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Some nondifferentiable multiobjective programming problems
J. Math. Anal. Appl. 316 (2006) 472–482 www.elsevier.com/locate/jmaa
Some nondifferentiable multiobjective programming problems ✩ S.K. Mishra a,∗ , M.A. Noor b a Department of Mathematics, Statistics and Computer Science, College of Basic Sciences and Humanities,
G.B. Pant University of Agriculture and Technology, Pantnagar-263 145, India b Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
Received 27 April 2005 Available online 25 July 2005 Submitted by William F. Ames
Abstract In this paper it is shown that a relaxation defining the class of generalized d-V-type-I functions leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar non-differentiable case, and avoids the major difficulty of verifying that the inequality holds for the same kernel function. The results obtained in this paper generalize and extend the previously known results in this area. 2005 Elsevier Inc. All rights reserved. Keywords: Multi-objective programming; Duality; Pareto efficient solution; Generalized d-V-invexity
1. Introduction The field of multi-objective programming, also known as vector programming, has grown remarkably in different directions in the settings of optimality conditions and du✩ This research is supported by the Department of Science and Technology, Ministry of Science and Technology, Government of India, under the SERC Fast Track Scheme for Young Scientists 2001–2002 through grant No. SR/FTP/MS-22/2001. * Corresponding author. E-mail addresses: [email protected] (S.K. Mishra), [email protected] (M.A. Noor).
0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.04.067
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ality theory since the 1980s. It has been enriched by the applications of various types of generalizations of convexity theory, with and without differentiability assumptions, and in the framework of continuous time programming, fractional programming, inverse vector optimization, saddle point theory, symmetric duality, variational problems and variational inequality problems, etc. Convexity plays a vital role in many aspects of mathematical programming including sufficient optimality condition and duality theorems see, for example, Mangasarian [13] and Bazaraa et al. [3]. To relax convexity assumptions imposed on the functions in theorems on sufficient optimality and duality, various generalized convexity notions have been proposed. Hanson [8] introduced the class of invex functions, see also [5]. Later, Hanson and Mond [10] defined two new classes of functions called type-I and type-II functions, and sufficient optimality conditions were established by using these concepts. Rueda and Hanson [24] further extended type-I functions to the classes of pseudo-type-I and quasitype-I functions and obtained sufficient optimality criteria for a nonlinear programming problem involving these functions. Kaul et al. [12] considered a multiple objective nonlinear programming problem involving generalized type-I functions and obtained some results on optimality and duality, where the Wolfe and Mond–Weir duals are considered. Univex functions were introduced and studied by Bector et al. [4]. Rueda et al. [25] obtained optimality and duality results for several mathematical programs by combining the concepts of type-I and univex functions. Mishra [16] considered a multiple objective nonlinear programming problem and obtained optimality, duality and saddle point results of a vector-valued Lagrangian by combining the concepts of type-I, pseudo-type-I, quasi-type-I, quasi-pseudo-type-I, pseudo-quasi-type-I and univex functions. Aghezzaf and Hachimi [1] introduced new classes of generalized type-I vector-valued functions and derived various duality results for a nonlinear multiobjective programming problem. It is known that, despite substituting invexity for convexity, many theoretical problems in differentiable programming can also be solved, see Hanson [8], Egudo and Hanson [7], and Jeyakumar and Mond [11]. But the corresponding conclusions cannot be obtained in nondifferentiable programming with the aid of invexity introduced by Hanson [8] because the existence of a derivative is required in the definition of invexity. There exists a generalization of invexity to locally Lipschitz functions, with derivative replaced by the Clarke generalized gradient, see Craven [6], Reiland [23], Mishra and Mukherjee [18,19], Mishra [14,15], and Mishra and Giorgi [17]. However, Antczak [2] used directional derivative, in association with a hypothesis of an invex kind following Ye [28]. The necessary optimality conditions in Antczak [2] are different from those cited in the literature. In the present paper, we consider a nondifferentiable and multiobjective programming problem and derive some Karush–Kuhn–Tucker type of sufficient optimality conditions for a (weakly) Pareto efficient solution to the problem involving the new classes of directionally differentiable generalized type-I functions. Furthermore, the Mond–Weir type and general Mond–Weir type of duality results are also obtained in terms of right differentials of the aforesaid functions involved in the multiobjective programming problem.
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2. Preliminaries In this section, we extend the concepts of weak strictly-pseudo-quasi-type I, strong pseudo-quasi-type I, weak quasi-strictly-pseudo type I and weak strictly pseudo-type I functions introduced in Aghezzaf and Hachimi [1] in the setting of Antczak [2] and give some preliminaries. Let X be any subset of R n . Definition 2.1. A subset X is said to be an α-invex set, if there η : X × X → R n , α(x, u) : X × X → R+ such that u + λα(x, u)η(x, u) ∈ X,
∀x, u ∈ X, λ ∈ [0, 1].
