Printer Friendly

Nondifferentiable multiobjective programming under type I semi-d-univexity.

Abstract

In this paper, we extend the classes of generalized type I univexity introduced by S.K. Mishra, Shou-Yang Wang, K.K.Lai [J.Math.Anal.Appl.303(2005)315-326] to type I semi-d-univexity by combining the concepts of generalized d-univexity in S.K. Mishra, Shou-Yang Wang, K.K.Lai [Euro.J.Oper.Res.160(2005)218-226] and semi-preinvexity in X.Q.Yang, Guang-Ya Chen [J. Math.Anal.Appl.169(1992)359-373]. Using the new concepts, some optimality conditions and some duality results for the nondifferentiable multiobjective programming problem are established.

AMS subject classification: 65k99, 90C46.

Keywords: Nondifferentiable multiobjective programming, type I semi-d-univexity, optimality conditions, duality.

1. Introduction

Convexity and generalized convexity play a key role in many aspects of optimization, such as optimality conditions, saddle-point theorems, duality theorems, theorems of alternatives, and convergence of optimization algorithms, so the research on convexity and generalized convexity is one of the important aspects in mathematical programming.

In recent years, the concepts of convexity has been generalized and extended in several directions. Among them, Hanson [9] introduced the class of invex functions. Later, Hanson and Mond [10] defined two new classes of functions called type-I and type-II function, and sufficient optimality conditions were established by using these concepts. Rueda and Hanson [19] further extended type-I functions to the classes of pseudo-type-I and quasi-type-I functions and obtained sufficient optimality conditions for a nonlinear programming involving these classes of 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. [5]. Rueda et al. [22] obtained optimality and duality results for several mathematical programs by combining the concepts of type-I and univex functions. Mishra [15] considered a multiple objective nonlinear programming problem and obtained a few results on optimality, duality and saddle point 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, 2] introduced new classes of weak quasiinvex, strong pseudoinvex, weak quasiinvex, weak pseudoinvex and strong quasiinvex functions. Recently, Mishra et al. [6] extend the classes of generalized type I vector-valued functions [1] to generalized univex type I vector-valued functions and consider a multiple objective optimization problem involving generalized univex type I functions.

On the other hand, many theoretical problems in differentiable programming can be solved [8, 9, 11] by substituting invexity for convexity. But the corresponding conclusions cannot be obtained in nondifferentiable programming with the aid of invexity because of the existence of requiring a derivative in the definition of invexity. For this, some generalizations of invexity to locally Lipschitz functions were studied, for example, with derivative replaced by the clarke generalized gradient [7, 13, 14, 16-18], Antczak [3] used directional derivative, which associated with a hypothesis of an invex kind following Ye [21].

In this paper, we consider a nondifferentiable multiobjective programming problem. Some sufficient optimality conditions are obtained for an efficient solution to the problem involving the new classes of arcwise directional differentiable [22] semi-d-univex functions by combining the concepts of generalized d-univexity [4] and semi-preinvexity [22]. Moreover, the Mond-Weir type and Mixed type duality results are also derived. The results in this paper extend many works in the literature.

2. Preliminaries

To compare vectors, we will distinguish between [less than or equal to] and [??] or between [greater than or equal to] and [??]. Specifically.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similar notations are applied to distinguish between [greater than or equal to] and [??].

We consider the following multiobjective optimization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where f : X [right arrow] [R.sup.p] and g : X [right arrow] [R.sup.m] and X [subset or equal] [R.sup.n] is an open set.

Let [X.sub.0] be the set of all feasible solutions of (VP). We quote some definitions and also give some new results.

Definition 2.1 A point a [member of] [X.sub.0] is said to be an efficient solution of problem (VP) if there exists no x [member of] [X.sub.0] such that f(x) [less than or equal to] f(a).

