ∂F(x ∂gH F(x(0) )  ∂F(x  ; =  ; ∂x j  ∂x j ∂x j b) if, for a given j, the signed partial derivatives exist and are of opposite sign) then   e (0) )  b (0) ) ∂F(x ∂gH F(x(0) )  ∂F(x  ; =  ; ∂x j ∂x j ∂x−j

e (0) ) ∂F(x ∂x−j

= − ∂F(x ∂x+ e

(0)

)

(they

j

Proof. Assume that F is gH-differentiable at x(0) . Let’s take h = te j and t > 0; condition (7) of gH-differentiability ensures that e 0 + te j ) − F(x e (0) ) F(x = w ej + e ε(te j ) t and, taking the limit for t → 0+ we obtain e 0 + te j ) − F(x e (0) ) F(x = w e j ; lim t→0+ t analogously, if we take h = −e j , t > 0, we also have e 0 − te j ) − F(x e (0) ) F(x = w e j . lim t→0+ t so that e 0 − te j ) − F(x e (0) ) e 0 + te j ) − F(x e (0) ) F(x F(x = lim = w e j lim t→0+ t t t→0+ e (0) ) ∂F(x ∂x j

e 0 +te j )−F(x e (0) ) F(x exists and, from (23) and int e (0) ∂F(x ) . For b), the signed partial deriva∂x j ∂F(x e (0) e (0) ) e j = ∂x− ) = ∂F(x same absolute value w + ∂x j . j

= lim t→0 e j = equality ||a| − |b|| ≤ |a − b|, we obtain w

For a), the partial derivative

tives exist and, from (23), they have the

(23)

17

e exist with ∂F(x− ) , Remark 4. If, for a given j, the signed partial derivatives of F ∂x j (0) (0) (0) e e e ∂F(x ) ) (0) b e ∂F(x ) and ∂F(x ∂x+j ∂x−j , ∂x+j , then F = ( F; F) cannot be gH-differentiable at x . Indeed, from the proof of previous proposition, we have that, if the signed partial derivae (0) ) ∂F(x e (0) ) tives ∂F(x exist, then gH-differentiability requires that they have the same ∂x−j , ∂x+j ∂F(x e (0) ) e (0) e j = ∂x− ) = ∂F(x absolute value w ∂x+j . j e

(0)

An analogous result can be expressed in terms of the functions F and F. h i Proposition 11. Let F : K → KC , K an open subset of Rn . Assume that F = F, F is gH-differentiable at x(0) ∈ K and suppose that the signed partial derivatives of F and F exist at x(0) for a given index j ∈ {1, 2, ..., n}. Denote u−j = u−j = a)

∂F(x(0) ) ∂x−j if u−j

and u+j = =

u+j

∂F(x(0) ) ∂x+j .

=: m j and

∂F(x(0) ) ∂x−j ,

u+j =

∂F(x(0) ) ∂x+j ,

Then

u−j

= u+j =: n j , then

i  m j + n j m j − n j  ∂gH F(x(0) ) h = min{m j , n j }, max{m j , n j } = ; ∂x j 2 2 b)

If u−j = u+j =: p j and u−j = u+j =: q j , then i  p j + q j p j − q j  ∂gH F(x(0) ) h . = min{p j , q j }, max{p j , q j } = ; ∂x j 2 2

Proof. For a), we have that

b (0) ) ∂F(x ∂x j

=

m j +n j 2

and

e (0) ) ∂F(x ∂x j

=

n j −m j 2

and the conclusion b follows from Proposition 10, part a). For b), as F is differentiable, we have that b (0) ) e (0) ) e (0) ) p +q q −p ∂F(x = j 2 j and ∂F(x = − ∂F(x = j 2 j and the conclusion follows from Proposi∂x j ∂x−j ∂x+j tion 10, part b). Remark 5. If, for a given j, the signed partial derivatives of F and F exist and,h withi the notation of Proposition 11, u−j −u−j , u+j −u+j and u−j −u−j , u+j −u+j , then F = F, F cannot be gH-differentiable at x(0) . Indeed, in terms of the radius function, the partial     e (0) ) e (0) ) derivatives exist ∂F(x = 12 u−j − u−j and ∂F(x = 12 u+j − u+j ; on the other hand, the ∂x−j ∂x+j e (0) ) e (0) ) ∂F(x ∂F(x necessary condition ∂x− = ∂x+ is equivalent to u−j − u−j = u+j − u+j , i.e. to j j either u−j − u−j = u+j − u+j or u−j − u−j = u+j − u+j . For a gH-differentiable function, we have the following immediate connection between directional gH-derivative and partial derivatives of the midpoint and radius functions (if they exist). b F) e be an interval-valued function on an open set K ⊆ Rn and Theorem 4. Let F = (F; e has all the (standard) partial derivatives assume that F is gH-differentiable at x(0) ; if F 18