Note that, for α(x, u) = 1, α-invex set becomes the invex set. It is well known that the α-invex may not be convex sets, see Noor [21]. Definition 2.2. The function f on the α-invex set is said to be α-preinvex function, if there exist η : X × X → R n , α(x, u) : X × X → R+ such that f u + λα(x, u)η(x, u) ≤ (1 − λ)f (u) + λf (x), ∀x, u ∈ X, λ ∈ [0, 1]. Definition 2.3. The differentiable function f is said to be α-invex, if there exist functions η : X × X → R n , α(x, u) : X × X → R+ such that f (x) − f (u) ≥ α(x, u) f (u), η(x, u) , ∀x, u ∈ X. Here f (u) is the differential of the α-preinvex f at u ∈ X. It is obvious that the invex functions and preinvex functions are special cases of α-invex and α-preinvex functions. Clearly, every differentiable α-preinvex function is a α-invex function. The converse is also true under some suitable conditions, see Noor [21]. From now onward, we assume that the set X is an α-invex set, unless otherwise specified. Also f (u + λη(x, u)) − f (u) , α(x, u)f u, η(x, u) = lim λ λ→0+ where f (·,·) is the directional derivative of f . A similar notation is made for g (u, η(x, u)). Consider the following multiobjective programming problem: (P)
min f (x) s.t. g(x) 0,
x ∈ X,
where f : X → R k , g : X → R m , X is a nonempty open α-invex of R n , η : X × X → R n is a vector function.
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Let D = {x ∈ X: g(x) 0} be the set of all the feasible solutions for (P) and denote I = {1, . . . , k}, M = {1, 2, . . . , m}, J (x) = {j ∈ M: gj (x) = 0} and J˜(x) = {j ∈ M: gj (x) < 0}. It is obvious that J (x) ∪ J˜(x) = M. Throughout this paper, the following convention for vectors in R n will be followed: x>y
if and only if
xi > yi , i = 1, 2, . . . , n,
xy
if and only if
xi yi , i = 1, 2, . . . , n,
x≥y
if and only if xi yi , i = 1, 2, . . . , n, but x = y.
Definition 2.4. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist vector-functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, fi (x) − fi (u) αi (x, u)fi u, η(x, u) and
−gj (u) βj (x, u)gj u, η(x, u) .
Definition 2.5. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be weak strictlypseudoquasi-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
⇒
i=1
p
fi u, η(x, u) < 0
i=1
and −
m
βj (x, u)gj (u) 0
⇒
j =1
m
gj u, η(x, u) 0.
j =1
Definition 2.6. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be strong pseudoquasi-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
i=1
⇒
p
fi u, η(x, u) ≤ 0
i=1
and −
m j =1
βj (x, u)gj (u) 0
⇒
m
gj u, η(x, u) 0.
j =1
Definition 2.7. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be weak quasistrictly-pseudo-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist
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functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
⇒
i=1
p
fi u, η(x, u) 0
i=1
and −
m
βj (x, u)gj (u) 0
⇒
j =1
m
gj u, η(x, u) ≤ 0.
j =1
Definition 2.8. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be weak strictlypseudo-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
i=1
⇒
p
fi u, η(x, u) < 0
i=1
and −
m
βj (x, u)gj (u) 0
j =1
⇒
m
gj u, η(x, u) < 0.
j =1
Remark 2.1. The functions defined above are different from those in Aghezzaf and Hachimi [1], Antczak [2], Hanson et al. [9], Jeyakumar and Mond [11], Suneja et al. [26], Mishra et al. [20] and Rueda et al. [25]. For examples of differentiable generalized type functions, one can refer to Aghezzaf and Hachimi [1]. Definition 2.9. A point x¯ ∈ D is said to be a weak Pareto efficient solution for (P) if the relation f (x) ≮ f (x) ¯ holds for all x ∈ D. Definition 2.10. A point x¯ ∈ D is said to be a locally weak Pareto efficient solution for (P) if there is a neighborhood N (x) ¯ around x¯ such that f (x) ≮ f (x) ¯ holds for all x ∈ N(x) ¯ ∩ D. The following results from Antczak [2] and Weir and Mond [27] will be needed in next sections of the paper. Lemma 2.1. If x¯ is a locally weak Pareto or a weak Pareto efficient solution of (P) and if gj is continuous at x¯ for j ∈ J˜(x), ¯ then the following system of inequalities: ¯ η(x, x) ¯ < 0, gJ (x) x, ¯ η(x, x) ¯ <0 f x, has no solution for x ∈ X.
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Lemma 2.2. Let S be a nonempty set in R n and ψ : S → R p be a preinvex function on S. m, Then either ψ(x) < 0 has a solution x ∈ S, or λT ψ(x) 0 for all x ∈ S, or some λ ∈ R+ but both alternatives are never true. Lemma 2.3 (F. John type necessary optimality condition). Let x¯ be a weak Pareto efficient solution for (P). Moreover, we assume that gj is continuous for j ∈ J˜(x), ¯ f and g are directionally differentiable at x¯ with f (x, ¯ η(x, x)), ¯ and gJ (x) (x, ¯ η(x, x)) ¯ pre-invex funck,µ m such that (x, ¯ ∈ R+ ¯ ξ¯ , µ) ¯ satisfies the following tions of x on X. Then there exist ξ¯ ∈ R+ conditions: ¯ η(x, x) ¯ + µ¯ T g x, ¯ η(x, x) ¯ 0 ∀x ∈ X, ξ¯ T f x, µ¯ T g(x) ¯ = 0, g(x) ¯ 0. Definition 2.11. The function g is said to satisfy the generalized Slater’s constraint qualification at x¯ ∈ D if g is d-invex at x, ¯ and there exists x˜ ∈ D such that gj (x) ˜ < 0, j ∈ J (x). ¯ Definition 2.12. A f : X → R k be defined on X and directionally differentiable at u ∈ X is said to be α-d-invex at u ∈ X with respect to η if for any x ∈ X, f (x) − f (u) α(x, u)f u, η(x, u) . Lemma 2.4 (Karush–Kuhn–Tucker type necessary optimality condition). Let x¯ be a weak Pareto efficient solution for (P). Assume that g j is continuous for j ∈ J˜(x), ¯ f and g are directionally differentiable at x¯ with f (x, ¯ η(x, x)), ¯ and gJ (x) (x, ¯ η(x, x)) ¯ pre-invex functions on X. Moreover, we assume that g satisfies the general Slater’s constraint qualification m such that (x, at x. ¯ Then there exists µ¯ ∈ R+ ¯ µ) ¯ satisfies the following conditions: ¯ η(x, x) ¯ + µ¯ T g x, ¯ η(x, x) ¯ 0 ∀x ∈ X, (1) f x, µ¯ T g(x) ¯ = 0,
(2)
g(x) ¯ 0.
