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1.
Banach空间的p— Asplund 伴随空间 总被引:4,自引:1,他引:3
程立新 《应用泛函分析学报》2001,3(2):120-128
我们称一个定义在Banach空间E上的连续凸函数f具有Frechet可微性质(FDP),如果E上的每个实值凸函数g≤f均在E一个稠密的Gδ-子集上Frechet可微。本文主要证明了:对任何Banach空间E,均存在一个局部凸相容拓扑p使得1)(E,p)是Hausdorff局部凸空间;2) E上的每个范数连续具有FDP的凸函数均是p-连续的;3)每个p-连续的凸函数均具有FDP ;4)p等价某个范数拓扑当且仅不E是Asplund空间。 相似文献
2.
C. Zălinescu 《Journal of Global Optimization》2013,57(3):719-731
The support function is strongly related to the cost function in economics. Because the differentiability of the cost function is an important property, being related to the celebrated Shephard’s lemma, our propose is to study the differentiability of the support function of a nonempty closed and convex subset of a finite dimensional normed space. So we provide characterizations of the differentiability of the support function on three subsets of its domain. 相似文献
3.
Juan Pablo Rincón-Zapatero Manuel S. Santos 《Journal of Mathematical Analysis and Applications》2012,394(1):305-323
In this paper we provide some sufficient conditions for the differentiability of the value function in a class of infinite-horizon continuous-time models of convex optimization arising in economics. We dispense with the assumption of interior optimal paths. This assumption is quite unnatural in constrained optimization, and is usually hard to check in applications. The differentiability of the value function is used to prove Bellman’s equation as well as the existence and continuity of the optimal feedback policy. We also establish the uniqueness of the vector of dual variables. These results become useful for the characterization and computation of optimal solutions. 相似文献
4.
Alexander Shapiro 《Mathematical Programming》1995,70(1-3):149-157
In this paper, directional differentiability properties of the optimal value function of a parameterized semi-infinite programming problem are studied. It is shown that if the unperturbed semi-infinite programming problem is convex, then the corresponding optimal value function is directionally differentiable under mild regularity assumptions. A max-min formula for the directional derivatives, well-known in the finite convex case, is given. 相似文献
5.
In a normed vector space, we study the minimal time function determined by a moving target set and a differential inclusion, where the set-valued mapping involved has constant values of a bounded closed convex set U. After establishing a characterization of ?-subdifferential of the minimal time function, we obtain that the limiting subdifferential of the minimal time function is representable by virtue of the corresponding normal cones of sublevel sets of the function and level or sublevel sets of the support function of U. The known results require the set U to have the origin as an interior point and the target set is a fixed set. 相似文献
6.
In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L
1 approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation. 相似文献
7.
C. Zălinescu 《Optimization》2015,64(8):1795-1823
The notion of quasi-relative interior was introduced by Borwein and Lewis in 1992 and applied for duality results in partially finite convex optimization problems. In the last 10 years, several articles were dedicated to duality results in infinite-dimensional scalar, vector and set-valued optimization problems using this notion. The aim of this paper is to refine and discuss such results. We do this observing that the notion of quasi-relative interior is related to (non-proper) separation of a convex set and some of its elements, then pointing out the relation between the subdifferentiability of a function associated to a set of epigraph type at a certain point and the fact that a corresponding point is not in the quasi-relative interior of the closed convex hull of the set. 相似文献
8.
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
9.
Jonathan M. Borwein James V. Burke Adrian S. Lewis 《Proceedings of the American Mathematical Society》2004,132(4):1067-1076
Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with non-empty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lipschitz functions, and necessary. We show by way of example that most results fail more generally.
10.
Li Xin CHENG Yan Mei TENG 《数学学报(英文版)》2007,23(6):1063-1066
To guarantee every real-valued convex function bounded above on a set is continuous, how "thick" should the set be? For a symmetric set A in a Banach space E,the answer of this paper is: Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold: i) spanA has finite co-dimentions and ii) coA has nonempty relative interior. This paper also shows that a subset A C E satisfying every real-valued convex function bounded above on A is continuous on E if (and only if) every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E. 相似文献
11.
Tudor Zamfirescu 《Monatshefte für Mathematik》1987,103(3):241-247
In the sense of Baire categories, most convex curves on a smooth twodimensional closed convex surface are smooth. Moreover, if the set of all closed geodesics has empty interior in the space of all convex curves, then most convex curves are strictly convex.This paper was written during the author's visit at Western Washington University, whose substantial support is acknowledged. 相似文献
12.
In a general normed vector space, we study the minimal time function determined by a differential inclusion where the set-valued mapping involved has constant values of a bounded closed convex set U and by a closed target set S. We show that proximal and Fréchet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of U. The known results in the literature require the set U to have the origin as an interior point or U be compact. (In particular, if the set U is the unit closed ball, the results obtained reduce to the subdifferential of the distance function defined by S.) 相似文献
13.
In general normed spaces,we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior.We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function.Moreover,we provide necessary and suffcient conditions about the existence of weak(sharp) Pareto solutions. 相似文献
14.
S. Simons 《Journal of Optimization Theory and Applications》1991,71(1):127-136
In this paper, we give a direct proof of Rockafellar's result that the subdifferential of a proper convex lower-semicontinuous function on a Banach space is maximal monotone. Our proof is simpler than those that have appeared to date. In fact, we show that Rockafellar's result can be embedded in a more general situation in which we can quantify the degree of failure of monotonicity in terms of a quotient like the one that appears in the definition of Fréchet differentiability. Our analysis depends on the concepts of the least slope of a convex function, which is related to the steepest descent of optimization theory.The author would like to express his thanks to R. R. Phelps for reading a preliminary version of this paper and making some very valuable suggestions. 相似文献
15.
In this paper, we will study the differentiability on the boundary of solutions of elliptic non-divergence differential equations on convex domains. The results are divided into two cases: (i) at the boundary points where the blow-up of the domain is not the half-space, if the boundary function is differentiable then the solution is differentiable; (ii) at the boundary points where the blow-up of the domain is the half-space, the differentiability of the solution needs an extra Dini condition for the boundary function. Counterexample is given to show that our results are optimal. 相似文献
16.
17.
A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.This paper shows that the product of a GDS and a family of separable Prechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS. 相似文献
18.
Juan Enrique Martínez-Legaz José Vicente-Pérez 《Journal of Mathematical Analysis and Applications》2011,376(2):602-612
A subset of a locally convex space is called e-convex if it is the intersection of a family of open halfspaces. An extended real-valued function on such a space is called e-convex if its epigraph is e-convex. In this paper we introduce a suitable support function for e-convex sets as well as a conjugation scheme for e-convex functions. 相似文献
19.
We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the Borwein theory.For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs. 相似文献
20.
Dinh The Luc 《Proceedings of the American Mathematical Society》1997,125(2):555-567
We show in this paper that if a polyhedral convex set is defined by a parametric linear system with smooth entries, then it possesses local smooth representation almost everywhere. This result is then applied to study the differentiability of the solutions and the marginal functions of several classes of parametric optimization problems.