Definition 2.2 [22]A function l : X [right arrow] R is said to be arcwise directionally differentiable at [X.sub.0] if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

exists for each continuous arc [omega] : [0, 1] [right arrow] [R.sup.n] such that [omega] (0) = 0, [omega]'(0) = h.

Following [20], we define a type I semi-d-univex problem. In the following definitions [b.sub.0], [b.sub.1] : X x X x [0, 1] [right arrow] [R.sup.+], b(x, a) = [lim.sub.[lambda][right arrow]0] b(x, a, [lambda]) [greater than or equal to] 0, [[phi].sub.0], [[phi].sub.1] : R [right arrow] R and [eta] : X x X x [0, 1] [right arrow] [R.sup.n], [lim.sub.[lambda][right arrow]0] [lambda][eta](y, x, [lambda]) = 0, (d/d[lambda])[[lambda][eta](y, x, [lambda])][|.sub.[lambda]=0] = [??](y, x). Through the paper,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the sequel, we assume that [[phi].sub.0], [[phi].sub.1] : R [right arrow] R satisfy u [less than or equal to] 0 [??] [[phi].sub.0](u) [less than or equal to] 0, u [??] 0 [??] [[phi].sub.1](u) [??] 0 and u = 0 [??] [[phi].sub.1](u) [??] 0, and [b.sub.0](x, a) > 0 and [b.sub.1](x, a) [??] 0.

Definition 2.3 The problem (VP) is said to be weak strictly pseudo type I semi-d-univex at a [member of] [X.sub.0] if there exist real-valued functions [[phi].sub.0], [[phi].sub.1], [b.sub.0], [b.sub.1] and [eta] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] [X.sub.0] and for all i = 1, ..., p and j = 1, ..., m. If (VP) is weak strictly pseudo type I semi-d-univex at each a [member of] [X.sub.0], (VP) is said to be weak strictly pseudo type I semi-d-univex on X.

Definition 2.4 The problem (VP) is said to be strong pseudoquasi type I semi-d-univex at a [member of] [X.sub.0] if there exist real-valued functions [[phi].sub.0], [[phi].sub.1], [b.sub.0], [b.sub.1] and [eta] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] [X.sub.0] and for all i = 1, ..., p and j = 1, ..., m. If (VP) is strong pseudoquasi type I semi-d-univex at each a [member of] [X.sub.0], (VP) is said to be strong pseudoquasi type I semi-d-univex on X.

Definition 2.5 The problem (VP) is said to be weak quasistrictly pseudo type I semi-d-univex at a [member of] [X.sub.0] if there exist real-valued functions [[phi].sub.0], [[phi].sub.1], [b.sub.0], [b.sub.1] and [eta] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] [X.sub.0] and for all i = 1, ..., p and j = 1, ..., m. If (VP) is weak quasistrictly pseudo type I semi-d-univex at each a [member of] [X.sub.0], (VP) is said to be weak quasistrictly pseudo type I semi-d-univex on X.

Definition 2.6 The problem (VP) is said to be weak strictly pseudo-quasi type I semi-d-univex at a [member of] [X.sub.0] if there exist real-valued functions [[phi].sub.0], [[phi].sub.1], [b.sub.0], [b.sub.1] and [eta] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] [X.sub.0] and for all i = 1, ..., p and j = 1, ..., m. If (VP) is weak strictly pseudoquasi type I semi-d-univex at each a [member of] [X.sub.0], (VP) is said to be weak strictly pseudo-quasi type I semi-d-univex on X.

Remark 1 If we take [eta](y, x, [lambda]) = [eta](y, x), then the above definitions reduced to the ones in [4].

Remark 2 If we take [eta](y, x, [lambda]) = [eta](y, x) and if the functions f, g are all differentiable, then the above definitions reduced to the ones in [6].

3. Optimality conditions

In this section, we establish some sufficient optimality conditions for an a [member of] [X.sub.0] to be an efficient solution of problem (VP) under type I semi-d-univexity defined in the previous section.

