Nondifferentiable multiobjective programming under generalized dI -invexity Hachem Slimani ∗ , Mohammed Said Radjef Laboratory of Modeling and Optimization of Systems (LAMOS) Operational Research Department, University of Bejaia 06000, Algeria.
Abstract In this paper, we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints. We introduce new concepts of dI -invexity and generalized dI -invexity in which each component of the objective and constraint functions is directionally differentiable in its own direction di . New Fritz-John type necessary and Karush-Kuhn-Tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be weakly efficient, efficient or properly efficient. Moreover, we prove weak, strong, converse and strict duality results for a Mond-Weir type dual under various types of generalized dI -invexity assumptions. Key words: Multiobjective programming, Semi-directionally differentiable functions, generalized dI -invexity, Optimality, Duality, (Weakly or Properly) efficient point
1
Introduction
In optimization theory, convexity plays a very important role in many aspects of mathematical programming including sufficient optimality conditions and duality theorems see, for example, Mangasarian (1969) and Bazaraa et al. (2006). Several classes of functions have been defined for the purpose of weakening the hypothesis of convexity in mathematical programming. Hanson (1981) introduced the concept of invexity for the differentiable functions, generalizing the difference (x − x0 ) in the definition of convex function to any function η(x, x0 ). Further, Ben Israel and Mond (1986) considered a class of ∗ Corresponding author. Tel: 00213 34 21 08 00; Fax: 00213 34 21 51 88. Email addresses:
[email protected] (Hachem Slimani ),
[email protected] (Mohammed Said Radjef).
Preprint submitted to Elsevier Science
22 April 2009
functions called preinvex and also showed that the class of invex functions is equivalent to the class of functions whose stationary points are global minima, see also Craven and Clover (1985). Hanson and Mond (1987) introduced two new classes of functions called type I and type II functions for the scalar optimization problem, which were further generalized to pseudo-type I and quasi-type I by Rueda and Hanson (1988) and sufficient optimality conditions are obtained involving these functions. For other generalizations of invexity, see (Antczak 2003; Bector et al. 1993; Fulga and Preda 2009; Martin 1985; Pini and Singh 1997; Stancu-Minasian 2006) and the references cited therein. On the other hand, Kaul et al. (1994) considered a multiobjective problem involving generalized type-I functions, with scalarization, and obtained some results on optimality and duality, where the Wolfe and Mond-Weir duals are considered. Aghezzaf and Hachimi (2000) introduced new classes of generalized type-I vector-valued functions and, without scalarization, derived various duality results for a nonlinear multiobjective programming problem. Following Jeyakumar and Mond (1992) and Kaul et al. (1994), Hanson et al. (2001) introduced the V-type I problem with respect to η, including positive real-valued functions αi and βj in their definition, and they obtained optimality conditions and duality results under various types of generalized V-type I requirements. For nondifferentiable programming, the invexity was generalized to locally Lipschitz functions with gradient replaced by the Clarke generalized directional derivative given in Clarke (1983), see Craven (1986), Reiland (1990), Ye (1991), Suneja and Srivastava (1997), Antczak (2002), Mishra and Noor (2006) and Suneja et al. (2008). See also (Giorgi and Guerraggio 1996; Mishra et al. 2004, Govil and Mehra 2004, Mishra et al. 2005). However, this extension of invexity is that d-invex problems require the same direction d for the directional derivatives of the objective and constraint functions. This requirement restricts the field of application especially to the locally Lipschitz functions and avoids an important class of functions where the directional derivatives do not exist in all directions. In order to take over this later class of functions, in this paper, we introduce a concept of semi-directionally differentiable functions, where the directional derivatives exist finite in some directions, and we define a new wide class of nondifferentiable functions called dI -invexity, which contains the d-invex functions, in which each component of a vector function is directionally differentiable in its own direction di instead of a same direction d. Then, from Hanson et al. (2001), Mishra and Noor (2006), Suneja and Srivastava (1997), we introduce new classes of problems called dI -V-type I, quasi-, pseudo-, pseudo quasi-, quasi pseudo- dI -V-type I with respect to (ηi )i and (θj )j . New FritzJohn type necessary and Karush-Kuhn-Tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be weakly efficient, efficient or properly efficient. Moreover, a Mond-Weir type dual is formulated 2
and weak, strong, converse and strict duality results are proved under various types of generalized dI -invexity assumptions.
2
Preliminaries and definitions
The following conventions for equalities and inequalities will be used. If x = (x1 , ..., xn ), y = (y1 , ..., yn ) ∈ Rn , then x = y ⇔ xi = yi , i=1,...,n; x < y ⇔ xi < yi , i=1,...,n; x 5 y ⇔ xi 5 yi , i=1,...,n; x ≤ y ⇔ x 5 y and x 6= y. We also note Rq= (resp. Rq≥ or Rq> ) the set of vectors y ∈ Rq with y = 0 (resp. y ≥ 0 or y > 0). Definition 1 Antczak (2005). Let D be a nonempty subset of Rn , η : D×D → Rn and let x0 be an arbitrary point of D. The set D is said to be invex at x0 with respect to η, if for each x ∈ D, x0 + λη(x, x0 ) ∈ D, ∀ λ ∈ [0, 1].
(1)
D is said to be an invex set with respect to η, if D is invex at each x0 ∈ D with respect to the same η. Definition 2 Ben-Israel and Mond (1986). Let D ⊆ Rn be an invex set with respect to η : D × D → Rn . A function f : D → R is called pre-invex on D with respect to η, if for all x, x0 ∈ D, λf (x) + (1 − λ)f (x0 ) = f (x0 + λη(x, x0 )), ∀ λ ∈ [0, 1].
