In this paper, we are concerned with a nondifferentiable minimax fractional problem with inequality constraints. We introduce a new class of generalizedconvex function, that is, nonsmooth generalized (F, rho, theta)-...
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In this paper, we are concerned with a nondifferentiable minimax fractional problem with inequality constraints. We introduce a new class of generalizedconvex function, that is, nonsmooth generalized (F, rho, theta)-d-univex function. In the framework of the new concept, we derive Kuhn-Tucker type sufficient optimality conditions and establish weak, strong and converse duality theorems for the problem and its three different types of dual problems. (c) 2006 Elsevier Inc. All rights reserved.
Let f greater than or equal to 0 be a continuous, concave function on [a, b]. Let (f) over bar = (1/(b - a)) integral(a)(b) f (t) dt. Favard's inequality is that, when delta = (f) over bar 1/2delta integral((f) ov...
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Let f greater than or equal to 0 be a continuous, concave function on [a, b]. Let (f) over bar = (1/(b - a)) integral(a)(b) f (t) dt. Favard's inequality is that, when delta = (f) over bar 1/2delta integral((f) over bar-delta)((f) over bar+delta) psi(u) du greater than or equal to 1/b-a integral(a)(b) psi(f(t)) dt (1) for all convexfunctions, psi, defined on ((f) over bar - delta, (f) over bar + delta). We show there is a delta for which inequality (1) is valid for a class of nonconvexfunctions psi. Further, there is an optimal delta for which the reverse inequality of line (1) is true. The reverse inequality is strictly sharper (in this setting) then Jensen's inequality. (C) 2003 Elsevier Science Ltd. All rights reserved.
In this paper, we introduce a class of generalized invex n-set functions, called (rho,sigma,theta)-type-I and related non-convexfunctions, and then establish a number of parametric and semi-parametric sufficient opti...
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In this paper, we introduce a class of generalized invex n-set functions, called (rho,sigma,theta)-type-I and related non-convexfunctions, and then establish a number of parametric and semi-parametric sufficient optimality conditions for the primal problem under the aforesaid assumptions. This work partially extends an earlier work of Mishra et al. (Math. Methods Oper. Res. 67, 493-504, 2008) to a wider class of functions.
We introduce and study the notion of e-subdifferential for an e-convex function in locally convex spaces. Some relationships between the e-subdifferential and the Clarke-Rockafellar subdifferential are established. Be...
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We introduce and study the notion of e-subdifferential for an e-convex function in locally convex spaces. Some relationships between the e-subdifferential and the Clarke-Rockafellar subdifferential are established. Besides, the e-convexfunctions are characterized. Moreover, some results regarding the locally Lipschitz properties of e-convexfunctions are obtained.
Some properties of sigma-convexfunctions are studied. Then some relations between the notions of sigma-subdifferentials and sigma-convexfunctions are established. Also, we present some results regarding the sum form...
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Some properties of sigma-convexfunctions are studied. Then some relations between the notions of sigma-subdifferentials and sigma-convexfunctions are established. Also, we present some results regarding the sum formula for the sigma-subdifferential. Moreover, we obtain some particular relationships between the sigma-subdifferential and the (sigma, y)-conjugate. Finally, by imposing some assumptions on f and sigma, the maximal 2 sigma-monotonicity of sigma-subdifferential is studied. Indeed, the maximality of the usual Fenchel subdifferentials of lower semicontinuous convexfunctions follows from our result.
A new class of generalized convex functions, called the functions with pseucloconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is ...
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A new class of generalized convex functions, called the functions with pseucloconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets. (c) 2008 Elsevier Inc. All rights reserved.
In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of...
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In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Frechet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order n.
Some properties of strongly convexfunctions are presented. A characterization of pairs of functions that can be separated by a strongly convex function and a Hyers-Ulam stability result for strongly convexfunctions ...
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Some properties of strongly convexfunctions are presented. A characterization of pairs of functions that can be separated by a strongly convex function and a Hyers-Ulam stability result for strongly convexfunctions are given. An integral Jensen-type inequality and a Hermite-Hadamard-type inequality for strongly convexfunctions are obtained. Finally, a relationship between strong convexity and generalizedconvexity in the sense of Beckenbach is shown.
In this article we study a recently introduced notion of non-smooth analysis, namely convexifactors. We study some properties of the convexifactors and introduce two new chain rules. A new notion of non-smooth pseudoc...
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In this article we study a recently introduced notion of non-smooth analysis, namely convexifactors. We study some properties of the convexifactors and introduce two new chain rules. A new notion of non-smooth pseudoconvex function is introduced and its properties are studied in terms of convexifactors. We also present some optimality conditions for vector minimization in terms of convexifactors.
We introduce two pairs of nondifferentiable multiobjective second order symmetric dual problems with cone constraints over arbitrary closed convex cones, which is different from the one proposed by Mishra and Lai [12]...
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We introduce two pairs of nondifferentiable multiobjective second order symmetric dual problems with cone constraints over arbitrary closed convex cones, which is different from the one proposed by Mishra and Lai [12]. Under suitable second order pseudo-invexity assumptions we establish weak, strong and converse duality theorems as well as self-duality relations. Our symmetric duality results include an extension of the symmetric duality results for the first order case obtained by Kim and Kim [7] to the second order case. Several known results are abtained as special cases.
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