at x(0) and d = (d1 , d2 , ..., dn ) is any fixed direction, then the directional gH-derivative of F at x(0) is given by   n n X (0) X (0)  b e ∂F(x ) ∂F(x )  (0) 0 (24) ; dj FgH (x(0) ; d) =  d j  = DgH F(x )(d). ∂x ∂x j j j=1 j=1 Proof. Follows immediately from gH-differentiability and from the existence of all b and F. e partial derivatives of F b F) e be an interval-valued function on an open set K ⊆ Rn . If Theorem 5. Let F = (F; F is gH-differentiable at x(0) , then all the interval-valued partial gH-derivatives of F exist and, following the notation used in Definition 6,   ∂gH F(x(0) ) 0 e j = [b e j , w e j ], b j ; w b j + w = FgH (x(0) ; e j ) = w w j − w ∂x j for j = 1, . . . , n. Furthermore, all directional gH-derivatives of F exist and   n n X  X 0 b j ; d j w e j  . FgH (x(0) ; d) =  d j w j=1 j=1 Proof. According to Definition 6, F satisfies conditions (6) and (7) for all directions d; in particular, for d = e j , j = 1, . . . , n; and the proof is immediate The results above are analogous to the well-known properties of gH-derivative for a single variable interval-valued function, as in [29], [4] and, more detailed, in [10]. b 1 , x2 ) = x2 +x2 , F(x e 1 , x2 ) = Example 2. Consider F(x1 , x2 ) = (x12 +x2 ; |x1 −x22 |), i.e., F(x 1 2 2 2 2 2 (0) |x1 − x2 |, F(x1 , x2 ) = x1 + x2 −|x1 − x2 | and F(x1 , x2 ) = x1 + x2 +|x1 − x2 |. Let x = (1, 1); b (0) ) b (0) ) b is differentiable at x(0) (with ∂F(x e F and F are not. F = 2 and ∂F(x = 1) while F, ∂x1 ∂x2 (0) e e1 = 1 We see that F satisfies condition (7) of gH-differentiability at x = (1, 1) with w e 1) = 0 and e2 = −2; indeed, F(1, and w e + h1 , 1 + h2 ) − F(1, e 1) = 1 + h1 − (1 + h2 )2 F(1 = h − 2h − h2 1

=

2

2

ε(h)| |h1 − 2h2 + khk e

−h2

with e ε(h) = khk2 −→ 0 for h −→ 0. Then, F is differentiable at (1, 1) with partial gH-derivatives ∂gH F(1, 1) ∂x1 ∂gH F(1, 1) ∂x2

= (2; |1|) = [1, 3] = (1; |−2|) = [−1, 3]. 19

e at x(0) are The signed partial derivatives of F e (0) ) ∂F(x ∂x2+

e (0) ) ∂F(x ∂x1−

= −1,

e (0) ) ∂F(x ∂x1+

= 1,

e (0) ) ∂F(x ∂x2−

= −2

and = 2. Consider direction d = (−2, 1); according to the last proposition, the directional derivative is 0 FgH (x(0) ; (−2, 1)) = (2d1 + d2 ; |−d1 − 2d2 | = (−3; 4).

Analogously, considering the direction d = (2, −1), we obtain 0 FgH (x(0) ; (2, −1)) = (2d1 + d2 ; |d1 − 2d2 | = (3; 4).

b 1 , x2 ) = x 2 + x2 , Example 3. Consider F(x1 , x2 ) = (x12 + x2 ; |x1 − x2 | + x12 ), i.e., F(x 1 b e 1 , x2 ) = |x1 − x2 | + x2 . F b is differentiable at any x = (x1 , x2 ) (with ∂F(x) = 2x1 and F(x ∂x1