(2)
Definition 3 Antczak (2002). Let D ⊆ Rn be an invex set with respect to η : D × D → Rn . A m-dimensional vector valued function Ψ : D → Rm is pre-invex with respect to η, if each of its components is pre-invex on D with respect to the same function η. Definition 4 Clarke (1983). Let D be a nonempty open set in Rn . A function f : D → R is said to be locally Lipschitz at x0 ∈ D, if there exist a neighborhood v(x0 ) of x0 and a constant K > 0 such that |f (y) − f (x)| 5 Kky − xk, ∀ x, y ∈ v(x0 ), where k.k denotes the Euclidean norm. We say that f is locally Lipschitz on D if it is locally Lipschitz at any point of D. Definition 5 Clarke (1983). If f : D ⊆ Rn → R is locally Lipschitz at 3
x0 ∈ D, the Clarke generalized directional derivative of f at x0 in the direction d ∈ Rn , denoted by f 0 (x0 ; d), is given by f 0 (x0 ; d) = y→x lim sup 0
t→0+
f (y+td)−f (y) . t
And the usual one-sided directional derivative of f at x0 in the direction d is defined by f 0 (x0 ; d) = lim+ λ→0
f (x0 +λd)−f (x0 ) , λ
whenever this limit exists. Obviously, f 0 (x0 ; d) = f 0 (x0 ; d). We say that f is directionally differentiable at x0 , if its directional derivative f 0 (x0 ; d) exists finite for all d ∈ Rn . Definition 6 Ye (1991). Let f : D → RN be a function defined on a nonempty open set D ⊂ Rn and directionally differentiable at x0 ∈ D. f is called d-invex at x0 on D with respect to η, if there exists a vector function η : D × D → Rn such that for any x ∈ D, fi (x) − fi (x0 ) = fi0 (x0 ; η(x, x0 )), for all i = 1, ..., N,
(3)
where fi0 (x0 ; η(x, x0 )) denotes the directional derivative of fi at x0 in the di0 ))−fi (x0 ) rection η(x, x0 ): fi0 (x0 ; η(x, x0 )) = lim+ fi (x0 +λη(x,x . λ λ→0
If inequalities (3) are satisfied at any point x0 ∈ D, then f is said to be d-invex on D with respect to η. Now, we introduce a new concept of semi-directionally differentiable functions. Definition 7 Let D be a nonempty set in Rn and φ : D ×D → Rn a function. • We say that f : D → R is semi-directionally differentiable at x0 ∈ D, if there exists a nonempty subset S ⊂ Rn such that f 0 (x0 ; d) exists finite for all d ∈ S. • We say that f is semi-directionally differentiable at x0 ∈ D in the direction φ(x, x0 ), if its directional derivative f 0 (x0 ; φ(x, x0 )) exists finite for all x ∈ D. Proposition 8 Let f : D → R be a function defined on a nonempty set D ⊂ Rn . If f is directionally differentiable at x0 ∈ D, then it is semi-directionally differentiable at x0 but the converse is not true. Example 9Consider the function f : R2 → R defined by f (x1 , x2 ) =
0,
if x1 = 0 or x2 = 0;
1 + x2 , otherwise. 2
Clearly, the function f is not locally Lipschitz at x0 = (0, 0), because it is not continuous at that point, and it is not directionally differentiable at x0 4
because for d = (a, b) ∈ R2 (a 6= 0 and b 6= 0), f 0 (x0 ; d) = +∞. But f is semi-directionally differentiable at x0 in the direction φ(x, x0 ) = (0, 2 + x21 ). Ye (1991) has defined the invexity for a nondifferentiable vector function f by taking all its components directionally differentiable in the same direction d. In the following definition, we define the dI -invexity at a point x0 for f , where each component fi , i = 1, ..., N is semi-directionally differentiable in its own direction di . Definition 10 Let f : D → RN be a function defined on a nonempty open set D ⊂ Rn and for all i = 1, ..., N, fi is semi-directionally differentiable at x0 ∈ D in the direction ηi : D × D → Rn . f is called dI -invex at x0 on D with respect to (ηi )i=1,N , if for any x ∈ D, fi (x) − fi (x0 ) = fi0 (x0 ; ηi (x, x0 )), for all i = 1, ..., N,
(4)
where fi0 (x0 ; ηi (x, x0 )) denotes the directional derivative of fi at x0 in the di0 ))−fi (x0 ) rection ηi (x, x0 ): fi0 (x0 ; ηi (x, x0 )) = lim+ fi (x0 +ληi (x,x . λ λ→0
If inequalities (4) are satisfied at any point x0 ∈ D, then f is said to be dI -invex on D with respect to (ηi )i=1,N . Remark 11 In the above definition, it does not require that the directional derivative exists finite in all directions for each component fi , i = 1, ..., N . Further, it is not necessary that all the components to be semi-directionally differentiable in the same direction, which may widen the field of applications. In order to illustrate the definition 10, we give the following example. Example 12 Consider the function f = (f1 , f2 ) : R2 → R2 defined by 0,
f1 (x1 , x2 ) =
f2 (x1 , x2 ) =
if x1 = 0 or x2 = 0;
1 + x4 , otherwise. 1 1 + x21 , if x1 6= 0 and x2 = 0;
1 + x22 , if x1 = 0 and x2 6= 0; 0, otherwise.
