WEIERSTRASS' THEOREM IN THE TWENTIETH CENTURY: A SELECTION

Abstract

We survey some of the extensions and ramifications of Weierstrass' theorem in the twentieth century, discussing first qualitative theory (can we approximate?) and then quantitative theory (how fast can we approximate?).     

A modified inexact implicit method for mixed variational inequalities

In this paper, we suggest and analyze an inexact implicit method with a variable parameter for mixed variational inequalities by using a new inexactness restriction. Under certain conditions, the global convergence of the proposed method is proved. Some preliminary computational results are given to illustrate the efficiency of the new inexactness restriction. The results proved in this paper may be viewed as improvement and refinement of the previously known results.     

A two-stage prediction–correction method for solving monotone variational inequalities

In this paper, we present a two-stage prediction–correction method for solving monotone variational inequalities. The method generates the two predictors which should satisfy two acceptance criteria. We also enhance the method with an adaptive rule to update prediction step size which makes the method more effective. Under mild assumptions, we prove the convergence of the proposed method. Our proposed method based on projection only needs the function values, so it is practical and the computation load is quite tiny. Some numerical experiments were carried out to validate its efficiency and practicality.     

Interpolating varieties for entire functions of minimal type

We obtain both necessary and sufficient conditions for a discrete variety in Cⁿ to be an interpolating variety for entire functions of minimal type.     

Deciding uniqueness in norm maximization

NP-hardness is established for the problem whose instance is a system of linear inequalities defining a polytopeP, and whose question is whether, onP, the global maximum of the Euclidean norm is attained at more than one vertex ofP. The NP-hardness persists even for the restricted problem in whichP is a full-dimensional parallelotope with one vertex at the origin. This makes it possible to establish NP-hardness for other uniqueness problems, including some from pseudoboolean programming and computational convexity.Research of the first author was supported in part by the Deutsche Forschungsgemeinschaft. Research of the second author was supported in part by the National Science Foundation.     

A stochastic programming model to find optimal sample sizes to estimate unknown parameters in an LP

An LP is considered where the technology coefficients are unknown and random samples are taken to estimate them. A stochastic programming problem is formulated to find the optimal sample sizes where it is required that a confidence interval should cover the unknown deterministic optimum value by a given probability and the cost of sampling be minimum.     

A row-action method for convex programming

We present a new method for minimizing a strictly convex function subject to general convex constraints. Constraints are used one at a time, no changes are made in the constraint functions (thus the row-action nature of the algorithm) and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations. Convergence of the algorithm is established and the method is compared with other row-action algorithms for several relevant particular cases.Corresponding author. Research of this author was partially supported by CNPq grant No. 301280/86.     

Positive semidefinite quadratic forms on unitary matrices

Let

     

A generalized Newton method for absolute value equations associated with second order cones

In this paper, we introduce the absolute value equations associated with second order cones (SOCAVE in short), which is a generalization of the absolute value equations discussed recently in the literature. It is proved that the SOCAVE is equivalent to a class of second order cone linear complementarity problems (SOCLCP in short). In particular, we propose a generalized Newton method for solving the SOCAVE and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions. We also report some preliminary numerical results of the proposed method for solving the SOCAVE and the SOCLCP, which show the efficiency of the proposed method.     

SOME CONSEQUENCES OF THE TARSKI-KANTOROVITCH ORDERING THEOREM IN METRIC FIXED POINT THEORY

Abstract

We show that the Tarski-Kantorovitch Principle for continuous maps on a partially ordered set yields some fixed point theorems for contractive maps on a uniform space. Our proofs do not depend on the Axiom of Choice.     

Self-adaptive projection method for co-coercive variational inequalities

In some real-world problems, the mapping of the variational inequalities does not have any explicit forms and only the function value can be evaluated or observed for given variables. In this case, if the mapping is co-coercive, the basic projection method is applicable. However, in order to determine the step size, the existing basic projection method needs to know the co-coercive modulus in advance. In practice, usually even if the mapping can be characterized co-coercive, it is difficult to evaluate the modulus, and a conservative estimation will lead an extremely slow convergence. In view of this point, this paper presents a self-adaptive projection method without knowing the co-coercive modulus. We also give a real-life example to demonstrate the practicability of the proposed method.     

An improved Goldstein's type method for a class of variant variational inequalities

This paper aims at presenting an improved Goldstein's type method for a class of variant variational inequalities. In particular, the iterate computed by an existing Goldstein's type method [He, A Goldstein's type projection method for a class of variant variational inequalities J. Comput. Math. 17(4) (1999) 425–434]. is used to construct a descent direction, and thus the new method generates the new iterate by searching the optimal step size along the descent direction. Some restrictions on the involving functions of the existing Goldstein's type methods are relaxed, while the global convergence of the new method is proved without additional assumptions. The computational superiority of the new method is verified by the comparison to some existing methods.     

On semidefinite representations of non-closed sets

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinitely representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinitely representable. So far, all results focus on the case of closed sets.In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinitely representable set is shown to be semidefinitely representable. More general, one can remove faces of a semidefinitely representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.     

Ascoli-Arzela定理的一个推广及应用

给出了无穷区间上一类抽象连续函数族相对紧性判断的一个充分必要条件，并应用它获得了抽象空间上一阶微分方程的终值问题和Ｆｒｅｄｈｏｌｍ型积分方程解的存在性。     

A filled function method for global optimization

A novel filled function with one parameter is suggested in this paper for finding a global minimizer for a general class of nonlinear programming problems with a closed bounded box. A new algorithm is presented according to the theoretical analysis. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.     

Computational complexity of norm-maximization

This paper discusses the problem of maximizing a quasiconvex function over a convex polytopeP inn-space that is presented as the intersection of a finite number of halfspaces. The problem is known to beNP-hard (for variablen) when is thep ^th power of the classicalp-norm. The present reexamination of the problem establishesNP-hardness for a wider class of functions, and for thep-norm it proves theNP-hardness of maximization overn-dimensionalparallelotopes that are centered at the origin or have a vertex there. This in turn implies theNP-hardness of {–1, 1}-maximization and {0, 1}-maximization of a positive definite quadratic form. On the good side, there is an efficient algorithm for maximizing the Euclidean norm over an arbitraryrectangular parallelotope.The authors are indebted to J. O'Rourke, P, Pardalos and R. Freund for useful references. The second and third authors are indebted to the Institute for Mathematics and its Applications in Minneapolis, where much of this paper was written: they acknowledge additional support from the Alexander von Humboldt Stiftung and the National Science Foundation, respectively.     

Approximation by complex genuine Durrmeyer type polynomials in compact disks

In this paper, the order of simultaneous approximation and Voronovskaja kind results with quantitative estimate for the complex genuine Durrmeyer polynomials attached to analytic functions on compact disks are obtained. In this way, we put in evidence the overconvergence phenomenon for the genuine Durrmeyer polynomials, namely the extensions of the approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.     

THE ABSTRACT CAUCHY PROBLEM AND A GENERALIZATION OF THE LUMER-PHILLIPS THEOREM

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