(3)
3. Sufficient optimality conditions In this section, we establish a Karush–Kuhn–Tucker type sufficient optimality condition. Theorem 3.1. Let x¯ be a feasible solution for (P) at which conditions (1)–(3) are satisfied. Moreover, if any of the following conditions are satisfied: (a) (fi , m i=1 µj gj ) is strong pseudo-quasi-d-V-type-I at x¯ with respect to η, αi (x, u) and βj (x, u);
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(b) (fi , m i=1 µj gj ) is weak strictly pseudo-quasi-d-V-type-I at x¯ with respect to η, αi (x,u) and βj (x, u); (c) (fi , m i=1 µj gj ) is weak strictly pseudo-d-V-type-I at x¯ with respect to η, αi (x, u) and βj (x, u). Then x¯ is a weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Suppose that x¯ is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that fi (x) < fi (x) ¯ for any i ∈ {1, 2, . . . , p}. Therefore, from positivity of αi (x, x), ¯ we get p
αi (x, x)f ¯ i (x) <
i=1
p
αi (x, x)f ¯ i (x). ¯
(4)
i=1
By condition (a) and (4), we get p
¯ η(x, x) ¯ < 0. fi x,
(5)
i=1
¯ > 0, from (2) and condition (a), we get Since βj (x, x) m
¯ η(x, x) ¯ ≤ 0. µj gj x,
(6)
j =1
From (5) and (6), we get p
m ¯ η(x, x) ¯ + ¯ η(x, x) ¯ < 0, fi x, µj gj x, j =1
i=1
which contradicts (1). By condition (b), from (4), (2) and βj (x, u) > 0, we get p
m ¯ η(x, x) ¯ + ¯ η(x, x) ¯ < 0, fi x, µj gj x, j =1
i=1
which is a contradiction to (1). By condition (c), from (4), (2) and βj (x, u) > 0, we get p
¯ η(x, x) ¯ <0 fi x,
i=1
and
m
j =1 µj gj
p i=1
x, ¯ η(x, x) ¯ < 0. By these two inequalities, we get
m ¯ η(x, x) ¯ + ¯ η(x, x) ¯ < 0, fi x, µj gj x, j =1
which is a contradiction to (1). This completes the proof.
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4. Mond–Weir duality Now, in relation to (P) we consider the following dual problem, which is in the format of Mond–Weir [22]: (MWD) max f (y) = f1 (y), f2 (y), . . . , fk (y) subject to (ξ T f + µT g ) y, η(x, y) 0 for all x ∈ D, (7) µj gj (y) 0,
j = 1, . . . , m,
(8)
ξ T e = 1, ξ
k, ∈ R+
Let
m, µ ∈ R+
where e
(9)
= (1, 1, . . . , 1) ∈ R k .
W = (y, ξ, µ) ∈ X × R k × R m : (ξ T f + µT g ) y, η(x, y) 0, µj gj (y) 0, k m , ξ T e = 1, µ ∈ R+ j = 1, . . . , m, ξ ∈ R+
denote the set of all the feasible solutions of (MWD). We denote by prX W the projection of set W on X. Theorem 4.1 (Weak Duality). Let x and (y, ξ, µ) be feasible solutions for (P) and (MWD), respectively. Moreover, we assume that any one of the following conditions holds: (a) (fi , m i=1 µj gj ) is strong pseudo-quasi-d-V-type-I at y with respect to η, αi (x, u) and u) βj (x, and ξ > 0; (b) (fi , m i=1 µj gj ) is weak strictly pseudo-quasi-d-V-type-I at y with respect to η, αi (x,u) and βj (x, u); (c) (fi , m i=1 µj gj ) is weak strictly pseudo-d-V-type-I at y with respect to η, αi (x, u) and βj (x, u) at y on D ∪ prX W . Then the following cannot hold: f (x) ≤ f (y). Proof. We proceed by contradiction. Suppose that f (x) ≤ f (y).
(10)
Since αi (x, y) > 0 (10) implies that p
αi (x, y)fi (x) ≤
i=1
p
αi (x, y)fi (y).
(11)
i=1
Since (y, ξ, µ) is feasible for (MWD), it follows that −
m j =1
µj gj (y) 0.
(12)
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Since βj (x, y) > 0, (12) implies that −
m
βj (x, y)µj gj (y) 0.
(13)
j =1
By condition (a), (11) and (13) imply f y, η(x, y) ≤ 0, m
µj gj y, η(x, y) 0.
(14) (15)
j =1
Since ξ > 0, the above two inequalities give k
m ξi fi y, η(x, y) + µj gj y, η(x, y) < 0,
(16)
j =1
i=1
which contradicts (5). By condition (b), (11) and (13) imply p
fi y, η(x, y) < 0
(17)
i=1
and −
m
µj gj y, η(x, y) 0.
