Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.     

关于转移开、闭映射的注记

介绍转移闭值[转移开值]映射及其相关的概念,说明它们之间的相互关系,然后得出若干个转移闭值[转移开值]的性质.     

Convex extensions of the convex set valued maps

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.     

A coradiant based scalarization to characterize approximate solutions of vector optimization problems with variable ordering structures

This paper investigates some properties of approximate efficiency in variable ordering structures where the variable ordering structure is given by a special set valued map. We characterize $\epsilon$-minimal and $\epsilon$- nondominated elements as approximate solutions of a multiobjective optimization problem with a variable ordering structure and give necessary and sufficient conditions for these solutions, via scalarization.     

Second order optimality conditions for the extremal problem under inclusion constraints

In this paper, we establish second order optimality conditions for the problem of minimizing a function f on the solution set of an inclusion 0∈F(x), where f and the support function of set valued map F have compact second order approximations at x?.     

Informative barycentres in statistics

Barycentres of a discrete probability measure on a dually flat statistical manifold are introduced. They are shown to be unique and to behave as barycentres in Euclidean space. The estimation of these barycentres is studied. Potential applicative usefulness of informative barycentres include the problem of interpolating a statistical manifold valued map and the problem of model merging, which consists in merging several statistical models into a unique one. The results are illustrated on the exponential family, for which a projection theorem is proved.     

Benson真有效意义下向量集值优化的广义Fritz-John条件

借助Clarke切锥并用上图引入了关于集值映射的Clarke切导数.借助于一种新的择一性定理建立了向量集值优化问题在弱Benson真有效意义下的广义Fritz-John最优性条件,而且证明在一种伪凸的假设下,这种最优性条件还为充分的.     

A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

非紧域上集值映象的重合点与不动点

在赋范线性空间的非空非紧凸集上建立了集值映象对的一个重合点定理,然后用这一定理改进了文献［１］中的集值映象内向集定理与外向集定理,并得到几个集值映象不动点定理．     

Benson 真有效意义下向量优化的最优性条件

对向量值映射引入了Clarke切导数,A djacent切导数及Contingent导数的概念,讨论了它们的性质,并用其建立了束向量优化有效解在Benson真有效意义下的Fritz John及Kunh-Tucker条件。     

向量均衡问题强有效解的稳定性

在可行集扰动而向量值映射不扰动、可行集与向量值映射均扰动、可行集扰动而集值映射不扰动以及可行集与集值映射均扰动四种情况下,分别讨论了向量均衡问题强有效解的稳定性.     

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some relationships among gw-subdifferential,directional derivative and radial epiderivative for nonconvex vector functions

Abstract

In this work, we give some characterizations of gw-subdifferentiability of a vector-valued function by using its directional derivative and radial epiderivative. Moreover, under some assumptions, we proved that the directional derivative and radial epiderivative of a vector-valued function are the elements of the supremum set of gw-subgradients of it. Finally, without any convexity assumption, we proved that the epigraph of contingent derivative of a set valued map is included in the epigraph of contingent epiderivative of this set-valued map.     

A Jacobi-Davidson method for two real parameter nonlinear eigenvalue problems arising from delay differential equations

The critical delays of a delay-differential equation can be computed by solving a nonlinear two-parameter eigenvalue problem. For large scale problems, we propose new correction equations for a Jacobi-Davidson type method, that also forces real valued critical delays. We present two different equations: one complex valued equation using a direct linear system solver, and one Jacobi-Davidson style correction equation which is suitable for an iterative linear system solver. A numerical example of a large scale problem arising from PDEs shows the effectiveness of the method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

区间值函数的可微性及其在区间值规划中的应用

利用实值函数的全微分思想,讨论了区间值函数的可微性,建立了区间值函数的$D$-可微性的概念及其一些基本性质. 通过讨论无约束区间规划的最优性条件,给出了一类约束函数为实值函数的约束区间值规划问题取得最优解的必要条件. 同时给出了具有实值函数约束的凸区间值规划问题取得最优解的充分条件.     

On the full list of finite‐valued solutions of the Euler‐Poisson equations having four first integrals

Only 13 general and partial solutions of the problem of a heavy rigid body's movement have been found from the discovery of the first one by Euler in 1758 till now. All these solutions are finite‐valued in the classical Euler and Poisson's variables except the one in the Hess case. In this paper the problem of finding the full list of the finite‐valued solutions is considered.     

关于集值映射的极大极小定理

在锥序线性空间中建立了关于集值映射的不动点型极大极小定理,从而扩大了关于极大极小定理的应用范围     

Banach空间中集值测度的表示定理

本文在可分自反Ｒａｎａｃｈ空的的情形下，给出了任何一列两两等比、一致有界的矢值测度可以生成一个有界闭凸值集值测度的所谓表示定理，而这个定现对κ空间首先在［３］中建立。同时，找到了由一列两两等比、一致有界变差矢值测度所生成集值测度与这列矢值测度Ｒａｄｏｎ－Ｎｉｋｏｄｙｍ导数之间的关系。