B-vex functions

A class of functions, called b-vex functions, is introduced by relaxing the definition of convexity of a function. Both the differentiable and nondifferentiable cases are presented. Members of this class satisfy most of the basic properties of convex functions. This class forms a subset of the sets of both semistrictly quasiconvex as well as quasiconvex functions, but are not necessarily included in the class of preinvex functions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319. The authors are also grateful to an anonymous referee for the constructive criticism of the first version of the paper, to Dr. Collen Knickerbocker of St. Lawrence University, and to Mrs. Meena K. Bector for their help in sharpening Examples 2.1 and 2.2, respectively.  

混合单调算子的不动点存在唯一性定理及其应用

本文首先讨论了一类混合单调算子方程组解的存在唯一性及非对称迭代逼近问题,得到了若干不具有连续性和紧性条件的有关混合单调算子、增算子和减算子的新不动点定理.其次研究了具有a-凹和-a-凸的不具有连续性和紧性条件的混合单调算子的不动点,并得到了一个新结果.最后,我们将所得结果应用于RN上的Hammerstein积分方程之中(参见文[1-12]).  

E-Convex Sets, E-Convex Functions, and E-Convex Programming

A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.  

An Iterative Approach to Quadratic Optimization

Assume that C ₁, . . . , C _N are N closed convex subsets of a real Hilbert space H having a nonempty intersection C. Assume also that each C _i is the fixed point set of a nonexpansive mapping T _i of H. We devise an iterative algorithm which generates a sequence (x _n ) from an arbitrary initial x ₀H. The sequence (x_n) is shown to converge in norm to the unique solution of the quadratic minimization problem min _xC (1/2)Ax, x–x, u, where A is a bounded linear strongly positive operator on H and u is a given point in H. Quadratic–quadratic minimization problems are also discussed.  

二阶非线性常微分方程的正周期解

本文应用Ｋｒａｓｎｏｓｅｌｓｋｉｉ锥映射不动点定理，研究了二阶非线性常微分方程的ω-周期解的存在性，获得了若干正ω-周期解的存在性与多重性结果．  

Smoothing methods for convex inequalities and linear complementarity problems

A smooth approximationp (x, ) to the plus function max{x, 0} is obtained by integrating the sigmoid function 1/(1 + e^–x ), commonly used in neural networks. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for sufficiently large. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite. Speedup over MINOS 5.4 was as high as 1142 times for linear inequalities of size 2000 × 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems are converted into a system of smooth nonlinear equations and are solved by a quadratically convergent Newton method. For monotone LCPs with as many as 10 000 variables, the proposed approach was as much as 63 times faster than Lemke's method.This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grants CCR-9101801 and CCR-9322479.  

Optimality Conditions and Duality for a Class of Nonlinear Fractional Programming Problems

In this paper, we present sufficient optimality conditions and duality results for a class of nonlinear fractional programming problems. Our results are based on the properties of sublinear functionals and generalized convex functions.  

Contingent epiderivatives and set-valued optimization

In this paper we introduce the concept of the contingent epiderivative for a set-valued map which modifies a notion introduced by Aubin [2] as upper contingent derivative. It is shown that this kind of a derivative has important properties and is one possible generalization of directional derivatives in the single-valued convex case. For optimization problems with a set-valued objective function optimality conditions based on the concept of the contingent epiderivative are proved which are necessary and sufficient under suitable assumptions.  

一类减算子的不动点定理及其应用

本文研究了一类具有凹凸性的减算子，给出了这种减算子的不动点的存在性、唯一性和迭代收敛性的一个新的结果，而且在几种方程中得到了应用．  

Proper efficiency with respect to cones

Strict separation by a cone is used here to redefine proper efficiency. Two versions of the properness, which unify and generalize known definitions, are presented. Necessary and sufficient conditions for the existence of the set of properly efficient decisions and characterization of this set in terms of the supports of the decision set are given.The research was done while the author was a visiting professor at the University of British Columbia, Vancouver, British Columbia, Canada.