Theorem 3.1 Suppose that

(i) a [member of] [X.sub.0];

(ii) there exist [[tau].sup.0] [member of] [R.sup.p], [[tau].sup.0] > 0, [[lambda].sup.0] [member of] [R.sup.m] and [[lambda].sup.0] [??] 0 such that

(a) f'[(a, [??](a, x)).sup.T] [[tau].sup.0] + g'[(a, [??](a, x)).sup.T] [[lambda].sup.0] = 0,

(b) g[(a).sup.T][[lambda].sup.0] = 0,

(c) [e.sup.T][[lambda].sup.0] = 1, where e = [(1, ..., 1).sup.T] [member of] [R.sup.p];

(iii) problem (VP) is strong pseudoquasi type I semi-d-univex at a [member of] [X.sub.0] with respect to some [b.sub.0], [b.sub.1], [[phi].sub.0], [[phi].sub.1] and [eta] for all feasible x. Then a is an efficient solution to (VP).

Proof. Suppose that a is not an efficient solution to (VP). Then there exists a feasible solution x to (VP) such that

f(x) [less than or equal to] f(a).

From the above inequality and the definitions of [b.sub.0](x, a), [[phi].sub.0] (x, a), we have that

[b.sub.0](x, a)[[phi].sub.0][f(x) - f(a)] [less than or equal to] 0. (3.1)

In view of conditions (ii)(b) and the definitions of [b.sub.1](x, a), [[phi].sub.1] (x, a), we have that

-[b.sub.1](x, a)[[phi].sub.1] [g[(a).sup.T][[lambda].sup.0]] 5 0. (3.2)

By inequality (3.1) and (3.2), condition (iii), we have

f'(a, [??](a, x)) [less than or equal to] 0, g'(a, [??](a, x)) [??] 0.

Since [[tau].sup.0] > 0, the above inequalities give

f'[(a, [??](a, x)).sup.T] [[tau].sup.0] + g'[(a, [??](a, x)).sup.T][[lambda].sup.0] < 0, (3.3)

which contradicts conditions(ii)(a). This completes the proof. []

Theorem 3.2 Suppose that

(i) a [member of] [X.sub.0];

(ii) there exist [[tau].sup.0] [member of] [R.sup.p], [[tau].sup.0] [greater than or equal to] 0, [[lambda].sup.0] [member of] [R.sup.m] and [[lambda].sup.0] [??] 0 such that

(a) f'[(a, [??](a, x)).sup.T] [[lambda].sup.0] + g'[(a, [??](a, x)).sup.T] [[lambda].sup.0] = 0,

(b) g[(a).sup.T][[lambda].sup.0] = 0,

(c) [e.sup.T][[lambda].sup.0] = 1, where e = [(1, ..., 1).sup.T] [member of] [R.sup.p],

(iii) problem (VP) is weak strictly pseudoquasi type I semi-d-univex at a [member of] [X.sub.0] with respect to some [b.sub.0], [b.sub.1], [[phi].sup.0], [[phi].sup.1] and [eta] for all feasible x. Then a is an efficient solution to (VP).

Proof. Suppose that a is not an efficient solution to (VP). Then there exists a feasible solution x to (VP) such that

f(x) [less than or equal to] f(a):

From the above inequality and the definitions of [b.sub.0](x, a), [[phi].sup.0] (x, a), we obtain (3.1). In view of conditions (ii)(b) and the definitions of [b.sub.1](x, a), [[phi].sup.1] (x, a), we get (3.2). By inequality (3.1) and (3.2), condition (iii), we have

f'(a, [??](a, x)) < 0, g'(a, [??](a, x)) < 0.