1

b ∂F(x) ∂x2

e is differentiable at any (x1 , x2 ) , (a, a). = 1) while F e e at (a, a) are ∂F(a,a) The signed partial derivatives of F = 2a − 1, ∂x− e ∂F(a,a) ∂x2−

1

e ∂F(a,a) ∂x2+

e ∂F(a,a) ∂x1+

= 2a + 1,

= −1 and = 1. e We see that F does not satisfy condition (7) of gH-differentiability at (a, a) , (0, 0) (compare with Remark 4) because its signed partial derivatives are not of opposite sign. e1 = 1 and w e2 = −1; If a = 0, i.e., at point x(0) = (0, 0), F is gH-differentiable with w e 0) = 0 and indeed, F(0, e 1 , h2 ) − F(0, e 0) = |h1 − h2 | + h21 F(h = h − 2h − h2 1

= with

 2 h     khk1 e ε(h) =  h21    − khk

2

2

ε(h)| |h1 − 2h2 + khk e

if

h1 ≥ h2

if

h1 < h2

, lime ε(h) = 0. h→0

In conclusion, (1) F is gH-differentiable at (x1 , x2 ) , (a, a), a ∈ R with partial gH-derivatives ( ∂gH F(x1 , x2 ) ∂gH F(x1 , x2 ) (2x1 ; |2x1 + 1|) if x1 > x2 = , = (1, 1); (2x1 ; |2x1 − 1|) if x1 < x2 ∂x1 ∂x2 (2) F is not gH-differentiable at (x1 , x2 ) = (a, a), a , 0 , and (3) F is gH-differentiable at (x1 , x2 ) = (0, 0) with partial gH-derivative ∂gH F(0, 0) ∂x1 ∂gH F(0, 0) ∂x2

=

(0; |1|) = [−1, 1]

=

(1; |−1|) = [0, 2].

20

4.1. The fuzzy case The directional LgH-derivatives (and directional differentiability), as well as partial derivatives for a fuzzy-valued function can be defined by extending the previous definitions. Definition 11. Let K be an non-empty open subset of Rn and F : K → FC be a fuzzy function, with x(0) ∈ K. If d ∈ Rn is any admissible direction at x(0) ∈ K, then given α ∈ [0, 1], the directional level-wise generalized derivative (directional LgHderivative, for short) of the corresponding interval-valued function Fα : K → KC at x(0) in the direction d is defined as 0 (x(0) ; d) = lim+ F Lgh,α h→0

Fα (x(0) + h · d) gH Fα (x(0) ) h

(25)

if it exists. 0 (1) If F Lgh,α (x(0) ; d) ∈ KC exists for all α ∈ [0, 1], then F is said to have the directional LgH-derivative at x(0) in direction d; (2) We say that F is directionally (or weak) level-wise generalized differentiable (directionally or weak LgH-differentiable) at x(0) if F admits directional LgH-derivatives at 0 x(0) in any direction d ∈ Rn and for all α ∈ [0, 1]; the family of intervals {F Lgh,α (x(0) ; d) : α ∈ [0, 1]} is the directional Lgh-derivative of F at x(0) in direction d, denoted as 0 F Lgh (x(0) ; d); (3) We say that F is directionally (weak) gH-differentiable at x(0) if it is directionally (weak) LgH-differentiable at x(0) in any direction d and the directional LgH-derivative 0 0 F Lgh (x(0) ; d) defines a fuzzy interval (i.e., the intervals F Lgh,α (x(0) ; d) define the levelcuts of a fuzzy interval); (4) F is said directionally (weak) LgH-differentiable on K if it is directionally LgHdifferentiable at each point x(0) ∈ K and is said directionally (weak) gH-differentiable on K if it is directionally gH-differentiable at each point x(0) ∈ K. Theorem 6. F : X → FC be a fuzzy function on an open set X ⊆ Rn , with Fα = cα ; F fα ) an interval-valued function, for each α ∈ [0, 1]. If F is LgH-differentiable at (F x(0) , then all the interval-valued partial gH-derivatives of Fα exist and, following the notation used in Definition 9,   ∂gH Fα (x(0) ) 0 bα, j + w eα, j ], eα, j = [b eα, j , w bα, j ; w wα, j − w = FgH,α (x(0) ; e j ) = w ∂x j for all α ∈ [0, 1], for j = 1, . . . , n. Furthermore, all directional gH-derivatives of Fα , for each α ∈ [0, 1], exist and   n n X  X 0 eα, j  . bα, j ; d j w FgH,α (x(0) ; d) =  d j w j=1 j=1 Proof. According to Definition 9, F satisfies conditions (20) and (21) for all directions d; in particular, for d = e j , j = 1, . . . , n; and the proof is immediate.