Clearly, the functions f1 and f2 are not locally Lipschitz and not differentiable at x0 = (0, 0), but only semi-directionally differentiable functions at that point. Further, there exists no function η(x, x0 ) ≡ / (0, 0) such that for any x ∈ R2 , 0 0 f1 (x0 ; η(x, x0 )) and f2 (x0 ; η(x, x0 )) exist and finite, it follows that there exists no function η(x, x0 ) ≡ / (0, 0) for which f is d-invex at x0 on Rn . But f1 is semi-directionally differentiable at x0 in the direction η1 (x, x0 ) = (0, γ(x, x0 )) (or η1 (x, x0 ) = (γ(x, x0 ), 0)) with γ(x, x0 ) 6= 0, ∀ x ∈ R2 , f2 is semidirectionally differentiable at x0 in the direction η2 (x, x0 ) = (α(x, x0 ), β(x, x0 )) 5
with α(x, x0 ) 6= 0, β(x, x0 ) 6= 0, ∀ x ∈ R2 and f is dI -invex at x0 on Rn with respect to (ηi )i=1,2 . Consider the following multiobjective programming problem (M P ) Minimize f (x) = (f1 (x), ..., fN (x)), subject to
g(x) 5 0,
x ∈ D, where f : D → RN , g : D → Rk , D is a nonempty open subset of Rn . Let X = {x ∈ D : g(x) 5 0} be the set of feasible solutions of (MP). For x0 ∈ D, we denote by J(x0 ) the set {j ∈ {1, ..., k} : gj (x0 ) = 0}, J = |J(x0 )| and ˜ 0 ) (resp. J(x ¯ 0 )) the set {j ∈ {1, ..., k} : gj (x0 ) < 0 (resp. gj (x0 ) > 0)}. by J(x ˜ 0 ) ∪ J(x ¯ 0 ) = {1, ..., k} and if x0 ∈ X, J(x ¯ 0 ) = ∅. We have J(x0 ) ∪ J(x We recall some optimality concepts, the most often studied in the literature, for the problem (MP). Definition 13 A point x0 ∈ X is said to be a local weakly efficient solution of the problem (MP), if there exists a neighborhood N (x0 ) around x0 such that f (x) ≮ f (x0 ), for all x ∈ N (x0 ) ∩ X.
(5)
Definition 14 A point x0 ∈ X is said to be a weakly efficient (an efficient) solution of the problem (MP), if there exists no x ∈ X such that f (x) < f (x0 ) (f (x) ≤ f (x0 )).
(6)
Definition 15 An efficient solution x0 ∈ X of (MP) is said to be properly efficient, if there exists a positive real number M such that the inequality fi (x0 ) − fi (x) 5 M [fj (x) − fj (x0 )],
(7)
is verified for all i ∈ {1, ..., N } and x ∈ X such that fi (x) < fi (x0 ), and for a certain j ∈ {1, ..., N } such that fj (x) > fj (x0 ). Now, following Hanson et al. (2001), Mishra and Noor (2006), Suneja and Srivastava (1997) and the definition 10, we define the generalized d-V-type I problems. Definition 16 We say that the problem (MP) is of dI -V-type I at x0 ∈ D with respect to (ηi )i=1,N and (θj )j=1,k , if there exist (N + k) vector functions ηi : X × D → Rn , i = 1, N and θj : X × D → Rn , j = 1, k such that for all x ∈ X: fi (x) − fi (x0 ) = fi0 (x0 ; ηi (x, x0 )), ∀ i = 1, ..., N, (8) 6
−gj (x0 ) = gj0 (x0 ; θj (x, x0 )), ∀ j = 1, ..., k. (9) If the inequalities in (8) are strict (whenever x = 6 x0 ), we say that (MP) is of semi strictly dI -V-type I at x0 with respect to (ηi )i=1,N and (θj )j=1,k . Definition 17 We say that the problem (MP) is of quasi dI -V-type I at x0 ∈ D with respect to (ηi )i=1,N and (θj )j=1,k , if there exist (N + k) vector functions ηi : X × D → Rn , i = 1, N and θj : X × D → Rn , j = 1, k such that for some k vectors µ ∈ RN = and λ ∈ R= : N X
µi [fi (x) − fi (x0 )] 5 0 ⇒
i=1 k X
λj gj (x0 ) = 0 ⇒
j=1
N X i=1 k X
µi fi0 (x0 ; ηi (x, x0 )) 5 0, ∀ x ∈ X, (10) λj gj0 (x0 ; θj (x, x0 )) 5 0, ∀ x ∈ X. (11)
j=1
If the second (implied) inequality in (10) is strict (x 6= x0 ), we say that (MP) is of semi strictly-quasi dI -V-type I at x0 with respect to (ηi )i=1,N and (θj )j=1,k . Definition 18 We say that the problem (MP) is of pseudo dI -V-type I at x0 ∈ D with respect to (ηi )i=1,N and (θj )j=1,k , if there exist (N + k) vector functions ηi : X × D → Rn , i = 1, N and θj : X × D → Rn , j = 1, k such k that for some vectors µ ∈ RN = and λ ∈ R= : N X
µi fi0 (x0 ; ηi (x, x0 ))
=0 ⇒
i=1
N X
µi [fi (x) − fi (x0 )] = 0, ∀ x ∈ X, (12)
i=1 k X
λj gj0 (x0 ; θj (x, x0 ))
=0 ⇒
j=1
k X
λj gj (x0 ) 5 0, ∀ x ∈ X. (13)
j=1
If the second (implied) inequality in (12) (resp. (13)) is strict (x 6= x0 ), we say that (MP) is of semi strictly-pseudo dI -V-type I in f (resp. g) at x0 with respect to (ηi )i=1,N and (θj )j=1,k . If the second (implied) inequalities in (12) and (13) are both strict, we say that (MP) is of strictly-pseudo dI -V-type I at x0 with respect to (ηi )i=1,N and (θj )j=1,k . Definition 19 We say that the problem (MP) is of quasi pseudo dI -V-type I at x0 ∈ D with respect to (ηi )i=1,N and (θj )j=1,k , if there exist (N + k) vector functions ηi : X × D → Rn , i = 1, N and θj : X × D → Rn , j = 1, k such that k for some vectors µ ∈ RN = and λ ∈ R= the relations (10) and (13) are satisfied. If the second (implied) inequality in (13) is strict (x 6= x0 ), we say that (MP) is of quasi strictly-pseudo dI -V-type I at x0 with respect to (ηi )i=1,N and (θj )j=1,k . Definition 20 We say that the problem (MP) is of pseudo quasi dI -V-type I at x0 ∈ D with respect to (ηi )i=1,N and (θj )j=1,k , if there exist (N + k) vector 7
functions ηi : X × D → Rn , i = 1, N and θj : X × D → Rn , j = 1, k such that k for some vectors µ ∈ RN = and λ ∈ R= the relations (12) and (11) are satisfied. If the second (implied) inequality in (12) is strict (x 6= x0 ), we say that (MP) is of strictly-pseudo quasi dI -V-type I at x0 with respect to (ηi )i=1,N and (θj )j=1,k . Remark 21 The problems defined above are different from those in Antczak (2002), Hanson et al. (2001), Mishra and Noor (2006), Mishra et al. (2004), Mishra et al. (2005), Suneja and Srivastava (1997).