Since [[tau].sup.0] [greater than or equal to] 0, the above inequalities give

f'[(a, [??](a, x)).sup.T] [[tau].sup.0] + g'[(a, [??](a, x)).sup.T][[lambda].sup.0] < 0, (3.4)

which contradicts conditions(ii)(a). This completes the proof. []

Theorem 3.3 Suppose that

(i) a [member of] [X.sub.0];

(ii) there exist [[tau].sup.0] [member of] [R.sup.p], [[tau].sup.0] [??] 0, [[lambda].sup.0] [member of] [R.sup.m] and [[lambda].sup.0] [??] 0 such that

(a) f'[(a, [??](a, x)).sup.T] [[tau].sup.0] + g'[(a, [??](a, x)).sup.T][[lambda].sup.0] = 0,

(b) g[(a).sup.T][[lambda].sup.0] = 0,

(c) [e.sup.T][[lambda].sup.0] = 1, where e = [(1, ..., 1).sup.T] [member of] [R.sup.p],

(iii) problem (VP) is weak strictly pseudo type I semi-d-univex at a [member of] [X.sub.0] with respect to some [b.sub.0], [b.sub.1], [[phi].sub.0], [[phi].sub.1] and [eta] for all feasible x. Then a is an efficient solution to (VP).

Proof. Suppose that a is not an efficient solution to (VP). Then there exists a feasible solution x to (VP) such that

f(x) [less than or equal to] f(a).

From the above inequality and the definitions of [b.sub.0](x, a), [[phi].sub.0] (x, a), we obtain (3.1). In view of conditions (ii)(b) and the definitions of [b.sub.1](x, a), [[phi].sub.1] (x, a), we get (3.2). By inequality (3.1) and (3.2), condition (iii), we have

f'(a, [??](a, x)) < 0, g'(a, [??](a, x)) [??] 0.

Since [[tau].sup.0] [??] 0, the above inequalities give

f'[(a, [??](a, x)).sup.T] [[tau].sup.0] + g'[(a, [??](a, x)).sup.T][[lambda].sup.0] < 0, (3.5)

which contradicts conditions(ii)(a). This completes the proof. []

4. Mond-Weir type duality

In this section, we present some weak and strong duality theorems for (VP) and the following Mond-Weir dual type problem (See [25]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where e = [(1, 1, ..., 1).sup.T] [member of] [R.sup.p].

Denote by [Y.sup.0] the set of all the feasible solutions of problem (MWD), i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 4.1 (Weak duality) Suppose that

(i) x [member of] [X.sub.0];

(ii) (y, [tau], [lambda]) [member of] [Y.sup.0] and [tau] > 0;

(iii) problem (VP) is strong pseudoquasi type I semi-d-univex at y with respect to some [b.sub.0], [b.sub.1], [[phi].sup.0], [[phi].sup.1] and [eta].

Then f(x) [??] f(y).

Proof. Suppose contrary to the result, i.e.,

f(x) [less than or equal to] f(y).

From the above inequality and the definitions of [b.sub.0](x, y), [[phi].sub.0](x, y), we have that

[b.sub.0](x, y)[[phi].sub.0][f(x) - f(y)] [less than or equal to] 0. (4.1)

In view of (y, [tau], [lambda]) [member of] [Y.sup.0] and the definitions of [b.sub.1](x, y), [[phi].sub.1](x, y), we have that

-[b.sub.1](x, y)[[phi].sup.1][g[(y).sup.T][lambda]] [??] 0. (4.2)

By inequality (4.1) and (4.2), condition (iii), we have

f'(y, [??](y, x)) [less than or equal to] 0, g'(y, [??](y, x)) [??] 0.

Since [tau] > 0, the above inequalities give

f'[(y, [eta](y, x)).sup.T] [tau] + g'[(y, [??](y, x)).sup.T][lambda] < 0,

which contradicts the first constraint for (MWD). This completes the proof. []

Theorem 4.2 (Weak duality) Suppose that

(i) x [member of] [X.sub.0];

(ii) (y, [tau], [lambda]) [member of] [Y.sup.0] and [tau] [greater than or equal to] 0;

(iii) problem (VP) is weak strictly pseudoquasi type I semi-d-univex at y with respect to [b.sub.0]; [b.sub.1], [[phi].sub.0], [[phi].sub.1] and [eta].

