Generalized Fenchel’s Conjugation Formulas and Duality for Abstract Convex Functions

In this paper, we present a generalization of Fenchel’s conjugation and derive infimal convolution formulas, duality and subdifferential (and ε-subdifferential) sum formulas for abstract convex functions. The class of abstract convex functions covers very broad classes of nonconvex functions. A nonaffine global support function technique and an extended sum-epiconjugate technique of convex functions play a crucial role in deriving the results for abstract convex functions. An additivity condition involving global support sets serves as a constraint qualification for the duality. Work of Z.Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.     

Some Formulas for the Canonical Kernel Function

In this paper we present some expressions for the canonical kernel function of a bounded symmetric domain. We show how it is related both to the reproducing kernel of a holomorphic discrete series representation and to the contravariant form on highest weight modules over semisimple Lie algebras of Hermitian type.     

Asymptotic Formulas and Inequalities for the Gamma Function in Terms of the Tri-Gamma Function

Results in Mathematics - In the paper, the authors establish some asymptotic formulas and double inequalities for the factorial n! and the gamma function Γ in terms of the tri-gamma function...     

Subdifferential test for optimality

We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is monotone absorbing, hence maximal monotone when the function is convex.     

Evaluation Formulas for Conditional Function Space Integrals I

Abstract

In this article, we establish several explicit conditional function space integration formulas for functionals defined on a very general function space C _a,b [0,T]. The formulas we obtain are rather simple and don't involve function space integrals. In particular we obtain a formula for the conditional function space integral of each of the functionals exp{∫₀ ^T x(t)db(t)}, exp{?[∫₀ ^T x(t) db(t)]²}, and exp{? ∫₀ ^T x² (t) db(t)} which arise naturally in quantum mechanics.     

Quadrature Formulas Based on the Scaling Function

The scaling function corresponding to the Daubechies wavelet with two vanishing moments is used to derive new quadrature formulas. This scaling function has the smallest support among all orthonormal scaling functions with the properties M ₂ = M ₁ ² and M ₀ = 1. So, in this sense, its choice is optimal. Numerical examples are given.This work was partially supported by DFG grant GR 1777/2, by the Grant No 201/01/1200 of the CSF, by the grant MSMT 113200007 and by the grant IGS 116/5130/1 of FP TUL.     

Variational Subdifferential for Quasiconvex Functions

It is shown that a convenient subdifferential for the class of quasiconvex functions is variational. This property combines a variational principle with a kind of weak fuzzy sum rule. It entails a number of useful properties. The subdifferential considered here is the lower subdifferential at the origin (in the sense of Plastria) of the incident derivative or inner epiderivative of the function.     

Fenchel decomposition for stochastic mixed-integer programming

This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an FD algorithm for SMIP and establish finite convergence for binary first-stage. Second, we derive FD cuts based on the second-stage variables only and use an idea from disjunctive programming to lift the cuts to the higher dimension space including the first-stage variables. We then devise an alternative algorithm (FD-L algorithm) based on the lifted FD cuts. Finally, we report on computational results based on several test instances from the literature involving the special structure of knapsack problems with nonnegative left-hand side coefficients. The results are promising and show that both algorithms can outperform a standard direct solver and a disjunctive decomposition algorithm on large-scale instances. Furthermore, the FD-L algorithm provides better performance than the FD algorithm in general. Since Fenchel cuts can be computationally expensive in general and are best suited for problems with special structure, both algorithms exploit the special structure of the test instances by reducing the size of the cut generation problems based on the number of nonzero components in the non-integer solution that needs to be cut off.     

Subdifferential calculus for convex operators

Well-known scalar results on the Subdifferential of composite functions are extended to the framework of ordered topological vector spaces. This is done by using the sandwich theorem for convex operators which is derived from an extension of the Hahn-Banach theorem.     

New Tour on the Subdifferential of Supremum via Finite Sums and Suprema

Journal of Optimization Theory and Applications - This paper provides new characterizations for the subdifferential of the pointwise supremum of an arbitrary family of convex functions. The main...     

Hunting for a Smaller Convex Subdifferential

Certain useful basic results of the gradient (in the smooth case), the Clarkesubdifferential, the Michel–Penot subdifferential, which is also known asthe "small" subdifferential, and the directional derivative(in the nonsmooth case) are stated and discussed. One of the advantages ofthe Michel–Penot subdifferential is the fact that it is in general "smaller"than the Clarke subdifferential. In this paper it is shown that there existsubdifferentials which may be smaller than the Michel–Penot subdifferentialandwhich have certain useful calculus. It isfurther shown that in the case of quasidifferentiability, the Michel–Penotsubdifferential enjoys calculus whichhold for the Clarke subdifferential only in the regular case.     

一个精细的Bishop-Phelps定理

讨论了凸函数的次微分映射和凸集的支撑点集之间的内在关系，由此本给出了由凸函数的次微分映射所刻划的一个精细的Bishop-Phelps定理．     

Representation Formulas for Analytic Function Non－exponential in a Half Plane

ＲｅｐｒｅｓｅｎｔａｔｉｏｎＦｏｒｍｕｌａｓｆｏｒＡｎａｌｙｔｉｃＦｕｎｃｔｉｏｎＮｏｎ－ｅｘｐｏｎｅｎｔｉａｌｉｎａＨａｌｆＰｌａｎｅＤｅｎｇＧｕａｎｔｉｅ（邓冠铁）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕａｚｈｏｎｇＵｎｉｖｅｒｓｉ...     

共轭A-调和张量新的双权积分不等式

共轭A-调和张量的一些局部Aλr3(λ1,λ2,Ω)-加权积分不等式得到了证明,它们可看作是共轭调和函数和p-调和函数相应结果的推广.这些结果可用来研究共轭调和函数的可积性并估计它们的积分.同时也给出上述结果在拟正则映射中的应用.     

解无约束优化的一种新的共轭梯度法

本文提出了一种新的解无约束优化的共轭梯度算法,分析了算法的收敛性,并对算法进行了数值实验.数值实验的结果表明算法是有效的.     

On a Problem of Fenchel

We give a necessary and sufficient condition for a given set of positive real numbers to be the dihedral angles of a hyperbolic n -simplex in this note. This answers a question of W. Fenchel raised in his book, Elementary Geometry in Hyperbolic Space, (De Gruyter, Berlin, 1989, p. 174) where he obtained some necessary conditions for which six numbers have to satisfy in order to be the dihedral angles of a hyperbolic tetrahedron. We also present a simple proof of the known necessary and sufficient condition for the dihedral angles of Euclidean n-simplexes.