(18)
j =1
Since ξ 0, (17) and (18) imply (16), again a contradiction to (5). By condition (c), (11) and (13) imply p
fi y, η(x, y) < 0
(19)
i=1
and −
m
µj gj y, η(x, y) < 0.
(20)
j =1
Since ξ 0, (19) and (20) imply (16), again a contradiction to (5). This completes the proof. 2 Theorem 4.2 (Strong Duality). Let x¯ be a locally weak Pareto efficient solution or weak Pareto efficient solution for (P) at which the generalized Slater’s constraint qualification is ¯ η(x, x)), ¯ and g (x, ¯ η(x, x)) ¯ satisfied, f and g be directionally differentiable at x¯ with f (x, m ˆ ¯ Then there exists µ¯ ∈ R+ preinvex functions on X and gj be continuous for j ∈ J (x). such that (x, ¯ 1, µ) ¯ is feasible for (MWD). If the weak duality between (P) and (MWD) in Theorem 4.1 holds, then (x, ¯ 1, µ) ¯ is a locally weak Pareto efficient solution for (MWD).
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m such that Proof. Since x¯ satisfies all the conditions of Lemma 2.4, there exists µ¯ ∈ R+ conditions (1)–(3) hold. By (1)–(3), we have that (x, ¯ 1, µ) ¯ is feasible for (MWD). Also, by the weak duality, it follows that (x, ¯ 1, µ) ¯ is locally weak Pareto efficient solution for (MWD). 2
Theorem 4.3 (Converse Duality). Let (y, ¯ ξ¯ , µ) ¯ be a weak Pareto efficient solution for (MWD). Moreover, we assume that the hypotheses of Theorem 4.1 hold at y¯ in D ∪ prX W , then y¯ is weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Suppose that y¯ is not weak Pareto efficient solution ˜ y) ¯ > 0, and condition for (P), that is, there exists x˜ ∈ D such that f (x) ˜ < f (y). ¯ Since αi (x, (a) of Theorem 4.1 holds, we get k
¯ η(x, ˜ y) ¯ < 0. ξ¯i fi y,
(21)
i=1
Since βj (x, ˜ y) ¯ > 0 and the feasibility of x˜ for (P) and (y, ¯ ξ¯ , µ) ¯ for (MWD), respectively, we have −
m
βj (x, ˜ y) ¯ µ¯ j gj (y) ¯ 0,
j =1
which in light of condition (a) of Theorem 4.1 yields m
¯ η(x, ˜ y) ¯ 0. µ¯ j gj y,
(22)
j =1
By (21) and (22), we get k
m ξ¯i fi y, ¯ η(x, ˜ y) ¯ + ¯ η(x, ˜ y) ¯ < 0. µ¯ j gj y,
(23)
j =1
i=1
This contradicts the dual constraint (5). By condition (b), we get k
¯ η(x, ˜ y) ¯ < 0 and fi y,
m
¯ η(x, ˜ y) ¯ 0. µ¯ j gj y,
j =1
i=1
Since ξ¯i 0, the above two inequalities imply (23), again a contradiction to (5). By condition (c), we have k i=1
fi
y, ¯ η(x, ˜ y) ¯ < 0 and
m
¯ η(x, ˜ y) ¯ < 0. µ¯ j gj y,
j =1
Since ξ¯i ≥ 0, the above two inequalities imply (23), again a contradiction to (5). This completes the proof. 2
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References [1] B. Aghezzaf, M. Hachimi, Generalized invexity and duality in multiobjective programming problems, J. Global Optim. 18 (2000) 91–101. [2] T. Antczak, Multiobjective programming under d-invexity, European J. Oper. Res. 137 (2002) 28–36. [3] M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley, New York, 1991. [4] C.R. Bector, S.K. Suneja, S. Gupta, Univex functions and univex nonlinear programming, in: Proc. Administrative Sci. Assoc. Canada, 1992, pp. 115–124. [5] B.D. Craven, Invex functions and constrained local minima, Bull. Austral. Math. Soc. 24 (1981) 357–366. [6] B.D. Craven, Nondifferentiable optimization by smooth approximation, Optimization 17 (1986) 3–17. [7] R.R. Egudo, M.A. Hanson, Multi-objective duality with invexity, J. Math. Anal. Appl. 126 (1987) 469–477. [8] M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545–550. [9] M.A. Hanson, R. Pini, C. Singh, Multiobjective programming under generalized type-I invexity, J. Math. Anal. Appl. 261 (2001) 562–577. [10] M.A. Hanson, B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Program. 37 (1987) 51–58. [11] V. Jeyakumar, B. Mond, On generalized convex mathematical programming, J. Aust. Math. Soc. Ser. B 34 (1992) 43–53. [12] R.N. Kaul, S.K. Suneja, M.K. Srivastava, Optimality criteria and duality in multiple objective optimization involving generalized invexity, J. Optim. Theory Appl. 80 (1994) 465–482. [13] O.L. Mangasarian, Nonlinear Programming, McGraw–Hill, New York, 1969. [14] S.K. Mishra, Lagrange multipliers saddle points and scalarizations in composite multiobjective nonsmooth programming, Optimization 38 (1996) 93–105. [15] S.K. Mishra, On sufficiency and duality in nonsmooth multiobjective programming, Opsearch 34 (1997) 221–231. [16] S.K. Mishra, On multiple objective optimization with generalized univexity, J. Math. Anal. Appl. 224 (1998) 131–148. [17] S.K. Mishra, G. Giorgi, Optimality and duality with generalized semi-univexity, Opsearch 37 (2000) 340– 350. [18] S.K. Mishra, R.N. Mukherjee, Generalized convex composite multiobjective nonsmooth programming and conditional proper efficiency, Optimization 34 (1995) 53–66. [19] S.K. Mishra, R.N. Mukherjee, On generalized convex multiobjective nonsmooth programming, J. Aust. Math. Soc. Ser. B 38 (1996) 140–148. [20] S.K. Mishra, S.Y. Wang, K.K. Lai, Optimality and duality in nondifferentiable and multiobjective programming under generalized d-invexity, J. Global Optim. 29 (2004) 425–438. [21] M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math. 5 (2004) 1–9. [22] B. Mond, T. Weir, Generalized concavity and duality, in: S. Schaible, W.T. Ziemba (Eds.), Generalized Concavity Optimization and Economics, Academic Press, New York, 1981, pp. 263–280. [23] T.W. Reiland, Nonsmooth invexity, Bull. Austral. Math. Soc. 42 (1990) 437–446. [24] N.G. Rueda, M.A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl. 130 (1988) 375–385. [25] N.G. Rueda, M.A. Hanson, C. Singh, Optimality and duality with generalized convexity, J. Optim. Theory Appl. 86 (1995) 491–500. [26] S.K. Suneja, M.K. Srivastava, Optimality and duality in nondifferentiable multiobjective optimization involving d-type I and related functions, J. Math. Anal. Appl. 206 (1997) 465–479. [27] T. Weir, B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl. 136 (1988) 29–38. [28] Y.L. Ye, d-invexity and optimality conditions, J. Math. Anal. Appl. 162 (1991) 242–249.