3
Optimality conditions
In (Antczak 2002; Mishra and Noor 2006; Mishra et al. 2004, Mishra et al. 2005; Suneja and Srivastava 1997; Ye 1991), sufficient conditions have been given for a feasible point to be weakly efficient. In the following theorems, we give sufficient conditions for a feasible point to be efficient or properly efficient for (MP) under various types of generalized dI -V-type I requirements. Theorem 22 (Sufficiency) Let x0 be a feasible solution for (MP) and suppose that there exist (N + J) vector functions ηi : X × X → Rn , i = 1, N ; θj : X × X → Rn , j ∈ J(x0 ) and scalars µi = 0, i = 1, N , such that the following relation is satisfied: N X i=1
X
µi fi0 (x0 ; ηi (x, x0 )) +
N X
µi = 1; λj = 0, j ∈ J(x0 )
i=1
λj gj0 (x0 ; θj (x, x0 )) = 0, ∀ x ∈ X.
(14)
j∈J(x0 )
Moreover, assume that one of the following conditions is satisfied: (a) the problem (MP) is quasi strictly-pseudo dI -V-type I at x0 with respect to (ηi )i=1,N , (θj )j∈J(x0 ) and for µ and λ; (b) the problem (MP) is semi strictly-quasi dI -V-type I at x0 with respect to (ηi )i=1,N , (θj )j∈J(x0 ) and for µ and λ; (c) the problem (MP) is strictly-pseudo dI -V-type I at x0 with respect to (ηi )i=1,N , (θj )j∈J(x0 ) and for µ and λ. Then x0 is an efficient solution for (MP). PROOF. Suppose that x0 is not an efficient solution of (MP). Then there exists a feasible solution x ∈ X of (MP) such that f (x) ≤ f (x0 ), which implies that N X
µi [fi (x) − fi (x0 )] 5 0.
i=1
8
(15)
From the above inequality and the condition (a), we obtain N X
µi fi0 (x0 ; ηi (x, x0 )) 5 0.
(16)
i=1
By using the relation (14), we deduce that X
λj gj0 (x0 ; θj (x, x0 )) = 0,
(17)
j∈J(x0 )
which implies, from the condition (a) (in view of definition 19), that X
λj gj (x0 ) < 0.
j∈J(x0 )
The last inequality contradicts the fact that gj (x0 ) = 0, ∀ j ∈ J(x0 ) and hence the conclusion follows. The proof of the part (b) is very similar to the proof of part (a), except that for this case the inequality (16) becomes strict () and, using the reverse implication in (11), we get the contradiction again. By condition (c), from gj (x0 ) = 0, λj = 0, ∀ j ∈ J(x0 ), in view of the reverse implication in (13), we obtain X
λj gj0 (x0 ; θj (x, x0 )) < 0, ∀ x ∈ X \ {x0 }.
j∈J(x0 )
By using the relation (14), we deduce N X
µi fi0 (x0 ; ηi (x, x0 )) > 0, ∀ x ∈ X \ {x0 },
i=1
which implies, according to the relation (12) (for strictly-pseudo dI -V-type I problem), that N X
µi [fi (x) − fi (x0 )] > 0, ∀ x ∈ X \ {x0 }.
(18)
i=1
Thus (15) and (18) contradict each other, hence x0 is an efficient solution of (MP). This completes the proof. ¥ In order to illustrate the result obtained, we shall give an example of multiobjective optimization problem in which the efficient solution will be obtained by the application of theorem 22, whereas it will be impossible to apply for this purpose the sufficient optimality conditions given in (Antczak 2002; Mishra and Noor 2006; Mishra et al. 2004, Mishra et al. 2005; Suneja and Srivastava 1997; Ye 1991). 9
Example 23 We consider the following multiobjective optimization problem Minimize f (x) = (f1 (x), f2 (x), f3 (x)), subject to g(x) 5 0,
(19)
where f : R2 → R3 and g : R2 → R defined by 0,
f1 (x1 , x2 ) =
if x1 = 0 or x2 = 0;
−1 + x4 , otherwise. 1 −1 + x21 , if x1 6= 0 and x2 = 0;
f2 (x1 , x2 ) = −1 + x22 , if x1 = 0 and x2 6= 0; 0, 2,
f3 (x1 , x2 ) = g(x1 , x2 ) =
otherwise. if x1 = 0 and x2 = 0;
2 + x2 + x2 , otherwise. 2 1 0, if x1 = 0 or x2 = 0; −1 + x2 , otherwise. 1
The set X of feasible solutions of problem is nonempty. Clearly, the functions f1 , f2 and g are not locally Lipschitz and not differentiable at x0 = (0, 0) ∈ X, but only semi-directionally differentiable functions at that point. There exists no function η : R2 × R2 → R2 , η ≡ / (0, 0), and λ ∈ R= such that 3 X
fi0 (x0 ; η(x, x0 )) + λg 0 (x0 ; η(x, x0 )) = 0, ∀ x ∈ X.