21

According to this, we have that the gH-gradient of a LgH-differentiable function for each α-level function is given by ! ∂gH Fα (x(0) ) ∂gH Fα (x(0) ) ,..., ∂x1 ∂xn  (1) (2)  (1) (w ; w ), ..., (w ; w(2) ) ,

∇˜ gH Fα (x(0) ) = =

α,1

α,1

α,n

α,n

bα , F eα are differentiable in the ordinary sense, we have and if F   cα (x(0) ) ∂F fα (x(0) ) cα (x(0) ) ∂F fα (x(0) )   ∂F ∂ F (0) ) . ∇˜ gH Fα (x ) = ( ), ..., ( ; ; ∂x1 ∂xn  ∂x1 ∂xn In this case, using interval notation, the partial interval-valued gH-derivative is   cα (x(0) ) ∂F fα (x(0) ) ∂F cα (x(0) ) ∂F fα (x(0) )  ∂gH Fα (x(0) )  ∂F  .  = − , + ∂x j ∂x j  ∂x j ∂x j ∂x j We define the ith partial LgH-derivative of F at x(0) as the family, if exists, ( ) ∂Lgh F(x(0) ) ∂Lgh Fα (x(0) ) = : α ∈ [0, 1] . ∂xi ∂xi And we define the LgH-gradient of F at x0 as follows. ! ∂Lgh F(x(0) ) ∂Lgh F(x(0) ) , ..., . ∂x1 ∂xn

∇˜ Lgh F(x(0) ) =

(26)

In the one-dimensional case, we can state a rule to calculate the directional LgHderivative via the LgH-derivative, as follows. Theorem 7. Let K be a non-empty open subset of R and F : K → FC be an fuzzy function. If F is LgH-differentiable, then F is directionally LgH-differentiable on K, and 0 0 F Lgh,α (t0 ; d0 ) = F Lgh,α (t0 ) · d0 , (27) for all t0 ∈ K, d0 ∈ R and α ∈ [0, 1]. Proof. Given α ∈ [0, 1], by Definition 11, we have to prove that 0 F Lgh,α (t0 ; d0 ) = lim+ h→0

Fα (t0 + h · d0 ) gH Fα (t0 ) h

(28)

exists. Since F is LgH-differentiable, then, by Definition 4, there exists the following limit: Fα (t0 + h) gH Fα (t0 ) 0 F Lgh,α (t0 ) = lim+ , h→0 h where F 0 (t0 ) ∈ FC . In consequence, we have the following equalities: lim+

h→0

Fα (t0 + h) gH Fα (t0 ) Fα (t0 + h) gH Fα (t0 ) 0 = F Lgh,α (t0 ) = lim− h→0 h h 22

(29)

From equations (28) and (29), it follows: 0 F Lgh,α (t0 ; d0 )

=

= 0

h =h·d0

=

Fα (t0 + h · d0 ) gH Fα (t0 ) · d0 h · d0 Fα (t0 + h0 ) gH Fα (t0 ) lim+ · d0 , h→0 h0 0 Fα (t0 + h ) gH Fα (t0 ) · d0 , lim h→0− h0

lim

h→0+

            

if d0 > 0

(30)

if d0 < 0

0 F Lgh,α (t0 ) · d0

But the reverse of the previous theorem is not true, such as we can see in the following example. Example 4. Let F : K =] − 5, 5[→ FC be a fuzzy function defined as follows: F(t) = h1, 2, 3i|t − 2|; its α-levels, α ∈ [0, 1], are defined through the functions Fα (t) = [h1, 2, 3i|t − 2|]α = [(1 + α)|t − 2|, (3 − α)|t − 2|]. bα ) = 2|t − 2| and F eα ) = (1 − α)|t − 2|. In terms of midpoint notation, we have F b For all α ∈ [0, 1], F is not differentiable at t0 = 2 so that F is not LgH-differentiable at t0 . But F is directionally (weak) LgH-differentiable at t0 in any direction d; indeed, we have lim

h→0+

1 1 [Fα (2 + h · d) gH Fα (2)] = lim+ [1 − α, 3 − α]|h · d| = [1 − α, 3 − α]|d| h→0 h h

and F is directionally LgH-differentiable at t0 in any direction d ∈ R, with 0 F Lgh,α (t0 , d) = [1 − α, 3 − α]|d|.