Some nondifferentiable multiobjective programming problems ✩ S.K. Mishra a,∗ , M.A. Noor b a Department of Mathematics, Statistics and Computer Science, College of Basic Sciences and Humanities,
G.B. Pant University of Agriculture and Technology, Pantnagar-263 145, India b Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
Received 27 April 2005 Available online 25 July 2005 Submitted by William F. Ames
Abstract In this paper it is shown that a relaxation defining the class of generalized d-V-type-I functions leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar non-differentiable case, and avoids the major difficulty of verifying that the inequality holds for the same kernel function. The results obtained in this paper generalize and extend the previously known results in this area. 2005 Elsevier Inc. All rights reserved. Keywords: Multi-objective programming; Duality; Pareto efficient solution; Generalized d-V-invexity
1. Introduction The field of multi-objective programming, also known as vector programming, has grown remarkably in different directions in the settings of optimality conditions and du✩ This research is supported by the Department of Science and Technology, Ministry of Science and Technology, Government of India, under the SERC Fast Track Scheme for Young Scientists 2001–2002 through grant No. SR/FTP/MS-22/2001. * Corresponding author. E-mail addresses: [email protected] (S.K. Mishra), [email protected] (M.A. Noor).
0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.04.067
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ality theory since the 1980s. It has been enriched by the applications of various types of generalizations of convexity theory, with and without differentiability assumptions, and in the framework of continuous time programming, fractional programming, inverse vector optimization, saddle point theory, symmetric duality, variational problems and variational inequality problems, etc. Convexity plays a vital role in many aspects of mathematical programming including sufficient optimality condition and duality theorems see, for example, Mangasarian [13] and Bazaraa et al. [3]. To relax convexity assumptions imposed on the functions in theorems on sufficient optimality and duality, various generalized convexity notions have been proposed. Hanson [8] introduced the class of invex functions, see also [5]. Later, Hanson and Mond [10] defined two new classes of functions called type-I and type-II functions, and sufficient optimality conditions were established by using these concepts. Rueda and Hanson [24] further extended type-I functions to the classes of pseudo-type-I and quasitype-I functions and obtained sufficient optimality criteria for a nonlinear programming problem involving these functions. Kaul et al. [12] considered a multiple objective nonlinear programming problem involving generalized type-I functions and obtained some results on optimality and duality, where the Wolfe and Mond–Weir duals are considered. Univex functions were introduced and studied by Bector et al. [4]. Rueda et al. [25] obtained optimality and duality results for several mathematical programs by combining the concepts of type-I and univex functions. Mishra [16] considered a multiple objective nonlinear programming problem and obtained optimality, duality and saddle point results of a vector-valued Lagrangian by combining the concepts of type-I, pseudo-type-I, quasi-type-I, quasi-pseudo-type-I, pseudo-quasi-type-I and univex functions. Aghezzaf and Hachimi [1] introduced new classes of generalized type-I vector-valued functions and derived various duality results for a nonlinear multiobjective programming problem. It is known that, despite substituting invexity for convexity, many theoretical problems in differentiable programming can also be solved, see Hanson [8], Egudo and Hanson [7], and Jeyakumar and Mond [11]. But the corresponding conclusions cannot be obtained in nondifferentiable programming with the aid of invexity introduced by Hanson [8] because the existence of a derivative is required in the definition of invexity. There exists a generalization of invexity to locally Lipschitz functions, with derivative replaced by the Clarke generalized gradient, see Craven [6], Reiland [23], Mishra and Mukherjee [18,19], Mishra [14,15], and Mishra and Giorgi [17]. However, Antczak [2] used directional derivative, in association with a hypothesis of an invex kind following Ye [28]. The necessary optimality conditions in Antczak [2] are different from those cited in the literature. In the present paper, we consider a nondifferentiable and multiobjective programming problem and derive some Karush–Kuhn–Tucker type of sufficient optimality conditions for a (weakly) Pareto efficient solution to the problem involving the new classes of directionally differentiable generalized type-I functions. Furthermore, the Mond–Weir type and general Mond–Weir type of duality results are also obtained in terms of right differentials of the aforesaid functions involved in the multiobjective programming problem.
