Global Optimization: Local Minima and Transition Points

We consider the minimization of smooth functions of the Euclidean space with a finite number of stationary points having moderate asymptotic behavior at infinity. The crucial role of transition points of first order (i.e., saddle points of index 1) is emphasized. It is shown that (generically) any two local minima can be connected via an alternating sequence of local minima and transition points of first order. In particular, the graph with local minima as its nodes and first order transition points representing the edges turns out to be connected (Theorem A). On the other hand, any connected (finite) graph can be realized in the above sense by means of a smooth function of three variables having a minimal number of stationary points (Theorem B).     

On comparison of jump point detection for an exchange rate series

The main purpose of this paper is to investigate the detection of jump points of a discontinuous function in the presence of a noise by the wavelet approach. A computing algorithm of our method is proposed and then applied to the daily exchange rate of US Dollar against Deutsche Mark. All the points detected by our method reflect very strong economic and political impacts. Other statistical methods to detect jump points have also been applied to the same exchange rate data. Our proposed method has produced more convincing empirical results than others.     

Multimodality analysis of a class of multistage allocation problems

Multiple optimum solutions of a multistage allocation problem, well-known to chemical engineers, are analyzed. The number of local optima becomes greater with a decrease in the initial-condition value of the first stage or with an increase in the total stage number. The fact that this behavior is closely related to the flat portion of the profile of a curvef(x), which determines the objective function, is revealed. A construction method by Aris is used to give an excellent insight into this behavior. Moreover, the construction curves ensure that all stationary points are found. Finally, a theorem to discriminate local optima from stationary points, without evaluating second-order derivatives, is presented.The authors would like to thank Dr. I. Hashimoto and Dr. H. Nishitani for valuable discussions. Computations were carried out with the assistance of Messrs. H. Unno, H. Nakano, Y. Era, and Y. Ueno. The authors are indebted to the computing centers of Osaka University, Kyoto University, and Nagoya University for the use of their facilities.     

一种改进的超收敛与外推的方法

1.引 言 由于采用高精度算法能大大提高有限元计算的精度,因此有许多专家对它进行了多方面研究,取得了一批卓有成效的成果[16]研究有限元高精度的方法主要有两种: （1）美国H.A.Schatz.B.wahlbin[4,5]等发现的直接考察u-uh或　（u-uh）在局部对称点所具有的超收敛性的方法. （2）中国林群,朱起定[1,2],陈传淼[3]等所发现的通过研究uI-uh或　（uI-uh）所具有的整体超收敛与外推性质来得到u-uh或　（u-uh）在剖分点与其他某些特殊点的超收敛与外推性质.     

A local min-orthogonal method for finding multiple saddle points

The purpose of this paper is twofold. The first is to remove a possible ill-posedness related to a local minimax method developed in SIAM J. Sci. Comput. 23 (2001) 840-865, SIAM J. Sci. Comput. 24 (2002) 840-865 and the second is to provide a local characterization for nonminimax type saddle points. To do so, a local L-⊥ selection is defined and a necessary and sufficient condition for a saddle point is established, which leads to a min-orthogonal method. Those results exceed the scope of a minimax principle, the most popular approach in critical point theory. An example is given to illustrate the new theory. With this local characterization, the local minimax method in SIAM J. Sci. Comput. 23 (2001) 840-865, SIAM J. Sci. Comput. 24 (2002) 840-865 is generalized to a local min-orthogonal method for finding multiple saddle points. In a subsequent paper, this approach is applied to define a modified pseudo gradient (flow) of a functional for finding multiple saddle points in Banach spaces.     

A method for the computation of nonsimple turning points corresponding to cusps

A direct method is described for the approximation of nonsimple turning points, corresponding to cusp points, of nonlinear operator equations depending on two parameters. The procedure is based on the application of a special projection method to the computation of simple turning points of a suitable augmented system. Numerical examples illustrate the features of the proposed algorithm.     

A method for minimizing the sum of a convex function and a continuously differentiable function

This paper presents a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function. As in the convex case developed by the author, the algorithm is a descent method which generates successive search directions by solving quadratic programming subproblems. An inexact line search ensures global convergence of the method to stationary points.     

双点局部极大算子的双权弱型不等式

给出了双点局部极大算子在加双权意义下的弱有界性.     

LOCAL AND GLOBAL EXPOSED POINTS

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Critical Points Index for Vector Functions and Vector Optimization

In this work, we study the critical points of vector functions from ℝ ⁿ to ℝ ^m with n≥m, following the definition introduced by Smale in the context of vector optimization. The local monotonicity properties of a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We propose a classification of critical points through the introduction of a generalized Morse index for a critical point, consisting of a triplet of nonnegative integers. The proposed index is based on the sign of an appropriate invariant vector-valued second-order differential.     

