用次微分表达的对偶问题

本利用次微分建立了多目标规划的一个新的对偶问题，并给出其弱、强和逆对偶性，得到了一个新的次梯度的定义，并用其建立了一个新的对偶问题。     

Slow Passage through a Saddle-Center Bifurcation

Received September 24, 1998; accepted August 9, 1999     

Generalized quasiconvexities,cone saddle points,and minimax theorem for vector-valued functions

We establish the following theorems: (i) an existence theorem for weak type generalized saddle points; (ii) an existence theorem for strong type generalized saddle points; (iii) a generalized minimax theorem for a vector-valued function. These theorems are generalizations and extensions of the author's recent results. For such extensions, we propose new concepts of convexity and continuity of vector-valued functions, which are weaker than ordinary ones. Some of the proofs are based on a few key observations and also on the Browder coincidence theorem or the Tychonoff fixed-point theorem. Also, the minimax theorem follows from the existence theorem for weak type generalized saddle points. The main spaces with mathematical structures considered are real locally convex spaces and real ordered topological vector spaces.This paper is dedicated to Professor Kensuke Tanaka on his sixtieth birthday.This paper was written when the author was a visitor at the Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata, Japan. The author is indebted to Prof. K. Tanaka for suggesting this work.The author is very grateful to Prof. P. L. Yu for his useful encouragement and suggestions and to the referees for their valuable suggestions and comments.     

Saddle Point Optimality Conditions in Fuzzy Optimization Problems

The fuzzy-valued Lagrangian function of fuzzy optimization problem via the concept of fuzzy scalar (inner) product is proposed. A solution concept of fuzzy optimization problem, which is essentially similar to the notion of Pareto solution in multiobjective optimization problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. Under these settings, the saddle point optimality conditions along with necessary and sufficient conditions for the absence of a duality gap are elicited.     

Optimal growth and recursive utility: Phase diagram analysis

The dynamics of the one-sector optimal growth model with recursive utility is analyzed through the use of a phase diagram. The steady state uniquely exists and is a saddle point. An increase in recursivity lowers both the steady-state capital and steady-state consumption. The model differs from the constant discount rate model in that a reduction in the population growth rate or a Hicks-neutral technical progress increases the steady-state consumption but not necessarily the steady-state capital.This research was supported in part by Indiana University through an Outstanding Young Faculty Award for which the author is grateful.The author is particularly indebted to Robert Becker and John Boyd for a careful reading of and detailed comments on an earlier draft, and to Frank Raymond and an anonymous referee for suggesting various expositional improvements. He is also indebted to Roy Gardner, Nori Hashimoto, and Nicolas Spulber for their suggestions.     

A saddle-point theorem with application to structural optimization

The relaxation for optimal complicance design is independent of whether the underlying elastic problem is formulated in terms of displacements or strains. For the purposes of numerical experimentation and computation, it may be advantageous to formulate optimal design problems in terms of displacements as is done in Ref. 1. The relaxed problem delivered by the displacement-based formulation is of min-min-max type. Because of this, efficient numerical implementation is hampered by the order of the last two min-max operations. We show here that the last two min-max operations may be exchanged, facilitating an efficient numerical algorithm. We remark that the rigorous results given here corroborate the numerical methods and experiments given in Ref. 1.This work was supported by NSF Grant DMS-92-05158.     

Analysis of iterative methods for saddle point problems: a unified approach

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

     

Generalized G-KKM Theorems in Generalized Convex Spaces and Their Applications

Some new generalized G-KKM and generalized S-KKM theorems are proved under the noncompact setting of generalized convex spaces. As applications, some new minimax inequalities, saddle point theorems, a coincidence theorem, and a fixed point theorem are given in generalized convex spaces. These theorems improve and generalize many important known results in recent literature.     

非线性问题严格互补极点－鞍点定理

本文给出两个具有新的非线性型式问题的严格互补极点－鞍点定理，同时论述了建议的二次变分－凸分析方法的一般性和有效性。其一是旋转叶轮问题，通过引进周期性自由面变量，揭示了隐含的一类对偶变分原理，从而给出了可能存在的全部互补原理。其二是带转动变量的非Ｋ－Ｌ壳体，具有超出常规的几何非线性性质，目前只知道使用这里的方法可以建立互补原理。