i=1
Then, the sufficient optimality conditions given in (Antczak 2002; Mishra and Noor 2006; Mishra et al. 2004, Mishra et al. 2005; Suneja and Srivastava 1997; Ye 1991) are not applicable. However, there exist vector functions η1 (x, x0 ) = (1+x21 , 0), η2 (x, x0 ) = (1+x21 , 1+x22 ), η3 (x, x0 ) = (x1 , x2 ), θ(x, x0 ) = (0, 1+x22 ) and scalars µ1 = µ2 = 0, µ3 = λ = 1 such that the relation (14) is satisfied and the problem (19) is semi strictly-quasi dI -V-type I at x0 with respect to (ηi )i=1,2,3 , θ and for µ = (µ1 , µ2 , µ3 ) and λ. It follows that, by theorem 22, x0 is an efficient solution for the given multiobjective optimization problem. In the following theorem, we give sufficient optimality conditions for a feasible solution to be properly efficient for (MP). Theorem 24 (Sufficiency) Let x0 be a feasible solution for (MP) and suppose that there exist (N + J) vector functions ηi : X × X → Rn , i = 1, N ; θj : X × X → Rn , j ∈ J(x0 ) and scalars µi > 0, i = 1, N ; λj = 0, j ∈ J(x0 ) such that the relation (14) of theorem 22 is satisfied. 10
Moreover, assume that one of the following conditions is satisfied: (a)
the problem (MP) is dI -V-type I at x0 with respect to (ηi )i=1,N and (θj )j∈J(x0 ) ; (b) the problem (MP) is pseudo quasi dI -V-type I at x0 with respect to (ηi )i=1,N , (θj )j∈J(x0 ) and for µ and λ; (c) the problem (MP) is semi strictly-pseudo dI -V-type I in g at x0 with respect to (ηi )i=1,N , (θj )j∈J(x0 ) and for µ and λ. Then x0 is a properly efficient solution for (MP).
PROOF. By condition (a), for all x ∈ X we have N X
µi fi (x) −
i=1
N X
(8)
N z}|{ X
µi fi (x0 ) =
i=1
µi fi0 (x0 ; ηi (x, x0 ))
i=1 (14)
z}|{
= −
X
λj gj0 (x0 ; θj (x, x0 ))
j∈J(x0 ) (9)
z}|{
=
X
λj gj (x0 ) = 0.
j∈J(x0 )
Thus
N X i=1
µi fi (x) =
N X
µi fi (x0 ) for all x ∈ X with µ > 0. Hence, from theorem
i=1
1 of Geoffrion (1968), x0 is a properly efficient solution for (MP). By condition (b), from gj (x0 ) = 0, λj = 0, ∀ j ∈ J(x0 ) (in view of definition 20), we obtain X
λj gj0 (x0 ; θj (x, x0 )) 5 0, ∀ x ∈ X.
j∈J(x0 )
From the above inequality and the relation (14), it follows that N X
µi fi0 (x0 ; ηi (x, x0 )) = 0, ∀ x ∈ X.
i=1
By using the relation (12) (in view of definition 20), we deduce that N X i=1
µi fi (x) =
N X
µi fi (x0 ), ∀ x ∈ X,
i=1
and the conclusion follows. For the proof of part (c), we proceed as in part (b) and using the reverse implication in (13), we get 11
X
λj gj0 (x0 ; θj (x, x0 )) < 0, ∀ x ∈ X \ {x0 }.
j∈J(x0 )
In the same manner as in (b), we get
N X
µi fi (x) =
i=1
N X
µi fi (x0 ), ∀ x ∈ X and
i=1
it follows that x0 is a properly efficient solution for (MP). This completes the proof. ¥
Now, we give an example of dI -V-type I problem in which the properly efficient solution will be obtained by the application of theorem 24. Besides, we prove that the d-invexity of the functions involved in this problem, f and g, is not verified. Example 25 We consider the following multiobjective optimization problem Minimize f (x) = (f1 (x), f2 (x)), subject to g(x) 5 0,
(20)
where f : R2 → R2 and g : R2 → R defined by f1 (x1 , x2 ) =
f2 (x1 , x2 ) =
0,
if x1 = 0 or x2 = 0;
1 + x2 , otherwise. 1 1 + x21 , if x1 6= 0 and x2 = 0;
1 + x22 , if x1 = 0 and x2 6= 0; 0,
0,
g(x1 , x2 ) =
otherwise.