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2. Preliminaries In this section, we extend the concepts of weak strictly-pseudo-quasi-type I, strong pseudo-quasi-type I, weak quasi-strictly-pseudo type I and weak strictly pseudo-type I functions introduced in Aghezzaf and Hachimi [1] in the setting of Antczak [2] and give some preliminaries. Let X be any subset of R n . Definition 2.1. A subset X is said to be an α-invex set, if there η : X × X → R n , α(x, u) : X × X → R+ such that u + λα(x, u)η(x, u) ∈ X,
∀x, u ∈ X, λ ∈ [0, 1].
Note that, for α(x, u) = 1, α-invex set becomes the invex set. It is well known that the α-invex may not be convex sets, see Noor [21]. Definition 2.2. The function f on the α-invex set is said to be α-preinvex function, if there exist η : X × X → R n , α(x, u) : X × X → R+ such that f u + λα(x, u)η(x, u) ≤ (1 − λ)f (u) + λf (x), ∀x, u ∈ X, λ ∈ [0, 1]. Definition 2.3. The differentiable function f is said to be α-invex, if there exist functions η : X × X → R n , α(x, u) : X × X → R+ such that f (x) − f (u) ≥ α(x, u) f (u), η(x, u) , ∀x, u ∈ X. Here f (u) is the differential of the α-preinvex f at u ∈ X. It is obvious that the invex functions and preinvex functions are special cases of α-invex and α-preinvex functions. Clearly, every differentiable α-preinvex function is a α-invex function. The converse is also true under some suitable conditions, see Noor [21]. From now onward, we assume that the set X is an α-invex set, unless otherwise specified. Also f (u + λη(x, u)) − f (u) , α(x, u)f u, η(x, u) = lim λ λ→0+ where f (·,·) is the directional derivative of f . A similar notation is made for g (u, η(x, u)). Consider the following multiobjective programming problem: (P)
min f (x) s.t. g(x) 0,
x ∈ X,
where f : X → R k , g : X → R m , X is a nonempty open α-invex of R n , η : X × X → R n is a vector function.
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Let D = {x ∈ X: g(x) 0} be the set of all the feasible solutions for (P) and denote I = {1, . . . , k}, M = {1, 2, . . . , m}, J (x) = {j ∈ M: gj (x) = 0} and J˜(x) = {j ∈ M: gj (x) < 0}. It is obvious that J (x) ∪ J˜(x) = M. Throughout this paper, the following convention for vectors in R n will be followed: x>y
if and only if
xi > yi , i = 1, 2, . . . , n,
xy
if and only if
xi yi , i = 1, 2, . . . , n,
x≥y
if and only if xi yi , i = 1, 2, . . . , n, but x = y.
Definition 2.4. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist vector-functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, fi (x) − fi (u) αi (x, u)fi u, η(x, u) and
−gj (u) βj (x, u)gj u, η(x, u) .
Definition 2.5. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be weak strictlypseudoquasi-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
⇒
i=1
p
fi u, η(x, u) < 0
i=1
and −
m
βj (x, u)gj (u) 0
⇒
j =1
m
gj u, η(x, u) 0.
j =1
Definition 2.6. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be strong pseudoquasi-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
i=1
⇒
p
fi u, η(x, u) ≤ 0
i=1
and −
m j =1
βj (x, u)gj (u) 0
⇒
m
gj u, η(x, u) 0.
j =1
Definition 2.7. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be weak quasistrictly-pseudo-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist
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functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
⇒
i=1
p
fi u, η(x, u) 0
i=1
and −
m
βj (x, u)gj (u) 0
⇒
j =1
m
gj u, η(x, u) ≤ 0.
j =1
Definition 2.8. (fi , gj ), i = 1, 2, . . . , p and j = 1, 2, . . . , m, is said to be weak strictlypseudo-d-V-type-I with respect to η, αi (x, u) and βj (x, u) at u ∈ X if there exist functions η : X × X → R n , αi (x, u) : X × X → R+ and βj (x, u) : X × X → R+ such that for all x ∈ X, p
αi (x, u)fi (x) ≤
i=1
p
αi (x, u)fi (u)
i=1
⇒
p
fi u, η(x, u) < 0
i=1
and −
m
βj (x, u)gj (u) 0
j =1
⇒
m
gj u, η(x, u) < 0.
j =1
Remark 2.1. The functions defined above are different from those in Aghezzaf and Hachimi [1], Antczak [2], Hanson et al. [9], Jeyakumar and Mond [11], Suneja et al. [26], Mishra et al. [20] and Rueda et al. [25]. For examples of differentiable generalized type functions, one can refer to Aghezzaf and Hachimi [1]. Definition 2.9. A point x¯ ∈ D is said to be a weak Pareto efficient solution for (P) if the relation f (x) ≮ f (x) ¯ holds for all x ∈ D. Definition 2.10. A point x¯ ∈ D is said to be a locally weak Pareto efficient solution for (P) if there is a neighborhood N (x) ¯ around x¯ such that f (x) ≮ f (x) ¯ holds for all x ∈ N(x) ¯ ∩ D. The following results from Antczak [2] and Weir and Mond [27] will be needed in next sections of the paper. Lemma 2.1. If x¯ is a locally weak Pareto or a weak Pareto efficient solution of (P) and if gj is continuous at x¯ for j ∈ J˜(x), ¯ then the following system of inequalities: ¯ η(x, x) ¯ < 0, gJ (x) x, ¯ η(x, x) ¯ <0 f x, has no solution for x ∈ X.