In the last two examples, we consider a fuzzy function F : K → FC defined on a subset K of Rn . In general, we can show that there exists no direct implication between directional (weak) LgH-differentiability of F at x0 ∈ K and the existence of its LgH-partial derivatives. In Example 5, we provide a directionally LgH-differentiable function at a point, which does not admit a partial LgH-derivative at that point. In Example 6, we illustrate the reverse situation, that is, the partial LgH-derivative exists but the function is not directionally LgH-differentiable. Example 5. Let F : R2 → FC be a fuzzy function defined as follows: F(x1 , x2 ) = h1, 2, 3i|x1 − 2| + x2 for any (x1 , x2 ) ∈ R2 . Its α-levels are, for every α ∈ [0, 1], Fα (x1 , x2 ) = [(1 − α)|t − 2| + x2 , (3 − α)|t − 2| + x2 ], 23

Given x(0) = (2, 0), we have that for any α ∈ [0, 1] and d = (d1 , d2 ) ∈ R2 , 0 F Lgh,α ((2, 0); (d1 , d2 )) = limh→0+ h1 [Fα ((2 + hd1 , hd2 ) gH Fα (2, 0)]

= limh→0+ h1 [(1 − α)|hd1 | + hd2 , (3 − α)|hd1 | + hd2 ] = [(1 − α)|d1 | + d2 , (3 − α)|d1 | + d2 ]

.

Then, F is directionally LgH-differentiable at x(0) = (2, 0) in any direction d ∈ R2 . However, if we take α = 0.5, and define e1 = (0, 1), we have that h i ∂LgH F0.5 (2, 0) = limh→0 h1 F0.5 ((2, 0) + he1 ) gH F0.5 (2, 0) ∂x1 = limh→0 h1 [1.5|h|, 2.5|h|] , which does not exist. In consequence, F does not admit the partial LgH-derivative at x(0) with respect to x1 . Example 6. Let us define a function F : R2 → FC by the means of its α-cuts, as follows. Given (x1 , x2 ) ∈ R2 , Fα (x1 , x2 ) =

[F α (x1 , x2 ), F α (x1 , x2 )]       (1−α)x1 x2  1 x2  , max 0, ,  min 0, (1−α)x 2 2 2 2 x1 +x2 x1 +x2    [0, 0]

=

if (x1 , x2 ) , (0, 0) if (x1 , x2 ) = (0, 0)

,

for all α ∈ [0, 1]. Note that it is well defined. Given α ∈ [0, 1] and x = (x1 , x2 ) ∈ R2 , if x1 , 0 and x2 , 0, then the real-valued functions F α and F α are differentiable with respect to its first and second components at x. This implies that F has partial LgHderivative at that x. If x1 , 0 and x2 = 0, then the directional LgH-derivative of F at x¯, if exists, is given by ∂LgH Fα (x) Fα (x + t · ei ) gH Fα (x) , = lim t→0 ∂xi t for all α ∈ [0, 1], with i ∈ {1, 2}. Given α ∈ [0, 1], the previous limits exists for i = 1, 2 and become as follows: ∂LgH Fα (x) ∂x1

= =

∂LgH Fα (x) ∂x2

=

=

Fα ((x1 , 0) + t(1, 0)) gH Fα (x1 , 0) t→0 t Fα ((x1 + t, 0)) gH Fα (x1 , 0) [0, 0] lim = lim = [0, 0] ∈ KC , t→0 t→0 t t lim

Fα (( x¯1 , 0) + t(0, 1)) gH Fα ( x¯1 , 0) Fα ( x¯1 , t) gH Fα ( x¯1 , 0) = lim t→0 t t      1t 1t min 0, (1−α)x , max 0, (1−α)x x12 +t2 x12 +t2 Fα (x1 , t) lim = lim = [0, 0] ∈ KC . t→0 t→0 t t lim t→0

24

If x1 = 0 and x2 , 0, and proceeding as before, we have that the directional LgH-derivative of F at x exists and is given by ∂LgH Fα (x) = [0, 0] ∈ KC , ∂xi

i = 1, 2.

