The notion of s-monotone polarity for s-subdifferential is introduced and studied. Also, the concept of Frechet s-subdifferential is introduced and then some results regarding this concept are obtained. In addition, s...
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The notion of s-monotone polarity for s-subdifferential is introduced and studied. Also, the concept of Frechet s-subdifferential is introduced and then some results regarding this concept are obtained. In addition, some particular relationships between the s-subdifferential and Frechet s-subdifferential are presented.
In this paper, we introduce a new second-order directional derivative and a second-order subdifferential of Hadamard type for an arbitrary nondifferentiable function. We derive second-order necessary and sufficient op...
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In this paper, we introduce a new second-order directional derivative and a second-order subdifferential of Hadamard type for an arbitrary nondifferentiable function. We derive second-order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second-order in unconstrained optimization. In particular, we obtain two results with strongly pseudoconvexfunctions. We also compare our conditions with the results of the recently published paper (Bednarik and Pastor, 2008) and a lot of other works, published in high level journals, and prove that they are particular cases of our necessary and sufficient ones. We prove that the necessary optimality conditions concern more functions than the conditions in terms of lower Dini directional derivative, even the optimality conditions with the last derivative can be applied to a function, which does not belong to some special class. At last, we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone-constrained vector optimization. (C) 2015 Elsevier Ltd. All rights reserved.
In this paper, we review and unify some classes of generalized convex functions introduced by different authors to prove minimax results in infinite-dimensional spaces and show the relations between these classes. We ...
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In this paper, we review and unify some classes of generalized convex functions introduced by different authors to prove minimax results in infinite-dimensional spaces and show the relations between these classes. We list also for the most general class already introduced by Jeyakumar (Ref. 1) an elementary proof of a minimax result. The proof of this result uses only a finite-dimensional separation theorem;although this minimax result was already presented by Neumann (Ref. 2) and independently by Jeyakumar (Ref. 1), we believe that the present proof is shorter and more transparent.
We firstly establish an identity involving local fractional integrals. Then, with the help of this equality, some new Newton-type inequalities for functions whose the local fractional derivatives in modulus and their ...
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We firstly establish an identity involving local fractional integrals. Then, with the help of this equality, some new Newton-type inequalities for functions whose the local fractional derivatives in modulus and their some powers are generalizedconvex are obtained. Some applications of these inequalities for Simpson's quadrature rules and generalized special means are also given.
The notion of strongly n-convexfunctions with modulus c > 0 is introduced and investigated. Relationships between such functions and n-convexfunctions in the sense of Popoviciu as well as generalizedconvex funct...
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The notion of strongly n-convexfunctions with modulus c > 0 is introduced and investigated. Relationships between such functions and n-convexfunctions in the sense of Popoviciu as well as generalized convex functions in the sense of Beckenbach are given. Characterizations by derivatives are presented. Some results on strongly Jensen n-convexfunctions are also given. (C) 2010 Elsevier Ltd. All rights reserved.
This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalizedconvexity properties of functions in the case of gradient maps. In the pre...
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This paper is a sequel to Ref. 1 in which several kinds of generalized monotonicity were introduced for maps. They were related to generalizedconvexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case.
In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We ach...
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In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalizedconvexity of a real valued function, which is obtained out of its local counterpart on some dense sets.
Known as well as new types of monotone and generalized monotone maps are considered. For gradient maps, these generalized monotonicity properties can be related to generalizedconvexity properties of the underlying fu...
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Known as well as new types of monotone and generalized monotone maps are considered. For gradient maps, these generalized monotonicity properties can be related to generalizedconvexity properties of the underlying function. In this way, pure first-order characterizations of various types of generalized convex functions are obtained.
Based on the strong stationary condition, we formulate second-order and higher order duals for a mathematical program with complementary constraints (MPCC). Under suitable generalizedconvexity assumptions, we then pr...
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Based on the strong stationary condition, we formulate second-order and higher order duals for a mathematical program with complementary constraints (MPCC). Under suitable generalizedconvexity assumptions, we then propose several duality theorems for second and higher order duality in MPCC problems, including the weak duality, strong duality, converse duality, and strict converse duality theorems. Moreover, we present examples to illustrate these results, where one of them reveals that a lower bound provided by our proposed second-order duality is tighter than the one given by the first-order duality.
We introduce a pair of multiobjective generalized second order symmetric dual programs where the objective function contains a support function. Weak, strong and converse duality theorems for these second order proble...
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We introduce a pair of multiobjective generalized second order symmetric dual programs where the objective function contains a support function. Weak, strong and converse duality theorems for these second order problems are established under suitable generalized second order convexity assumptions. Also, we give some special cases of our second order symmetric duality results.
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