Parametric Disjunctive Programming: One-Sided Differentiability of the Value Function

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.     

Sufficient conditions for one-sided operators

In this article we give sufficient conditions on a pair of weight (w, v) for some one-sided operators to be bounded from L^p (v^p) to L^p (w^p). The operators we deal with include the one-sided fractional maximal operator and the one-sided singular integrals. For the first operator, necessary and sufficient conditions are known (see [8, 6]). These conditions usually amount to checking the boundedness of the operator on functions that are powers of the weights and are hard to check. Our conditions are of A_p type and are therefore easy to verify. Similar results for two-sided operators were obtained by C. Pérez in [9] and [10].     

Über holomorphe und meromorphe einseitige Inverse

In this note a theorem of B. Gramsch [6] on one-sided meromorphic inverses of Semi-Fredholmoperator valued holomorphic functions is generalized to holomorphic functions on a Stein space with values in the set of Semi-Fredholm-operators between two Banach spaces. By the way, a theorem of G.R. Allan [1] on holomorphic one-sided inverses is generalized to holomorphic functions on a Stein space with values in certain paraalgebras (c.f. [5]). As an application of that a duality theorem for holomorphic bases of finite dimensional subspaces of (F)- and (DF)-spaces is proved (c.f. [3]).     

关于Banach空间中凸泛函的广义次梯度不等式

本文在前人^[1,2]的基础之上，以凸泛函的次梯度不等式为工具，将Jensen不等式推广到Banach空间中的凸泛函，导出了Banach空间中的Bochner积分型的广义Jensen不等式，给出其在Banach空间概率论中某些应用，从而推广了文献[3—6]的工作．     

Duality for optimization and best approximation over finite intersections

Recently Deutsch, Li and Swetits [2] have studied, in Hilbert space, a dual problem (Q_m ) to the primal problem (P) of minimization of a special class of convex functions f over the intersection of m closed convex sets, where m is finite. In the first part of this paper we obtain, in a locally convex space, some results on problem (Q_m ) and on its relations with the usual Lagrangian dual problem (Q) to (P) (studied in [9]), in the case when (P) has a solution. In the second part we give some applications to duality for the distance to the intersection of m closed convex sets in a normed linear space, in the case when a nearest point exists. Most of our results seem to be new even in the particular cases studied in [9] (the case m = 1), [l] (duality formulas for the distance to the intersection of m closed half-spaces in a normed linear space) and [2].     

When are solutions to nonlinear elliptic boundary value problems convex?

We extend and apply a concavity maximum principle from [10, 9, 7] to some nonlinear elliptic boundary problems and free boundary problems on convex domains Ω?IRⁿ. In particular we extend "convex dead core' results from n = 2 as in [4 ] to arbitrary n. We also show the convexity of the coincidence set in the obstacle problem under suitable assumptions.     

Anmerkungen zu einem lemma von hoàng tuy

In [1], [2] HOÀNG TUY gave an approach to the main theorems of the convex analysis and the convex optimization being based on a lemma; he has proved it by means o induction. In [1] the equivalence of the main theorems of convex optimization given in [1], [2] does not use a separation theorem or equivalent statements. In this note the author has proved that the lemma of HOÀNG TUY can be characterized as a special separation theorem and be obtained from a separation theorem of Eidelheit. That means that the lemma is equivalent to the theorem of Hahn-Banach.     

单侧导数和对称导数混合方式下的微分中值定理

微分中值定理是分析中的一个重要定理,文[1-2]用对称导数讨论该定理,文[3-4]用单侧导数讨论该定理,而本文把两种导数结合起来以混合方式给出该定理的三种形式,且条件更弱.     

Robust convex quadratically constrained programs

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. In contrast to [4], where it is shown that such robust problems can be formulated as semidefinite programs, our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second-order cone programs (SOCP). We propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit SOCP and present examples where these classes of uncertainty sets are natural. Research partially supported by DOE grant GE-FG01-92ER-25126, NSF grants DMS-94-14438, CDA-97-26385, DMS-01-04282 and ONR grant N000140310514.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

一类广义凸函数及其等价刻画

引入了一个广义凸函数一高阶强Pre-invex函数,它是Guneer-Bhatia介绍的高阶强凸函数的一种推广.在此基础上,讨论并证明了高阶强Pre-invex函数的一些等价刻画,这些结论是Guneer-Bhatia给出结论的推广,扩大了凸函数的应用范围.     