In a similar way, if x1 = 0 = x2 , we have that ∂LgH Fα (x) = [0, 0] ∈ KC , ∂xi

i = 1, 2.

Therefore, F has the partial LgH-derivatives. On the other hand, let us see that there does not exist the directional LgH-derivative of F at x = (0, 0). To this purpose, consider the direction d = (2, −1) and α ∈ [0, 1). We have that 0 (x; d) = F LgH,α

=

Fα ((0, 0) + h · (2, −1)) gH Fα (0, 0) h " # Fα (2h, −h) −2(1 − α) lim = lim+ ,0 , h→0+ h→0 h 5h lim

h→0+

and the limit does not exist. 5. Necessary optimality conditions We consider the following partial orders over interval sets and over fuzzy numbers. Definition 12. Given A = [a, a], B = [b, b] ∈ KC , we say that (i) A LU B if and only if a ≤ b and a ≤ b, (ii) A LU B if and only if A LU B and A , B, (iii) A ≺LU B if and only if a < b and a < b. Using midpoint notation A = (b a;e a), B = (b b; e b), the partial orders (i) and (iii) above can be expressed as (see [14])   b   b b a ≤ b a 0, there exists δ > 0 such that if ||x + h · d¯ − x|| = ||h · d|| ¯ H(Fα (x + h · d), Fα (x)) < . Then, it follows that there exists h0 > 0 such that ¯ gH F(x) ≺LU [0, 0], F(x + hd) ¯ gH Fα (x) ≺α−LU [0, 0], for all h¯ ∈]0, h0 ]. So given h¯ ∈]0, h0 ], we have that Fα (x + h¯ · d) that is, n o   ¯ − F (x), F α (x + h¯ · d) ¯ − F α (x) ,   min F α (x + h¯ · d) α o  ≺ n  [0, 0], ¯ − F (x), F α (x + h¯ · d) ¯ − F α (x)  α−LU max F α (x + h¯ · d) α what implies that (

¯ − F (x) F α (x + h¯ · d) α ¯ ¯ − F α (x) F α (x + h · d)

< 0 , < 0

¯ ≺α−LU Fα (x). But this is a contradiction with our what is equivalent to Fα (x + h¯ · d) initial assumption that x¯ is a weak α-LU-solution of F, and the proof is completed. Following the same construction, we propose a definition of α-convexity for fuzzyvalued functions under Lgh-directional differentiability. Definition 16. Let F be LgH-directional differentiable and α ∈ [0, 1]. We say that F is α-convex at x ∈ Rn on W ⊆ Rn if 0 F Lgh,α (x; x − x) α−LU Fα (x) gH Fα (x),

(31)

for all x ∈ W. We say that F is α-convex on W if it is α-convex at every x ∈ Rn on W. We say that F is α-convex at x ∈ Rn if it is α-convex on Rn . And we say that F is α-convex on Rn if it is α-convex at every x ∈ Rn . If F is α-convex for all α ∈ [0, 1] we say that it is LU-convex. Under α-convex, we provide the following sufficient optimality condition. Theorem 9 (Sufficient α-optimality condition). Let F be a directional LgH-differentiable fuzzy function. Let x ∈ Rn and α ∈ [0, 1]. If there exists no d ∈ Rn such that 0 F Lgh,α (x; d) ≺α−LU [0, 0],

and F is α-convex at x, then x is a weak α-LU-solution of F. 28

(32)

Proof. To prove the result, let us suppose the contrary, that is, there exists x ∈ Rn such that F(x) 0 for all x ∈ C \ {0}. If we define d¯ = −¯y, it follows that xT d¯ < 0, for all x ∈ C \ {0}. (44) In particular, from (38) and definitions of C(A) and C(G), it is satisfied ( z · d < 0, for all z ∈ ∇˜ Lgh Fα (x) ∇g j (x) · d < 0, j ∈ I(x).

(45)

So, from (45) we have that ∇˜ Lgh Fα (x) · d¯ ≺α−LU [0, 0]. 0 ¯ ⊆ Pn By hypothesis, we have that F LgH,α (x; d) i=1

0 ¯ ⊆ F LgH,α (x; d)

n X ∂LgH Fα ( x¯) i=1

∂xi

∂LgH Fα ( x¯) ∂xi

· d¯i , and then

· d¯i = ∇˜ Lgh Fα (x) · d¯ ≺α−LU [0, 0].