Subvexormal Functions and Subvex Functions

Subvexormal functions and subinvexormal functions are proposed, whose properties are shared commonly by most generalized convex functions and most generalized invex functions, respectively. A necessary and sufficient condition for a subvexormal function to be subinvexormal is given in the locally Lipschitz and regular case. Furthermore, subvex functions and subinvex functions are introduced. It is proved that the class of strictly subvex functions is equivalent to that of functions whose local minima are global and that, in the locally Lipschitz and regular case, both strongly subvex functions and strongly subinvex functions can be characterized as functions whose relatively stationary points (slight extension of stationary points) are global minima.     

求全局最优解的新算法

利用局部极大值点与动力系统的稳定奇点的对应性，计算代数方程的根、无约束极大值点、有约束极大值点、非线性规划解、及最小二乘解．我们采用了常微分方程数值解的Euler算法及网格初始点的循序迭代算法，并以具体的例子和程序说明创立的方法具有通用性，同时考虑了一些存在的问题以便在理论和算法上作进一步的改进。     

Application of modified information criterion to multiple change point problems

The modified information criterion (MIC) is applied to detect multiple change points in a sequence of independent random variables. We find that the method is consistent in selecting the correct model, and the resulting test statistic has a simple limiting distribution. We show that the estimators for locations of change points achieve the best convergence rate, and their limiting distribution can be expressed as a function of a random walk. A simulation is conducted to demonstrate the usefulness of this method by comparing the powers between the MIC and the Schwarz information criterion.     

Some Geometrical Aspects of Efficient Points in Vector Optimization

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phelps theorem. We study also the density of positive proper efficient point sets. This research was supported by a Central Research Grant of Hong Kong Polytechnic University, Grant G-T 507. Research of the first author was also supported by the National Natural Science Foundation of P.R. China, Grant 10361008, and the Natural Science Foundation of Yunnan Province, China, Grant 2003A002M. Research of the second author was also supported by the Natural Science Foundation of Chongqing. Research of the third author was supported by a research grant from Australian Research Counsil.     

Asymptotic of Eigenvalues and Lattice Points

dimension. elementary domains of In this work we study the spectral counting function for the p-Laplace operator in one We show the existence of a two-term Weyl-type asymptote. The method of proof is rather based on the Dirichlet lattice points problem, which enables us to obtain similar results for infinite measure.     

Global Optimality Conditions for Nonconvex Optimization

In this paper we give an analytical equivalent for the inclusion of a set to the Lebesque set of a convex function. Using this results, we obtain global optimality conditions (GOC) related to classical optimization theory for convex maximization and reverse-convex optimization. Several examples illustrate the effectiveness of these optimality conditions allowing to escape from stationary points and local extremums.     

Elimination of bounds in optimization problems by transforming variables

The problem of minimizing a nonlinear objective function ofn variables, with continuous first and second partial derivatives, subject to nonnegativity constraints or upper and lower bounds on the variables is studied. The advisability of solving such a constrained optimization problem by making a suitable transformation of its variables in order to change the problem into one of unconstrained minimization is considered. A set of conditions which guarantees that every local minimum of the new unconstrained problem also satisfies the first-order necessary (Kuhn—Tucker) conditions for a local minimum of the original constrained problem is developed. It is shown that there are certain conditions under which the transformed objective function will maintain the convexity of the original objective function in a neighborhood of the solution. A modification of the method of transformations which moves away from extraneous stationary points is introduced and conditions under which the method generates a sequence of points which converges to the solution at a superlinear rate are given.     

圆内亚纯函数与其微分多项式的公共Borel点

该文把A.P.Singh关于一类齐次微分多项式级的结果推广到更一般的微分多项式。并证明了：如果Q（f)≠0是圆内有限正级亚纯函数f的身长分多项式，则f^(k0Q(f)的Borel点必是f的Borel点，其中K0满足0≤K0≤min{K:f^(k)出现在Q（f)中}。     

About the Measure of Heavily Visited Points of the Brownian Motion

We consider those space points of the Wiener process, the local time of which is at least a constant multiple of the local time of the origin. The upper and lower class behavior of the Lebesgue measure of these points are investigated.     

The structure of admissible points with respect to cone dominance

We study the set of admissible (Pareto-optimal) points of a closed, convex setX when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions ofX, we extend a representation theorem of Arrow, Barankin, and Blackwell by showing that all admissible points are either limit points of certainstrictly admissible alternatives or translations of such limit points by rays in the closure of the preference cone. We also show that the set of strictly admissible points is connected, as is the full set of admissible points.Relaxing the convexity assumption imposed uponX, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element ofX to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.Several results from this paper were presented in less general form at the National ORSA/TIMS Meeting, Chicago, Illinois, 1975.This research was supported, in part, by the United States Army Research Office (Durham), Grant No. DAAG-29-76-C-0064, and by the Office of Naval Research, Grant No. N00014-67-A-0244-0076. The research of the second author was partially conducted at the Center for Operations Research and Econometrics (CORE), Université Catholique de Louvain, Heverlee, Belgium.The authors are indebted to A. Assad for several helpful discussions and to A. Weiczorek for his careful reading of an earlier version of this paper.