if x1 = 0 or x2 = 0;
−1 + x2 , otherwise. 1
The set X of feasible solutions of problem is nonempty. Clearly, the functions f1 , f2 and g are not locally Lipschitz and not differentiable at x0 = (0, 0) ∈ X, but only semi-directionally differentiable functions at that point. • There exist vector functions η1 (x, x0 ) = (1+x41 , 0), η2 (x, x0 ) = (1+x41 , 1+x42 ), θ(x, x0 ) = (0, 1 + x42 ) and scalars µ1 = µ2 = λ = 21 such that the relation (14) is satisfied and the problem (20) is dI -V-type I at x0 with respect to (ηi )i=1,2 and θ. It follows that, by theorem 24, x0 is a properly efficient solution for the given multiobjective optimization problem. • Let us see that f and g are not d-invex at x0 with respect to a same η. In fact, there exists no function η : R2 × R2 → R2 , η ≡ / (0, 0), for which f1 , f2 and g are d-invex at x0 . To prove the necessary condition, we need to prove the following lemma. 12
Lemma 26 Suppose that (i) x0 is a (local) weakly efficient solution for (MP); ˜ 0 ) and there exist vector functions ηi : (ii) gj is continuous at x0 for j ∈ J(x n X × D → R , i = 1, N , θj : X × D → Rn , j ∈ J(x0 ) which satisfy at x0 with respect to η : X × D → Rn the following inequalities, fi0 (x0 ; η(x, x0 )) 5 fi0 (x0 ; ηi (x, x0 )), ∀ x ∈ X, ∀ i = 1, N ,
(21)
gj0 (x0 ; η(x, x0 )) 5 gj0 (x0 ; θj (x, x0 )), ∀ x ∈ X, ∀ j ∈ J(x0 ).
(22)
Then the system of inequalities fi0 (x0 ; ηi (x, x0 )) < 0, i = 1, N ,
(23)
gj0 (x0 ; θj (x, x0 )) < 0, j ∈ J(x0 ),
(24)
has no solution x ∈ X.
PROOF. Let x0 ∈ X be a local weakly efficient solution for (MP) and suppose there exists x˜ ∈ X such that the inequalities (23) and (24) are true. For i ∈ {1, ..., N }, let ϕfi (x0 , x˜, τ ) = fi (x0 + τ η(˜ x, x0 )) − fi (x0 ). We observe that this function vanishes at τ = 0 and ϕ (x ,˜ x,τ )−ϕfi (x0 ,˜ x,0) = lim+ fi (x0 +τ η(˜xτ,x0 ))−fi (x0 ) lim fi 0 τ τ →0 τ →0+ fi0 (x0 ; ηi (˜ x, x0 )) < 0 using (21) and (23).
= fi0 (x0 ; η(˜ x, x0 )) 5
It follows that, for all i ∈ {1, ..., N }, ϕfi (x0 , x˜, τ ) < 0 if τ is in some open interval (0, δfi ). Thus, for all i ∈ {1, ..., N }, fi (x0 + τ η(˜ x, x0 )) < fi (x0 ), τ ∈ (0, δfi ).
Similarly, by defining ϕgj (x0 , x˜, τ ) = gj (x0 + τ η(˜ x, x0 )) − gj (x0 ), j ∈ J(x0 ) and using (22) and (24), we can prove that gj (x0 + τ η(˜ x, x0 )) < gj (x0 ), j ∈ J(x0 ), τ ∈ (0, δgj ) and by definition of J(x0 ), we obtain gj (x0 + τ η(˜ x, x0 )) < 0, j ∈ J(x0 ), τ ∈ (0, δgj ). ˜ 0 ), therefore, there exists δj such that Since gj is continuous at x0 for j ∈ J(x ˜ 0 ), τ ∈ (0, δj ). gj (x0 + τ η(˜ x, x0 )) < 0, j ∈ J(x ˜ 0 )}. Let δ0 = min{δfi , i = 1, N , δgj , j ∈ J(x0 ), δj , j ∈ J(x Then (x0 + τ η(˜ x, x0 )) ∈ Nδ0 (x0 ), τ ∈ (0, δ0 ), where Nδ0 (x0 ) is a neighborhood of x0 . Now (25) fi (x0 + τ η(˜ x, x0 )) < fi (x0 ), i = 1, N , gj (x0 + τ η(˜ x, x0 )) < 0, j = 1, k. 13
(26)
By (25) and (26), we get (x0 + τ η(˜ x, x0 )) ∈ Nδ0 (x0 ) ∩ X, τ ∈ (0, δ0 ). Hence (25) is a contradiction to the assumption that x0 is a (local) weakly efficient solution for (MP). Thus there exists no x ∈ X satisfying the system (23) and (24). ¥
The following lemma given by Weir and Mond (1988) will be used. Lemma 27 Let S be a nonempty invex set in Rn with respect to η : S × S → Rn and let ψ : S → Rm be a pre-invex function on S with respect to the same η. Then either ψ(x) < 0 has a solution x ∈ S, or pt ψ(x) = 0 for all x ∈ S, for some p ∈ Rm ≥, but both alternatives are never true. Antczak (2002) has given necessary conditions for x0 ∈ X to be a weakly efficient solution for (MP) by taking the functions fi , i = 1, N and gj , j ∈ J(x0 ) directionally differentiable in a same direction η(x, x0 ). Now we give necessary optimality criteria by considering each function fi , i = 1, N (resp. gj , j ∈ J(x0 )) semi-directionally differentiable in its own direction ηi (x, x0 ), i = 1, N (resp. θj (x, x0 ), j ∈ J(x0 )). Theorem 28 (Fritz John type necessary optimality conditions) Suppose that (i) x0 is a weakly efficient solution for (MP); ˜ 0 ) and there exist vector functions ηi : (ii) gj is continuous at x0 for j ∈ J(x X × D → Rn , i = 1, N , θj : X × D → Rn , j ∈ J(x0 ) which satisfy at x0 with respect to η : X × D → Rn the inequalities (21) and (22); (iii) for all i = 1, N , fi (for all j ∈ J(x0 ), gj ) is semi-directionally differentiable at x0 in the direction ηi (θj ), X is an invex set with respect to γ : X × X → Rn and the functions fi0 (x0 ; ηi (x, x0 )), i = 1, N and gj0 (x0 ; θj (x, x0 )), j ∈ J(x0 ) are pre-invex of x on X with respect to γ; N +|J(x0 )|
Then there exists (µ, λ) ∈ R≥ N X i=1
µi fi0 (x0 ; ηi (x, x0 )) +
X
such that (x0 , µ, λ) satisfies
λj gj0 (x0 ; θj (x, x0 )) = 0, ∀ x ∈ X.
