New optimality conditions and duality results of G type in differentiable mathematical programming

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Abstract

In this paper, a new class of differentiable functions, called G-invex functions with respect to η, is introduced by extending the definition of invex functions. New necessary optimality conditions of G-F. John and G-Karush–Kuhn–Tucker type are obtained for differentiable constrained mathematical programming problems. The G-invexity concept introduced is used to prove the sufficiency of these necessary optimality conditions. Further, a so-called G-Mond–Weir-type dual is formulated and various duality results are also established by assuming the functions involved to be G-invex with respect to the same function η.

Introduction

In the theory of constrained extremum problems, optimality conditions and duality results for differentiable nonlinear constrained problems are important theoretically as well as computationally and can be formulated in several different ways. In general, these criteria can be classified as either necessary or sufficient. The best-known necessary optimality criterion for a constrained mathematical programming problem is constituted by the Karush–Kuhn–Tucker optimality conditions (see, for example, [13]). However, the F. John criterion [13] is in a sense more general. In order for the Karush–Kuhn–Tucker optimality conditions to hold, one must impose some suitable constraint qualification [8], [13] on the constraints of the optimization problem. On the other hand, no such constraint qualification need be imposed on the constraints in order for the F. John optimality conditions to hold. In constrained optimization these necessary optimality conditions are also sufficient for optimality if the functions delimiting the mathematical programming problem are convex or satisfy certain generalized convexity properties such as pseudo-convexity or quasi-convexity (see, for example, [8], [13], [15].

However, in the recent years attempts have been made by several authors to define various classes of differentiable nonconvex functions and to study their duality and optimality criteria for solving such problems (see, for example, [1], [11], [12], [14], [18], [20], and others). One such generalization convex function is the invexity notion introduced by Hanson [11]. The term invex (which means invariant convex) was suggested by Craven [10]. Over the years, many generalizations of this concept have been given in the literature. For example, the concept of invexity of functions was also generalized to B-invex functions by Suneja, Singh and Bector [19]. Using the definition of a weighted r-mean (where r is a real number) for a sequence of positive numbers, Antczak [1] introduced new classes of (nonconvex) differentiable functions and called them (p,r)-invex with respect to η, and also called their subclass r-invex functions with respect to η [3]. The class of (p,r)-invex functions with respect to η is an extension of the class of invex functions with respect to η introduced by Hanson [11].

In this paper, we introduce a new class of nonconvex functions, called G-invex functions. We extend an invexity notion [11] since the defined class of functions contains many various invexity concepts. A characteristic global optimality property of various classes of invex functions is also proved in the case of a G-invex function. It turns out that every stationary point of a G-invex function is its global minimum point. A sufficient condition for G-invexity is also proved in the paper. Also, we state some relations between the introduced class of G-invex functions with respect to η and various classes of invex functions with respect to η.

Furthermore, we introduce new F. John-type and Karush–Kuhn–Tucker-type problems, called G-F. John and G-Karush–Kuhn–Tucker problems, respectively; they are defined assuming differentiability of the functions involved. Then, we apply the G-invexity notion introduced to develop optimality conditions of F. John type and Karush–Kuhn–Tucker type for constrained differentiable mathematical programming problems. We prove the sufficiency of the necessary optimality conditions of G type defined for differentiable constrained optimization problems involving G-invex functions with respect to the same function η, but not necessarily with respect to the same function G. In particular, we obtain optimality conditions of F. John type and Karush–Kuhn–Tucker type that are weaker than previous conditions presented in the literature.

Further considerations are devoted to duality in differentiable constrained mathematical programming problems. We define a new dual of Mond–Weir type, called the G-Mond–Weir dual, for the constrained optimization problem considered. Then, various duality theorems are proved between the mathematical programming problem considered and the G-Mond–Weir dual problem introduced. The main tool in proving these duality results is the concept of G-invexity introduced. As in the case of establishing the sufficiency property for optimization problems of this type, we assume that all functions involved in the original optimization problems areG-invex with respect to the same function η, but not necessarily with respect to the same function G. Also some examples are given to illustrate a nature of the class of nonconvex functions introduced.

Section snippets

G-invex functions

In this section, we provide some definitions and some results that we shall use in the sequel.

Definition 1

A function f:RR is said to be increasing if and only if x,yRx<yf(x)<f(y).

We shall also use a definition of an invex set with respect to η. The definition of a set of this type was given by Ben-Israel and Mond [9] and subsequently studied by many authors including Mohan and Neogy [16], Pini [17], Antczak [1] and [2].

Definition 2

Let X be a nonempty subset of Rn, η:X×XRn and let u be an arbitrary point of X.

Optimality conditions for nonlinear mathematical programming problems with G-invex functions

Now, we consider the following nonlinear mathematical programming problem (P): f(x)mingi(x)0,iJ={1,,m}, where f:XR, gi:XR, iJ, are (continuously) differentiable functions, and X is a nonempty open subset of Rn. We denote the set of all feasible solutions in (P) by D:={xX:gi(x)0,iJ}. Now, we give our statement of the modified F. John problem called the G-F. John problem:

G-F. John problem:Find (x¯,λ¯,ξ¯)D×R×Rm (if it exists) such thatλ¯Gf(f(x¯))f(x¯)+i=1mξ¯iGgi(gi(x¯))gi(x¯)=0,ξ¯i[

Duality

In this section, we consider a modified Mond–Weir duality for the mathematical programming problem considered in Section 3.

We define the following dual problem of Mond–Weir type f(y)maxλGf(f(y))f(y)+i=1mξiGgi(gi(y))gi(y)=0,(MWD)ξiGgi(gi(y))0,iJ.yX,λ>0,ξ0, where Gf and Ggi, iJ, are defined as in the preceding section. We call it the G-Mond–Weir-type dual problem.

Let W={(y,λ,ξ)X×R×Rm:λGf(f(y))f(y)+i=1mξiGgi(gi(y))gi(y)=0,ξiGgi(gi(y))0,iJ,λ>0,ξ0} be the set of all feasible

Conclusions

This work provides new optimality conditions and new duality results for differentiable mathematical problems. We have established new necessary optimality conditions of F. John and Karush–Kuhn–Tucker type, called G-F. John and G-Karush–Kuhn–Tucker conditions, respectively, for differentiable constrained optimization problems. It is pointed out that our statements of the F. John and Karush–Kuhn–Tucker necessary optimality conditions of G type are more general than the classical ones found for

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