共查询到20条相似文献,搜索用时 93 毫秒
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In this article the existence of the convex extension of convex set valued map is considered. Conditions are obtained, based on the notion of the derivative of set valued maps, which guarantee the existence of convex extension. The conditions are given, when the convex set valued map has no convex extension. The convex set valued map is specified, which is the maximal convex extension of the given convex set valued map and includes all other convex extensions. The connection between Lipschitz continuity and existence of convex extension of the given convex set valued map is studied. 相似文献
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Patrick N. Dowling 《Journal of Mathematical Analysis and Applications》2003,288(2):540-543
The ψ-direct sum of Banach spaces is strictly convex (respectively, uniformly convex, locally uniformly convex, uniformly convex in every direction) if each of the Banach spaces are strictly convex (respectively, uniformly convex, locally uniformly convex, uniformly convex in every direction) and the corresponding ψ-norm is strictly convex. 相似文献
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主要研究了两类近似凸集的关系和性质.首先,举例说明两类近似凸集没有相互包含关系.其次,在近似凸集(nearly convex)条件下,证明了在一定条件下函数上图是近似凸集与凸集的等价关系.同时,考虑了近似凸函数与函数上图是近似凸集的等价刻画、近似凸函数与函数水平集是近似凸集的必要性,并用例子说明近似凸函数与函数水平集是... 相似文献
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After a brief survey on condition numbers for linear systems of equalities, we analyse error bounds for convex functions and convex sets. The canonical representation of a convex set is defined. Other representations of a convex set by a convex function are compared with the canonical representation. Then, condition numbers are introduced for convex sets and their convex representations. 相似文献
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Marek Cezary Zdun 《Journal of Difference Equations and Applications》2018,24(5):773-783
AbstractA function f is said to be iteratively convex if f possesses convex iterative roots of all orders. We give several constructions of iteratively convex diffeomorphisms and explain the phenomenon that the non-existence of convex iterative roots is a typical property of convex functions. We show also that a slight perturbation of iteratively convex functions causes loss of iterative convexity. However, every convex function can be approximate by iteratively convex functions. 相似文献
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J. Brinkhuis 《Journal of Optimization Theory and Applications》2009,143(3):439-453
An attempt is made to justify results from Convex Analysis by means of one method. Duality results, such as the Fenchel-Moreau
theorem for convex functions, and formulas of convex calculus, such as the Moreau-Rockafellar formula for the subgradient
of the sum of sublinear functions, are considered. All duality operators, all duality theorems, all standard binary operations,
and all formulas of convex calculus are enumerated. The method consists of three automatic steps: first translation from the
given setting to that of convex cones, then application of the standard operations and facts (the calculi) for convex cones,
finally translation back to the original setting. The advantage is that the calculi are much simpler for convex cones than
for other types of convex objects, such as convex sets, convex functions and sublinear functions. Exclusion of improper convex
objects is not necessary, and recession directions are allowed as points of convex objects. The method can also be applied
beyond the enumeration of the calculi. 相似文献
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Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions. 相似文献
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Mohsen Alimohammady Yeol Je Cho Vahid Dadashi Mehdi Roohi 《Applied Mathematics Letters》2011,24(8):1289-1294
This paper deals with some basic notions of convex analysis and convex optimization via convex semi-closed functions. A decoupling-type result and also a sandwich theorem are proved. As a consequence of the sandwich theorem, we get a convex sub-differential sum rule and two separation results. Finally, the derived convex sub-differential sum rule is applied to solving the convex programming problem. 相似文献
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The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which is described by inequality constraints that are locally Lipschitz and not necessarily convex or differentiable. We show that if the Slater constraint qualification and a simple non-degeneracy condition is satisfied then the Karush–Kuhn–Tucker type optimality condition is both necessary and sufficient. 相似文献
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Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones. 相似文献
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关于K—极凸Banach空间 总被引:3,自引:0,他引:3
何仁义 《纯粹数学与应用数学》1998,14(2):19-22
引进K-极凸Banach空间,证明了XK-极凸当且仅当X自反、K-严格凸且有(H)性质,得到了K-极凸空间的一些性质,并讨论了K-极凸与K-K-强光滑、K-一致凸及完全K-凸的关系。 相似文献
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In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k-strongly convex space and k-very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory. 相似文献
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In this paper, we consider the composed convex optimization problem which consists in minimizing the sum of a convex function and a convex composite function. By using the properties of the epigraph of the conjugate functions and the subdifferentials of convex functions, we give some new constraint qualifications which completely characterize the strong Fenchel duality and the total Fenchel duality for composed convex optimiztion problem in real locally convex Hausdorff topological vector spaces. 相似文献
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In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron. 相似文献
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In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard. 相似文献
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Strong restricted-orientation convexity is a generalization of standard convexity. We explore the properties of strongly convex sets in multidimensional Euclidean space and identify major properties of standard convex sets that also hold for strong convexity. We characterize strongly convex flats and halfspaces, and establish the strong convexity of the affine hull of a strongly convex set. We then show that, for every point in the boundary of a strongly convex set, there is a supporting strongly convex hyperplane through it. Finally, we show that a closed set with nonempty interior is strongly convex if and only if it is the intersection of strongly convex halfspaces; we state a condition under which this result extends to sets with empty interior. 相似文献