(27)
j∈J(x0 )
PROOF. If the conditions (i) and (ii) are satisfied, then, by lemma 26 the system (23)-(24) has no solution for x ∈ X. Since, by hypothesis (iii), fi0 (x0 ; ηi (x, x0 )), i = 1, N and gj0 (x0 ; θj (x, x0 )), j ∈ 14
J(x0 ) are pre-invex of x on X with respect to γ, therefore, by lemma 27, there N +|J(x0 )| exists p = (µ, λ) ∈ R≥ such that the relation (27) is satisfied. ¥ Remark 29 For the above theorem, in the case where ηi = η, ∀ i = 1, N , θj = η, ∀ j ∈ J(x0 ), we obtain the theorem 10 of Antczak (2002). Now, we need a constraint qualification to prove the next result. Definition 30 The function g is said to satisfy the dI -constraint qualification at x0 ∈ X with respect to (θj )j∈J(x0 ) , if there exist x¯ ∈ X and θj : X × X → Rn , j ∈ J(x0 ) such that gj0 (x0 ; θj (¯ x, x0 )) < 0, ∀ j ∈ J(x0 ).
(28)
Note that in the dI -constraint qualification, we do not require that gj (¯ x) < 0, ∀ j ∈ J(x0 ) as in (Antczak 2002; Mishra and Noor 2006). Thus, the following Karush-Kuhn-Tucker type necessary optimality conditions for (MP) are satisfied. Theorem 31 (Karush-Kuhn-Tucker type necessary optimality conditions) Suppose that the conditions (i), (ii) and (iii) of theorem 28 are satisfied and the function g satisfies the dI -constraint qualification at x0 ∈ X with respect |J(x0 )| to (θj )j∈J(x0 ) . Then there exist µ ∈ RN such that (x0 , µ, λ) ≥ and λ ∈ R= satisfies the relation (27). N +|J(x )|
0 PROOF. Based on theorem 28, we obtain the existence of (µ, λ) ∈ R≥ such that (x0 , µ, λ) satisfies the relation (27). Now it is enough to prove that µ 6= 0. We proceed by contradiction. If µ = 0, then λ 6= 0 and (27) takes the following form
X
λj gj0 (x0 ; θj (x, x0 )) = 0, ∀ x ∈ X,
j∈J(x0 )
which contradicts the dI -constraint qualification. Hence µ 6= 0. ¥
4
Mond-Weir type duality
In relation to (MP) and using the relation (27), we formulate the following multiobjective dual problem, which is in the format of Mond-Weir (1981). It seems to be an open question to study Wolfe duality in our setting. But since the introduction of Mond-Weir duality, most of the results that have 15
appeared in the literature in recent years hold for both types of duals. (M W D) Maximize f (y) = (f1 (y), ..., fN (y)), subject to N X
µi fi0 (y; ηi (x, y)) +
i=1
X
λj gj0 (y; θj (x, y)) = 0, ∀ x ∈ X.
j∈J(y)
|J(y)|
y ∈ D, µ ∈ RN ≥ , λ ∈ R=
,
ηi : X × D → Rn , ∀ i = 1, ..., N, θj : X × D → Rn , ∀ j ∈ J(y). Let Ybe the set of feasible solutions of problem (MWD); ie,
Y = (y, µ, λ, (ηi )i , (θj )j ) :
N X
µi fi0 (y; ηi (x, y)) +
i=1
X
λj gj0 (y; θj (x, y)) = 0,
j∈J(y) |J(y)| R= ;
RN ≥,
∀ x ∈ X; y ∈ D, µ ∈ λ∈ ηi : X × D → Rn , ∀ i = 1, ..., N ; θj : X × D → Rn , ∀ j ∈ J(y)}. We denote by P rD Y the projection of set Y on D. Theorem 32 (Weak duality) Let x and (y, µ, λ, (ηi )i=1,N , (θj )j∈J(y) ) be feasible solutions for (MP) and (MWD) respectively. Moreover, assume that one of the following conditions is satisfied: (a) the problem (MP) is pseudo quasi dI -V-type I at y with respect to (ηi )i=1,N , (θj )j∈J(y) and for µ and λ (with µ > 0); (b) the problem (MP) is strictly-pseudo quasi dI -V-type I at y with respect to (ηi )i=1,N , (θj )j∈J(y) and for µ and λ. (c) the problem (MP) is quasi strictly-pseudo dI -V-type I at y with respect to (ηi )i=1,N , (θj )j∈J(y) and for µ and λ. Then f (x) f (y).
PROOF. By condition (a) (in view of definition 20), it follows that X
λj gj0 (y; θj (x, y)) 5 0.
(29)
j∈J(y)
Since (y, µ, λ, (ηi )i=1,N , (θj )j∈J(y) ) is feasible for (MWD), (29) implies that N X
µi fi0 (y; ηi (x, y)) = 0.
i=1
16
(30)
Condition (a) and inequality (30) give N X
µi [fi (x) − fi (y)] = 0.
