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1.
Richard A. Vitale 《Journal of Approximation Theory》1979,26(4):301-316
Approximation of set-valued functions is introduced and discussed under a convexity assumption. In particular, a theorem on positive linear operators is given. 相似文献
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Summary It is shown that every convex set-valued function defined on a cone with a cone-basis in a real linear space with compact values in a real locally convex space has an affine selection. Similar results can be obtained for midconvex set-valued functions. 相似文献
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Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399). 相似文献
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Generalized convex functions and vector variational inequalities 总被引:3,自引:0,他引:3
X. Q. Yang 《Journal of Optimization Theory and Applications》1993,79(3):563-580
In this paper, (, ,Q)-invexity is introduced, where :X ×X intR
m
+
, :X ×X X,X is a Banach space,Q is a convex cone ofR
m
. This unifies the properties of many classes of functions, such asQ-convexity, pseudo-linearity, representation condition, null space condition, andV-invexity. A generalized vector variational inequality is considered, and its equivalence with a multi-objective programming problem is discussed using (, ,Q)-invexity. An existence theorem for the solution of a generalized vector variational inequality is proved. Some applications of (, ,Q)-invexity to multi-objective programming problems and to a special kind of generalized vector variational inequality are given.The author is indebted to Dr. V. Jeyakumar for his constant encouragement and useful discussion and to Professor P. L. Yu for encouragement and valuable comments about this paper. 相似文献
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Hugo Leiva Nelson Merentes Kazimierz Nikodem José Luis Sánchez 《Journal of Global Optimization》2013,57(3):695-705
We introduce the notion of strongly $t$ -convex set-valued maps and present some properties of it. In particular, a Bernstein–Doetsch and Sierpiński-type theorems for strongly midconvex set-valued maps, as well as a Kuhn-type result are obtained. A representation of strongly $t$ -convex set-valued maps in inner product spaces and a characterization of inner product spaces involving this representation is given. Finally, a connection between strongly convex set-valued maps and strongly convex sets is presented. 相似文献
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Joël Benoist Nicolae Popovici 《Journal of Mathematical Analysis and Applications》2003,288(1):161-166
The aim of this paper is to show that under a mild semicontinuity assumption (the so-called segmentary epi-closedness), the cone-convex (respectively, cone-quasiconvex) set-valued maps can be characterized in terms of weak cone-convexity (respectively, weak cone-quasiconvexity), i.e., the notions obtained by replacing in the classical definitions the conditions of type “for all x,y in the domain and for all t in ]0,1[…” by the corresponding conditions of type “for all x,y in the domain there exists t in ]0,1[….” 相似文献
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《Optimization》2012,61(11):2227-2245
ABSTRACTIn this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base. 相似文献
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It is proved that ifX andY are linear spaces andF :X p(Y) is a set-valued map with convex graph such thatF(x) Ø for allx X andF(x
0) is a singleton for somex
0, thenF is single-valued and affine. Applications to metric projections and to adjoints of set-valued maps are given.Supported by NSF Grant DMS-9100228.The main result of this paper has been obtained while the second author was visiting the Pennsylvania State University in the framework of the exchange agreement between the Romanian Academy and the National Academy of Sciences of the U.S.A. 相似文献
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Selden Y. Trimble 《Mathematische Zeitschrift》1969,109(2):112-114
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In this paper, we study the approximate solutions for vector optimization problem with set-valued functions. The scalar characterization
is derived without imposing any convexity assumption on the objective functions. The relationships between approximate solutions
and weak efficient solutions are discussed. In particular, we prove the connectedness of the set of approximate solutions
under the condition that the objective functions are quasiconvex set-valued functions. 相似文献
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几乎锥-次类凸向量集值优化的Benson真有效性 总被引:3,自引:0,他引:3
§ 1 IntroductionIn [1 ],Yang ,LiandWangintroducedanewclassofgeneralizedconvexset valuedfunc tions,termednearlycone subconvexlikefunctions.In [2 ],ChenandRongstudiedBensonproperefficiencyinvectoroptimizationwithgeneralizedcone subconvexlikevector valuedfunc tions… 相似文献
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V. Yu. Protasov 《Functional Analysis and Its Applications》2011,45(1):46-55
We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x
1 + x
2) ⊂ φ(x
1) + φ(x
2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is
proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces. 相似文献
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Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings
In this paper, some gap functions for three classes of a system of generalized vector quasi-equilibrium problems with set-valued
mappings (for short, SGVQEP) are investigated by virtue of the nonlinear scalarization function of Chen, Yang and Yu. Three
examples are then provided to demonstrate these gap functions. Also, some gap functions for three classes of generalized finite
dimensional vector equilibrium problems (GFVEP) are derived without using the nonlinear scalarization function method. Furthermore,
a set-valued function is obtained as a gap function for one of (GFVEP) under certain assumptions.
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