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Lemma 2.2. Let S be a nonempty set in R n and ψ : S → R p be a preinvex function on S. m, Then either ψ(x) < 0 has a solution x ∈ S, or λT ψ(x) 0 for all x ∈ S, or some λ ∈ R+ but both alternatives are never true. Lemma 2.3 (F. John type necessary optimality condition). Let x¯ be a weak Pareto efficient solution for (P). Moreover, we assume that gj is continuous for j ∈ J˜(x), ¯ f and g are directionally differentiable at x¯ with f (x, ¯ η(x, x)), ¯ and gJ (x) (x, ¯ η(x, x)) ¯ pre-invex funck,µ m such that (x, ¯ ∈ R+ ¯ ξ¯ , µ) ¯ satisfies the following tions of x on X. Then there exist ξ¯ ∈ R+ conditions: ¯ η(x, x) ¯ + µ¯ T g x, ¯ η(x, x) ¯ 0 ∀x ∈ X, ξ¯ T f x, µ¯ T g(x) ¯ = 0, g(x) ¯ 0. Definition 2.11. The function g is said to satisfy the generalized Slater’s constraint qualification at x¯ ∈ D if g is d-invex at x, ¯ and there exists x˜ ∈ D such that gj (x) ˜ < 0, j ∈ J (x). ¯ Definition 2.12. A f : X → R k be defined on X and directionally differentiable at u ∈ X is said to be α-d-invex at u ∈ X with respect to η if for any x ∈ X, f (x) − f (u) α(x, u)f u, η(x, u) . Lemma 2.4 (Karush–Kuhn–Tucker type necessary optimality condition). Let x¯ be a weak Pareto efficient solution for (P). Assume that g j is continuous for j ∈ J˜(x), ¯ f and g are directionally differentiable at x¯ with f (x, ¯ η(x, x)), ¯ and gJ (x) (x, ¯ η(x, x)) ¯ pre-invex functions on X. Moreover, we assume that g satisfies the general Slater’s constraint qualification m such that (x, at x. ¯ Then there exists µ¯ ∈ R+ ¯ µ) ¯ satisfies the following conditions: ¯ η(x, x) ¯ + µ¯ T g x, ¯ η(x, x) ¯ 0 ∀x ∈ X, (1) f x, µ¯ T g(x) ¯ = 0,
(2)
g(x) ¯ 0.
(3)
3. Sufficient optimality conditions In this section, we establish a Karush–Kuhn–Tucker type sufficient optimality condition. Theorem 3.1. Let x¯ be a feasible solution for (P) at which conditions (1)–(3) are satisfied. Moreover, if any of the following conditions are satisfied: (a) (fi , m i=1 µj gj ) is strong pseudo-quasi-d-V-type-I at x¯ with respect to η, αi (x, u) and βj (x, u);
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(b) (fi , m i=1 µj gj ) is weak strictly pseudo-quasi-d-V-type-I at x¯ with respect to η, αi (x,u) and βj (x, u); (c) (fi , m i=1 µj gj ) is weak strictly pseudo-d-V-type-I at x¯ with respect to η, αi (x, u) and βj (x, u). Then x¯ is a weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Suppose that x¯ is not a weak Pareto efficient solution of (P). Then there is a feasible solution x of (P) such that fi (x) < fi (x) ¯ for any i ∈ {1, 2, . . . , p}. Therefore, from positivity of αi (x, x), ¯ we get p
αi (x, x)f ¯ i (x) <
i=1
p
αi (x, x)f ¯ i (x). ¯
(4)
i=1
By condition (a) and (4), we get p
¯ η(x, x) ¯ < 0. fi x,
(5)
i=1
¯ > 0, from (2) and condition (a), we get Since βj (x, x) m
¯ η(x, x) ¯ ≤ 0. µj gj x,
(6)
j =1
From (5) and (6), we get p
m ¯ η(x, x) ¯ + ¯ η(x, x) ¯ < 0, fi x, µj gj x, j =1
i=1
which contradicts (1). By condition (b), from (4), (2) and βj (x, u) > 0, we get p
m ¯ η(x, x) ¯ + ¯ η(x, x) ¯ < 0, fi x, µj gj x, j =1
i=1
which is a contradiction to (1). By condition (c), from (4), (2) and βj (x, u) > 0, we get p
¯ η(x, x) ¯ <0 fi x,
i=1
and
m
j =1 µj gj
p i=1
x, ¯ η(x, x) ¯ < 0. By these two inequalities, we get
m ¯ η(x, x) ¯ + ¯ η(x, x) ¯ < 0, fi x, µj gj x, j =1
which is a contradiction to (1). This completes the proof.
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4. Mond–Weir duality Now, in relation to (P) we consider the following dual problem, which is in the format of Mond–Weir [22]: (MWD) max f (y) = f1 (y), f2 (y), . . . , fk (y) subject to (ξ T f + µT g ) y, η(x, y) 0 for all x ∈ D, (7) µj gj (y) 0,
j = 1, . . . , m,
(8)
ξ T e = 1, ξ
k, ∈ R+
Let
m, µ ∈ R+
where e
(9)
= (1, 1, . . . , 1) ∈ R k .