(31)
i=1
Now suppose to the contrary that f (x) ≤ f (y). Then, since µ > 0, we obtain N X
µi [fi (x) − fi (y)] < 0,
(32)
i=1
which contradicts (31). Hence the conclusion follows. The proof of the parts (b) and (c) are very similar to the proof of part (a), except that: for the part (b), the inequality (31) becomes strict (>) and the inequality (32) becomes non strict (5). For the part (c), the inequality (29) becomes strict (). Since µ ≥ 0, then the inequality (32) becomes non strict (5). In the two cases, the inequalities (31) and (32) contradict each other always. This completes the proof. ¥ Remark 33 If we omit the assumption ”µ > 0” in the condition (a) or the word ”strictly” in the condition (b), we obtain, for this part of theorem, f (x) ≮ f (y). Theorem 34 (Strong duality) Let x0 be a weakly efficient solution for (MP) and suppose that the conditions (ii) and (iii) of theorem 28 are satisfied. Assume also that the function g satisfies the dI -constraint qualification at x0 |J(x0 )| with respect to (θj )j∈J(x0 ) . Then there exist µ ∈ RN such that ≥ and λ ∈ R= (x0 , µ, λ, (ηi )i=1,N , (θj )j∈J(x0 ) ) ∈ Y and the objective functions of (MP) and (MWD) have the same values at x0 and (x0 , µ, λ, (ηi )i=1,N , (θj )j∈J(x0 ) ), respectively. If, further, the weak duality between (MP) and (MWD) in theorem 32 holds with the condition (a) without ”µ > 0” (resp. with the condition (b) or (c)), then (x0 , µ, λ, (ηi )i=1,N , (θj )j∈J(x0 ) ) ∈ Y is a weakly efficient (resp. an efficient) solution of (MWD). |J(x0 )|
PROOF. By the theorem 31, there exist µ ∈ RN ≥ and λ ∈ R= N X i=1
µi fi0 (x0 ; ηi (x, x0 )) +
X
such that
λj gj0 (x0 ; θj (x, x0 )) = 0, ∀ x ∈ X.
j∈J(x0 )
It follows that (x0 , µ, λ, (ηi )i=1,N , (θj )j∈J(x0 ) ) ∈ Y . Trivially, the objective function values of (MP) and (MWD) are equal. Next suppose that (x0 , µ, λ, (ηi )i=1,N , (θj )j∈J(x0 ) ) ∈ Y is not a weakly efficient (resp. an efficient) solution of (MWD). Then there exists (y ∗ , µ∗ , λ∗ , (ηi∗ )i=1,N , (θj∗ )j∈J(y∗ ) ) ∈ Y such that f (x0 ) < f (y ∗ ) (resp. f (x0 ) ≤ 17
f (y ∗ )), which violates the weak duality theorem 32 (see the remark 33). Hence (x0 , µ, λ, (ηi )i=1,N , (θj )j∈J(x0 ) ) ∈ Y is indeed a weakly efficient (resp. an efficient) solution of (MWD). ¥ Theorem 35 (Converse duality) Let (y 0 , µ, λ, (ηi )i=1,N , (θj )j∈J(y0 ) ) be a feasible solution of (MWD) with y 0 ∈ X. Moreover, we assume that one of the hypothesis (a) − (c) of theorem 22 (resp. theorem 24 with µ > 0) holds at y 0 in P rX Y , then y 0 is an efficient (resp. a properly efficient) solution of (MP). PROOF. The proof of the above theorem is very similar to the proof of theorem 22 (resp. theorem 24), except that for this case we use the feasibility of (y 0 , µ, λ, (ηi )i=1,N , (θj )j∈J(y0 ) ) for (MWD) instead of the relation (14). ¥ Theorem 36 (Strict duality) Let x0 and (y 0 , µ, λ, (ηi )i=1,N , (θj )j∈J(y0 ) ) be feasible solutions for (MP) and (MWD) respectively, such that N X
µi fi (x0 ) =
i=1
N X
µi fi (y 0 ).
(33)
i=1
Moreover, assume that the problem (MP) is strictly-pseudo quasi dI -V-type I at y 0 with respect to (ηi )i=1,N , (θj )j∈J(y0 ) and for µ and λ. Then x0 = y 0 . PROOF. Suppose that x0 6= y 0 . Since (MP) is strictly-pseudo quasi dI -Vtype I at y 0 , from gj (y 0 ) = 0, λj = 0, ∀ j ∈ J(y 0 ) (in view of definition 20), we obtain X λj gj0 (y 0 ; θj (x0 , y 0 )) 5 0. (34) j∈J(y 0 )
The inequality (34) and the feasibility of (y 0 , µ, λ, (ηi )i=1,N , (θj )j∈J(y0 ) ) for (MWD) give N X
µi fi0 (y 0 ; ηi (x0 , y 0 )) = 0.
(35)
i=1
By inequality (35) and since (MP) is strictly-pseudo quasi dI -V-type I at y 0 , we obtain N X
µi [fi (x0 ) − fi (y 0 )] > 0,
i=1
which contradicts (33). Hence x0 = y 0 . ¥
5
Conclusion
In this paper, we have introduced a new concept of semi-directionally differentiable functions, which extend the locally Lipschitz functions, then we have 18
defined a new class of nondifferentiable functions called dI -invexity in which each component of a vector function is directionally differentiable in its own direction di instead of a same direction d. We have introduced new classes of problems called dI -V-type I, quasi-, pseudo-, pseudo quasi-, quasi pseudo- dI V-type I with respect to (ηi )i and (θj )j . In the framework of the new concepts, we established new Fritz-John type necessary and Karush-Kuhn-Tucker type necessary and sufficient optimality conditions for a feasible point to be weakly efficient, efficient or properly efficient. Moreover, we proved weak, strong, converse and strict duality results for a Mond-Weir type dual under various types of generalized dI -V-type I requirements. The results obtained in this paper generalize and extend the previously known results in this area.
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