Some fixed point theorems in convex metric spaces

The concept of a convex metric space was introduced by Takahashi [10]. He observed that it is possible to generalize fixed point theorems in Banach spaces. Subsequently, Machado [8], Itoh [5], Naimpally, Singh and Whitfield [9] and Beg and Azam [2], among others have studied fixed point theorems in convex metric spaces. This paper is a continuation of these investigations.     

Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

Filippov–Pliss lemma and m-dissipative differential inclusions

In the paper we prove a variant of the well known Filippov–Pliss lemma for evolution inclusions given by multivalued perturbations of m-dissipative differential equations in Banach spaces with uniformly convex dual. The perturbations are assumed to be almost upper hemicontinuous with convex weakly compact values and to satisfy one-sided Peron condition. The result is then applied to prove the connectedness of the solution set of evolution inclusions without compactness and afterward the existence of attractor of autonomous evolution inclusion when the perturbations are one-sided Lipschitz with negative constant.     

Remarks on convex functions and separable sequences,II

In this paper we generalize results on convex sequences and convex functions due to Wu and Debnath [S. Wu, L. Debnath, Inequalities for convex sequences and their applications, Comput. Math. Appl. 54 (2007) 525–534] by using some results on similarly separable sequences established in Niezgoda [M. Niezgoda, Remarks on convex functions and separable sequences, Discrete Math. 308 (2008) 1765–1773].     

Solving a class of multiplicative programming problems via c-programming

In this note we show that many classes of global optimization problems can be treated most satisfactorily by classical optimization theory and conventional algorithms. We focus on the class of problems involving the minimization of the product of several convex functions on a convex set which was studied recently by Kunoet al. [3]. It is shown that these problems are typical composite concave programming problems and thus can be handled elegantly by c-programming [4]–[8] and its techniques.     

FIXED POINT INDEX THEORY FOR WEAKLY INWARD MAPPINGS

1IntroductionLetEbeaBanachspace,PcEaclosedconvexsetandletA:finP~Ebecompletelycontinuous,wherefiCEisanopenboundedset.ItiswelLknownthataAsedpointindexi(A,nnP,P)callbedefinedwheneverA(flnP)cpalldAx/xonoffnP,whichresultsinmanyfixedpointtheoremstodealwithvariousproblems.AsimpleanalysisofthemethodofdefiningtheindexrevealsthattherestrictionthatAmapsfinPtoPitselfisessential.FOrdetailswereferto[1,pp.286--291]andtheintroductionsectionof[2].Hereafter,wewillsaythatsllchmappingsareconemappings.On…     

度量投影的收敛性（英文）

讨论了赋范空间中度量投影的收敛性．得到了在局部紧集控制下，Chebyshev凸集序列的度量投影的收敛性与K－M收敛，Wijsman收敛和Kuratowski收敛都等价．本文的结论完善了M．Tsukada在[1]和[2]结果．     

Some common fixed point theorems for weakly compatible mappings

Jungck's [G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci. 9 (1986) 771-779] notion of compatible mappings is further extended and used to prove some common fixed point theorems for weakly compatible non-self mappings in complete convex metric spaces. We improve on the method of proof used by Rhoades [B.E. Rhoades, A fixed point theorem for non-self set-valued mappings, Int. J. Math. Math. Sci. 20 (1997) 9-12] and Ahmed and Rhoades [A. Ahmed, B.E. Rhoades, Some common fixed point theorems for compatible mappings, Indian J. Pure Appl. Math. 32 (2001) 1247-1254] and obtain generalization of some known results. In particular, a theorem by Rhoades [B.E. Rhoades, A fixed point theorem for non-self set-valued mappings, Int. J. Math. Math. Sci. 20 (1997) 9-12] is generalized and improved.     

Mengerian quasi-graphical families and clutters

Balanceable clutters are clutters whose bipartite representation contains no odd wheel and no odd 3-path configuration as an induced subgraph (this is Truemper’s characterization of balanceable matrices). In this paper we study a proper subclass of balanceable clutters called quasi-graphical defined by forbidding one-sided even wheels and one-sided even 3-path configurations. We characterize Mengerian quasi-graphical clutters and, as a consequence, we show that a recent conjecture in Cornuéjols et al. (2000) [7] is true for quasi-graphical clutters.