Then f(x) [??] f(y).

Proof. Suppose contrary to the result, i.e.,

f(x) [less than or equal to] f(y).

From the above inequality and the definitions of [b.sub.0](x, y), [[phi].sub.0] (x, y), we have (4.1). In view of (y, [tau], [lambda]) [member of] [Y.sup.0] and the definitions of [b.sub.1](x, y), [[phi].sub.1] (x, y), we have (4.2). By inequality (4.1) and (4.2), condition (iii), we have

f'(y, [??](y, x)) < 0, g'(y, [??](y, x)) < 0.

Since [tau][greater than or equal to] 0, the above inequalities give

f'[(y, [??](y, x)).sup.T] [tau] + g'[(y, [??](y, x)).sup.T][lambda] < 0,

which contradicts the first constraint for (MWD). This completes the proof. []

Theorem 4.3 (Weak duality) Suppose that

(i) x [member of] [X.sub.0];

(ii) (y, [tau], [lambda]) [member of] [Y.sup.0];

(iii) problem (VP) is weak strictly pseudo type I semi-d-univex at y with respect to [b.sub.0], [b.sub.1], [[phi].sub.0], [[phi].sub.1] and [eta].

Then f(x) [??] f(y).

Proof. Suppose contrary to the result, i.e.,

f(x) [less than or equal to] f(y):

From the above inequality and the definitions of [b.sub.0](x, y), [[phi].sub.0] (x, y), we have (4.1). In view of (y, [tau], [lambda]) [member of] [Y.sup.0] and the definitions of [b.sub.1](x, y), [[phi].sub.1] (x, y), we have (4.2). By inequality (4.1) and (4.2), condition (iii), we have

f'(y, [??](y, x)) < 0, g'(y, [??](y, x)) [??] 0.

Since [tau] [??] 0, the above inequalities give

f'[(y, [??](y, x)).sup.T] [tau] + g'[(y, [??](y, x)).sup.T][lambda] < 0,

which contradicts the first constraint for (MWD). This completes the proof. []

Theorem 4.4 (Strong duality)Let [bar.x] be an efficient solution for (VP) and [bar.x] satisfies a constraint qualification for (VP) in [24]. Then there exist [bar.[tau]] [member of] [R.sup.p] and [bar.[lambda]] [member of] [R.sup.m] such that ([bar.x], [bar.[tau]], [bar.[lambda]]) is feasible for (MWD). If any of the weak duality in Theorem 4.1-4.3 also holds, then ([bar.x], [bar.[tau]], [bar.[lambda]]) is an efficient solution for (MWD).

Proof. Since [bar.x] is efficient for (VP) and satisfies the constraints qualification for (VP), then from the Kuhn-Tucker necessary optimality condition, we obtain [bar.[tau]] > 0 and [bar.[lambda]] [??] 0 such that

f'[([bar.x], [??]([bar.x], x)).sup.T] [bar.[tau]] + g'[([bar.x], [??]([bar.x], x)).sup.T][bar.[lambda]] = 0, [[bar.[lambda]].sup.T] g([bar.x]) = 0.

The vector [tau] may be normalized according to [[bar.[tau].sup.T] e = 1, [bar.[tau]] > 0, which gives that the triplet ([bar.x], [bar.[tau]], [bar.[lambda]]) is feasible for (MWD). The efficient of ([bar.x], [bar.[tau]], [bar.[lambda]]) for (MWD) follows from weak duality theorem. This completes the proof. []

5. Mixed type duality

In this section, we consider the following mixed type dual (See [23]) of (VP) defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

where e = [(1, 1, ..., 1).sup.T] [member of] [R.sup.p] and [J.sub.t], 0 [??] t [??] r, are partitions of the set M = {1, 2, ..., mg}.