W = (y, ξ, µ) ∈ X × R k × R m : (ξ T f + µT g ) y, η(x, y) 0, µj gj (y) 0, k m , ξ T e = 1, µ ∈ R+ j = 1, . . . , m, ξ ∈ R+
denote the set of all the feasible solutions of (MWD). We denote by prX W the projection of set W on X. Theorem 4.1 (Weak Duality). Let x and (y, ξ, µ) be feasible solutions for (P) and (MWD), respectively. Moreover, we assume that any one of the following conditions holds: (a) (fi , m i=1 µj gj ) is strong pseudo-quasi-d-V-type-I at y with respect to η, αi (x, u) and u) βj (x, and ξ > 0; (b) (fi , m i=1 µj gj ) is weak strictly pseudo-quasi-d-V-type-I at y with respect to η, αi (x,u) and βj (x, u); (c) (fi , m i=1 µj gj ) is weak strictly pseudo-d-V-type-I at y with respect to η, αi (x, u) and βj (x, u) at y on D ∪ prX W . Then the following cannot hold: f (x) ≤ f (y). Proof. We proceed by contradiction. Suppose that f (x) ≤ f (y).
(10)
Since αi (x, y) > 0 (10) implies that p
αi (x, y)fi (x) ≤
i=1
p
αi (x, y)fi (y).
(11)
i=1
Since (y, ξ, µ) is feasible for (MWD), it follows that −
m j =1
µj gj (y) 0.
(12)
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Since βj (x, y) > 0, (12) implies that −
m
βj (x, y)µj gj (y) 0.
(13)
j =1
By condition (a), (11) and (13) imply f y, η(x, y) ≤ 0, m
µj gj y, η(x, y) 0.
(14) (15)
j =1
Since ξ > 0, the above two inequalities give k
m ξi fi y, η(x, y) + µj gj y, η(x, y) < 0,
(16)
j =1
i=1
which contradicts (5). By condition (b), (11) and (13) imply p
fi y, η(x, y) < 0
(17)
i=1
and −
m
µj gj y, η(x, y) 0.
(18)
j =1
Since ξ 0, (17) and (18) imply (16), again a contradiction to (5). By condition (c), (11) and (13) imply p
fi y, η(x, y) < 0
(19)
i=1
and −
m
µj gj y, η(x, y) < 0.
(20)
j =1
Since ξ 0, (19) and (20) imply (16), again a contradiction to (5). This completes the proof. 2 Theorem 4.2 (Strong Duality). Let x¯ be a locally weak Pareto efficient solution or weak Pareto efficient solution for (P) at which the generalized Slater’s constraint qualification is ¯ η(x, x)), ¯ and g (x, ¯ η(x, x)) ¯ satisfied, f and g be directionally differentiable at x¯ with f (x, m ˆ ¯ Then there exists µ¯ ∈ R+ preinvex functions on X and gj be continuous for j ∈ J (x). such that (x, ¯ 1, µ) ¯ is feasible for (MWD). If the weak duality between (P) and (MWD) in Theorem 4.1 holds, then (x, ¯ 1, µ) ¯ is a locally weak Pareto efficient solution for (MWD).
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m such that Proof. Since x¯ satisfies all the conditions of Lemma 2.4, there exists µ¯ ∈ R+ conditions (1)–(3) hold. By (1)–(3), we have that (x, ¯ 1, µ) ¯ is feasible for (MWD). Also, by the weak duality, it follows that (x, ¯ 1, µ) ¯ is locally weak Pareto efficient solution for (MWD). 2
Theorem 4.3 (Converse Duality). Let (y, ¯ ξ¯ , µ) ¯ be a weak Pareto efficient solution for (MWD). Moreover, we assume that the hypotheses of Theorem 4.1 hold at y¯ in D ∪ prX W , then y¯ is weak Pareto efficient solution for (P). Proof. We proceed by contradiction. Suppose that y¯ is not weak Pareto efficient solution ˜ y) ¯ > 0, and condition for (P), that is, there exists x˜ ∈ D such that f (x) ˜ < f (y). ¯ Since αi (x, (a) of Theorem 4.1 holds, we get k
¯ η(x, ˜ y) ¯ < 0. ξ¯i fi y,
(21)
i=1
Since βj (x, ˜ y) ¯ > 0 and the feasibility of x˜ for (P) and (y, ¯ ξ¯ , µ) ¯ for (MWD), respectively, we have −
m
βj (x, ˜ y) ¯ µ¯ j gj (y) ¯ 0,
j =1
which in light of condition (a) of Theorem 4.1 yields m
¯ η(x, ˜ y) ¯ 0. µ¯ j gj y,
(22)
j =1
By (21) and (22), we get k
m ξ¯i fi y, ¯ η(x, ˜ y) ¯ + ¯ η(x, ˜ y) ¯ < 0. µ¯ j gj y,
(23)
j =1
i=1
This contradicts the dual constraint (5). By condition (b), we get k
¯ η(x, ˜ y) ¯ < 0 and fi y,
m
¯ η(x, ˜ y) ¯ 0. µ¯ j gj y,
j =1
i=1
Since ξ¯i 0, the above two inequalities imply (23), again a contradiction to (5). By condition (c), we have k i=1
fi
y, ¯ η(x, ˜ y) ¯ < 0 and
m
¯ η(x, ˜ y) ¯ < 0. µ¯ j gj y,
j =1
Since ξ¯i ≥ 0, the above two inequalities imply (23), again a contradiction to (5). This completes the proof. 2
482
S.K. Mishra, M.A. Noor / J. Math. Anal. Appl. 316 (2006) 472–482
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