32

(46)

Therefore, 0 ¯ = lim F LgH,α (x; d) + h→0

¯ gH Fα (x) Fα (x + h · d) ≺α−LU [0, 0] h

¯ < δ then This last implies that for any  > 0, there exists δ > 0 such that if ||h · d|| ¯ H(Fα (x + h · d), Fα (x)) < . It follows that there exists h0 > 0 such that ¯ gH F(x) ≺α−LU [0, 0], F(x + hd)

(47)

for all h ∈ (0, h0 ). Let us consider j ∈ I = {1, 2, . . . , m}: • If j < I(x), we have that g j (x) < 0. Since g is a continuous function, there exists ¯ < 0, for all h ∈ (0, h j ). h j > 0 such that g j (x + h j · d) • If j ∈ I(x) then g(x) = 0. From (45), it follows 0 > ∇g j (x) · d = lim+ h→0

¯ − g j (x) g j (x + h · d) . h

¯ − g j (x) < 0, ∀h ∈ (0, h j ), what then there exists h j > 0 such that g j (x + h · d) ¯ implies that g j (x + h · d) < 0, ∀h ∈ (0, h j ). n n oo ¯ < 0, for all j ∈ I, Now, we define h¯ = 21 min h0 , min h j : j ∈ I . So, g j (x + h¯ · d) ¯ ¯ what means that x + h · d is a feasible point for (FP). Therefore, x + h¯ · d¯ is feasible. On ¯ gH Fα (x) ≺α−LU [0, 0], that is, the other hand, by (47), we have that Fα (x + h¯ · d) n o   ¯ − F (x), F α (x + h¯ · d) ¯ − F α (x) ,   min F α (x + h¯ · d) α n o  ≺ [0, 0],  ¯ − F (x), F α (x + h¯ · d) ¯ − F α (x)  α−LU max F α (x + h¯ · d) α what implies that (

¯ − F (x) F α (x + h¯ · d) α ¯ ¯ − F α (x) F α (x + h · d)

< 0 , < 0

¯ ≺α−LU Fα (x). But this is a contradiction with our what is equivalent to Fα (x + h¯ · d) initial assumption that x¯ is a weak α-LU-solution of (FP). In consequence, there exist that λ ∈ R and µ ∈ Rm such that (33)–(35) are fulfilled, and the proof is completed. A consequence of the previous theorems is the following result. Theorem 11. Let us consider (FP), x ∈ D and α ∈ [0, 1], where is F LgH-partial differentiable and LgH-directional differentiable at x, such that 0 F LgH,α (x; d) ⊆

n X ∂LgH Fα ( x¯) i=1

∂xi

· di

for any d ∈ Rn . If x is a α-LU-solution of (FP), with {∇g j ( x¯) : i ∈ I( x¯)} a set of linearly independent vectors, then there exist λ ∈ R and µ = (µ1 , . . . , µm ) ∈ Rm such that conditions (33)-(35) are fulfilled. 33

Proof. The proof is immediate from Theorem 10, since every α-LU-solution of (FP) is weak α-LU-solution of (FP). Theorem 12. Let us consider (FP), x ∈ D and α ∈ [0, 1] such that F is LgH-partial differentiable and LgH-directional differentiable at x, and g j differentiable at x, for all j ∈ {1, . . . , m} and there exist λ ∈ R and µ = (µ1 , . . . , µm ) ∈ Rm , with (λ, µ) , 0, such that m X µ j ∇g j (x), (48) 0 ∈ λ∇˜ Lgh Fα (x) + j=1

µ j g j (x) = 0, (λ, µ) ≥ 0,

(49)

λ , 0,

(50)

or equivalently, X

0 ∈ λ∇˜ Lgh Fα (x) +

µ j ∇g j (x),

(51)

j∈I(x)

(λ, µI(x) ) ≥ 0,

λ , 0,

(52)

If F is α-convex at x, g j is convex at x on D, for j ∈ I( x¯), and there exists d = (d1 , . . . , dn ) ∈ Rn such that 0 F LgH,α (x; d) =

n X ∂LgH Fα (x) i=1

∂xi

· di

(53)

then x is a α-LU-solution of (FP). Proof. To prove the result let us suppose the contrary. So, there exists x ∈ D such that F(x)

Suggest Documents