Theorem 5.1 (Weak duality)Suppose that for all feasible x for (VP) and all feasible (y, [tau], [lambda]) for (MD):

(a) [tau] > 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is strong pseudoquasi type I semi-d-univex at y for each t, 1 [??] t [??] r, with respect to [b.sub.0], [b.sub.1], [[phi].sub.0], [[phi].sub.1] and [eta];

(b)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is weak strictly pseudoquasi type I semi-d-univex at y for each t, 1 [??] t [??] r, with respect to [b.sub.0], [b.sub.1], [[phi].sub.0], [[phi].sub.1] and [eta];

(c)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is weak strictly pseudo type I semi-d-univex at y for each t, 1 [??] t [??] r, with respect to [b.sub.0], [b.sub.1], [[phi].sub.0], [[phi].sub.1] and [eta]; then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Suppose contrary to the result. i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since x is feasible for (VP) and [lambda] [??] 0, the above inequality implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

Using the definitions of [[phi].sub.0], [[phi].sub.1] and (5.1), (5.2), one gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

Combining (5.3) and conditions (a), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [tau] > 0, the above inequality give

f'[(y, [??](y, x)).sup.T] [tau] + g'[(y, [??](y, x)).sup.T][lambda] < 0,,

which contradicts (5.1).

The remain proof is similar to the one of case (a). []

Theorem 5.2 (Strong duality)Let [bar.x] be an efficient solution for (VP) and [bar.x] satisfies a constraint qualification for (VP). Then there exist [bar.[tau]] [member of] [R.sup.p] and [bar.[lambda]] [member of] [R.sup.m] such that ([bar.x], [bar.[tau]], [bar.[lambda]]) is feasible for (MD). If the weak duality in Theorem 5.1 also holds, then ([bar.x], [bar.[tau]], [bar.[lambda]]) is an efficient solution for (MD).

Proof. Since [bar.x] is efficient for (VP) and satisfies the constraints qualification for (VP), then from the Kuhn-Tucker necessary optimality condition in [24], we obtain [bar.[tau]] > 0 and [bar.[lambda]] [??] 0 such that

f'[([bar.x], [??]([bar.x], x)).sup.T] [bar.[tau]] + g'[(x, [??]([bar.x], x)).sup.T][bar.[lambda]] = 0, [bar.[lambda].sup.T] g([bar.x]) = 0.

The vector [bar.[tau]] may be normalized according to [bar.[tau]]T e = 1, [bar.[tau]] > 0, which gives that the triplet ([bar.x], [bar.[tau]], [bar.[lambda]]) is feasible for (MD). The efficient of ([bar.x], [bar.[tau]], [bar.[lambda]]) for (MD) follows from weak duality theorem. This completes the proof. []

References

[1] B. Aghezzaf, M. Hachimi, generalized invexity and duality in multiobjective programming problems, Journal of Global Optimization. 18(2000), 91-101.

[2] B. Aghezzaf, M. Hachimi, Sufficient conditions and duality in multiobjective optimization involving generalized convexity, Numerical Functional Analysis and Optimization. 22(2002), 775-788.

[3] T. Antczak, R. Nehse, Mnltiobjective programming under d-invexity, European Journal of Operational Research. 137(2002), 28-36.

[4] S. K. Mishra, Shou-Yang Wang, K. K. Lai, Nondifferentiable Multiobjective programming under generalied d-uninvexity, European Journal of Operational Research. 160(2005), 218-226.

[5] C. R. Bector, S.K. Suneja, S. Gupta, Uninvex functions and uninvex nonlinear programming, in: Proceedings of the Administrative Science Association of Canada, 1992, pp. 115-124.

[6] S. K. Mishra, Shou-Yang Wang, K.K.Lai, Optimality and duality for multiobjective optimization under generilized type I uninvexity, Journal of Mathematical Analysis and Applications. 303(2005), 315-326.

