Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that th...
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Martin (Ref. 1) studied the optimality conditions of invex functions for scalar programming problems. In this work, we generalize his results making them applicable to vectorial optimization problems. We prove that the equivalence between minima and stationary points or Kuhn-Tucker points (depending on the case) remains true if we optimize several objective functions instead of one objective function. To this end, we define accurately stationary points and Kuhn-Tucker optimality conditions for multiobjective programming problems. We see that the Martin results cannot be improved in mathematical programming, because the new types of generalized convexity that have appeared over the last few years do not yield any new optimality conditions for mathematical programming problems.
In this note, we give new proofs for the well-known results including characterizations of prequasi-invex function and preinvex function under some assumptions.
In this note, we give new proofs for the well-known results including characterizations of prequasi-invex function and preinvex function under some assumptions.
We introduce a notion of a second-order invex function. A Frechet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Frechet differ...
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We introduce a notion of a second-order invex function. A Frechet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Frechet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.
In this article we introduce a notion of a higher order invex scalar function in terms of the higher order lower Dini directional derivatives. This notion differs from the respective ones, applied in duality theory. H...
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In this article we introduce a notion of a higher order invex scalar function in terms of the higher order lower Dini directional derivatives. This notion differs from the respective ones, applied in duality theory. Higher order invex functions are expanding classes of functions in the sense that every invex function of order n (n is a positive integer) is invex of order (n+1). We prove that a function f is invex of order n if and only if the set of stationary points of order n of f coincides with the set of global minimizers. We extend the known property that, if we do not specify the kernel , then a differentiable function is invex if and only if it is pseudoinvex, to a result, which includes higher order Dini derivatives. We introduce a notion of a pseudoinvex function of order n with respect to a known map . The pseudoinvex functions of order n are also expanding classes, intermediate between pseudoinvex and prequasiinvex functions. Further, we obtain characterizations of the solution set of the minimization problem of a pseudoinvex function of order n over an invex set, provided that a fixed solution is known. Some known results become particular cases of our theorems.
In this paper, a new class of generalized convex function is introduced, which is called the strongly alpha-preinvex function. We study some properties of strongly alpha-preinvex function. In particular, we establish ...
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In this paper, a new class of generalized convex function is introduced, which is called the strongly alpha-preinvex function. We study some properties of strongly alpha-preinvex function. In particular, we establish the equivalence among the strongly alpha-preinvex functions, strongly alpha-invex functions and strongly alpha eta-monotonicity under some suitable conditions. As special cases, one can obtain several new and previously known results for alpha-preinvex (invex) functions. (c) 2005 Elsevier Inc. All rights reserved.
We formulate two pairs of symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones. By using the concept of efficiency, we establish the weak, strong, converse and self-duality theorem...
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We formulate two pairs of symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones. By using the concept of efficiency, we establish the weak, strong, converse and self-duality theorems for our symmetric models. Several known results are obtained as special cases. (C) 2001 Elsevier Science B.V. All rights reserved.
In this paper we define two notions: Kuhn-Tucker saddle point invex problem with inequality constraints and Mond-Weir weak duality invex one. We prove that a problem is Kuhn-Tucker saddle point invex if and only if ev...
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In this paper we define two notions: Kuhn-Tucker saddle point invex problem with inequality constraints and Mond-Weir weak duality invex one. We prove that a problem is Kuhn-Tucker saddle point invex if and only if every point, which satisfies Kuhn-Tucker optimality conditions forms together with the respective Lagrange multiplier a saddle point of the Lagrange function. We prove that a problem is Mond-Weir weak duality invex if and only if weak duality holds between the problem and its Mond-Weir dual one. Additionally, we obtain necessary and sufficient conditions, which ensure that strong duality holds between the problem with inequality constraints and its Wolfe dual. Connections with previously defined invexity notions are discussed.
In this paper, several kinds of invariant monotone maps and generalized invariant monotone maps are introduced. Some examples are given which show that invariant monotonicity and generalized invariant monotonicity are...
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In this paper, several kinds of invariant monotone maps and generalized invariant monotone maps are introduced. Some examples are given which show that invariant monotonicity and generalized invariant monotonicity are proper generalizations of monotonicity and generalized monotonicity. Relationships between generalized invariant monotonicity and generalized invexity are established. Our results are generalizations of those presented by Karamardian and Schaible.
We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps...
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We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps is introduced;it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar.
In this paper the generalized invex monotone functions are defined as an extension of monotone functions. A series of sufficient and necessary conditions are also given that relate the generalized invexity of the func...
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In this paper the generalized invex monotone functions are defined as an extension of monotone functions. A series of sufficient and necessary conditions are also given that relate the generalized invexity of the function theta with the generalized invex monotonicity of its gradient function deltheta. This new class of functions will be important in order to characterize the solutions of the Variational-like Inequality Problem and Mathematical Programming Problem. (C) 2002 Elsevier Science B.V. All rights reserved.
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