[7] B. D. Craven, Nondifferentiable optimization by smooth approximation, Optimization. 17(1986), 3-17.

[8] R. R. Egudo, M. A. Hanson, Mnltiobjective duality with invexity, Journal of Mathematical Analysis and Applications. 126(1987), 469-477.

[9] M. A. Hanson, On sufficiency of the Kuhn-Tucker conditions, Journal of Mathematical Analysis and Applications. 80(1981), 545-550.

[10] M. A. Hanson, B. Mond, Necessary and sufficiency conditions in constrained optimization, Mathematical Programming. 37(1987), 51-58.

[11] V. Jeyakumar, B. Mond, On generalized convex mathematical programming, Journal of Australian Mathematical Society, Series B. 34(1992), 43-53. Nondifferentiable Multiobjective Programming 75

[12] R. N. Kaul, S. K. Suneja, S. K. Mishra, Optimality criteria and duality in multiobjective optimization involving generalized convexity, Journal of Optimization Theory and its Applications. 80(1994), 465-482.

[13] S. K. Mishra, Lagrange multipliers saddle points and scalarizations in composite multiobjective nonsmooth programming, Optimization. 38(1996), 93-105.

[14] S. K. Mishra, On sufficiency and duality in nonsmooth multiobjective programming, Opsearch. 34(1997), 221-231.

[15] S. K. Mishra, On multiobjective programming with generalized uninvexity, Journal of Mathematical Analysis and Applications. 224(1998), 131-148.

[16] S. K. Mishra, G. Giorgi, Optimization and duality with generalized semi-uninvexity, Opsearch. 37(2000), 340-350.

[17] S. K. Mishra, R. N. Mukherjee, On generalized convex multiobjective nonsmooth programming, Journal of Australian Mathematical Society, Series B. 38(1996), 140-148.

[18] T. W. Reiland, Nonsmooth invexity, Bulletin of Australian Mathematical Society. 42(1990), 437-446.

[19] N. G. Rueda, M. A. Hanson, Optimality criteria and duality in mathematical programming involving generalized invexity, Journal of Mathematical Analysis and Applications. 130(1988), 375-385.

[20] N. G. Rueda, M. A. Hanson, C. Singh, Optimality and duality with generalized convexity, Journal of Optimization Theory and its Applications. 86(1995), 491-500.

[21] Y. L. Ye, d-invexity and optimality conditions, Journal of Mathematical Analysis and Applications. 162(1991), 242-249.

[22] X. Q. Yang, Guang-Ya Chen, A class of nonconvex functions and pre-variational inequality, Journal of Mathematical Analysis and Applications. 169(1992), 359-373.

[23] Z. K. Xu, Mixed type duality in multiobjective programming problems, Journal of Mathematical Analysis and Applications. 198(1996), 621-635.

[24] I. Marusciac, On Fritz John optimality criterion in multiobjective optimization, Analysis Numerical Theory Approximation. 11(1982), 109-114.

[25] R. R. Egudo, Efficiency and generalized convex duality for multiobjective programs, Journal of Mathematical Analysis and Applications. 138(1989), 84-94.

Qing-jie Hu (1,2) and Yu Chen (2)

(1) College of Mathematics and Econometrics, Hunan University, 410082, Changsha, P.R. China E-mail: hqj0525@126.com

(2) Department of Information, Hunan Business College, 410205, Changsha, P.R. China
COPYRIGHT 2007 Research India Publications
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2007 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Qing-jie, Hu; Yu, Chen
Publication:Advances in Theoretical and Applied Mathematics
Article Type:Report
Geographic Code:9CHIN
Date:May 1, 2007
Words:4191
Previous Article:Some properties on generalizations of Lindelof spaces in bitopological space.
Next Article:Counting the number of Pythagorean triples in finite fields (1).
Topics:

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters