Cord-slope form of Taylor's expansion in univariate global optimization

Interval arithmetic and Taylor's formula can be used to bound the slope of the cord of a univariate function at a given point. This leads in turn to bounding the values of the function itself. Computing such bounds for the function, its first and second derviatives, allows the determination of intervals in which this function cannot have a global minimum. Exploiting this information together with a simple branching rule yields an efficient algorithm for global minimization of univariate functions. Computational experience is reported.The first and second authors have been supported by FCAR (Fonds pour la Formation de Chercheurs et l'Aide à la Recherche) Grant 92EQ1048 and AFOSR Grant 90-0008 to Rutgers University. The first author has also been supported by NSERC (Natural Sciences and Engineering Research Council of Canada) Grant to HEC and NSERC Grant GP0105574. The second author has been supported by NSERC Grant GP0036426, FCAR Grant 90NC0305, and a NSF Visiting Professorship for Women in Science at Princeton University. Work of the third author was done in part while he was a graduate student at the Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA and during a visit to GERAD, June–August 1991.     

Heuristic reliability optimization by tabu search

A new heuristic algorithm, based on the tabu search methodology, is proposed for constrained redundancy optimization in series and in complex systems. It has the advantage of not being blocked as soon as a local optimum is found. Results given by the new method are compared with those of previous heuristics on a series of examples.We are grateful to R. Bulfin for making the code for reliability optimization of series systems he wrote with C.-Y. Liu available to us.Work of the first author was supported by NSERC Grant No. GP0105574, FCAR Grant No. 92EQ1048 and AFOSR Grant No. 90-0008 to Rutgers University.Work of the second author was partly supported by AFOSR Grant No. 90-0008 to Rutgers University while he was a graduate student.     

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.     

An analytical approach to global optimization

Global optimization problems with a few variables and constraints arise in numerous applications but are seldom solved exactly. Most often only a local optimum is found, or if a global optimum is detected no proof is provided that it is one. We study here the extent to which such global optimization problems can be solved exactly using analytical methods. To this effect, we propose a series of tests, similar to those of combinatorial optimization, organized in a branch-and-bound framework. The first complete solution of two difficult test problems illustrates the efficiency of the resulting algorithm. Computational experience with the programbagop, which uses the computer algebra systemmacsyma, is reported on. Many test problems from the compendiums of Hock and Schittkowski and others sources have been solved.The research of the first and the third authors has been supported by AFOSR grants #0271 and #0066 to Rutgers University. Research of the second author has been supported by NSERC grant #GP0036426 and FCAR grants #89EQ4144 and #90NC0305.     

Local structure-preserving algorithms for partial differential equations

In this paper, we discuss the concept of local structure-preserving algorithms (SPAs) for partial differential equations, which are the natural generalization of the corresponding global SPAs. Local SPAs for the problems with proper boundary conditions are global SPAs, but the inverse is not necessarily valid. The concept of the local SPAs can explain the difference between different SPAs and provide a basic theory for analyzing and constructing high performance SPAs. Furthermore, it enlarges the applicable scopes of SPAs. We also discuss the application and the construction of local SPAs and derive several new SPAs for the nonlinear Klein-Gordon equation. This work was supported by the National Basic Research Program (Grant No. 2005CB321703). The first author was supported by the National Natural Science Foundation of China (Grant Nos. 40405019, 10471067) and the Major Research Projects of Jiangsu Province (Grant No. BK2006725); the second author was supported by the National Natural Science Foundation of China (Innovation Group) (Grant No. 40221503) and the third author was supported by the National Natural Science Foundation of China (Grant No. 10471145)     

Bernoullicity of solenoidal automorphisms and global fields

We show that ergodic automorphisms of solenoids are isomorphic to Bernoulli shifts by using the product formula for global fields. The authors gratefully acknowledge support by the Mathematical Sciences Research Institute. The first author was supported in part by NSF Grant DMS-9004253.     

Best simultaneous approximation to totally bounded sequences in Banach spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. The first author is supported in part by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 06C651); the second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 10671175, 10731060) and Program for New Century Excellent Talents in University; the third author is supported in part by Projects MTM2006-13997-C02-01 and FQM-127 of Spain     

Computing the geodesic center of a simple polygon

The geodesic center of a simple polygon is a point inside the polygon which minimizes the maximum internal distance to any point in the polygon. We present an algorithm which calculates the geodesic center of a simple polygon withn vertices in timeO(n logn).Work on this paper by the first author has been supported by National Science Foundation Grant No. DMS-8501947. Work on this paper by the second author has been supported by Office of Naval Research Grant No. N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. Part of the work on this paper by the first two authors has been carried out at the Workshop on Movable Separability of Sets at the Bellairs Research Institute of McGill University, Barbados, February 1986. Work on this paper by the third author has been supported by the Fonds zur Förderung der wissenschaftlichen Forschung (FWF), Project S32/01.     

On critical orientations in the kedem-sharir motion planning algorithm

We discuss a technical problem arising in the motion planning algorithm of Kedem and Sharir [KS], and propose a way to overcome it without increasing the asymptotic complexity of the algorithm The first author was supported by the Eshkol Grant 04601-90 from the Israeli Ministry of Science and Technology. The second author was partly supported by the Fund for Basic Research administered by the Israeli Academy of Sciences, by National Science Foundation Grants CCR-91-22103 and CCR-93-11127, and by grants from the U.S.-Israeli Binational Science Foundation, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. The third author was partly supported by the Interdisciplinary Program at Tel-Aviv University. The third author’s current address is: Department of Computer Science, MIT, Boston, MA, USA.     

Toeplitz Operators and Herz Spaces on the Half-Space

On the setting of the half-space we introduce the Schatten-Herz classes of Toeplitz operators and obtain characterizations for positive Toeplitz operators to belong to those classes. We also prove results concerning the boundedness and compactness of Toeplitz operators with Herz symbols. Such a study has been recently done on the ball. At a critical step of the proofs we employ a much simplified argument to extend the range of parameters for Herz spaces on which the Berezin transform is bounded. Our results show not only that most of results on the ball continue to hold, but also that there is some pathology caused by the unboundedness of the domain. The first author was in part supported by a Korea University Grant(2007), the second author was in part supported by Hanshin University Research Grant, and both authors were in part supported by KOSEF(R01-2003-000-10243-0).     

Primal-dual target-following algorithms for linear programming

In this paper, we propose a method for linear programming with the property that, starting from an initial non-central point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Together with the convergence analysis, we provide a general framework which enables us to analyze various primal-dual algorithms in the literature in a short and uniform way.This work was completed with the support of a research grant from SHELL. The first author is supported by the Dutch Organization for Scientific Research (NWO), Grant No. 611-304-028. The third author is on leave from the Eötvös University, Budapest, and partially supported by OTKA No. 2116. The fourth author is supported by the Swiss National Foundation for Scientific Research, Grant No. 12-34002.92.     

Analysis of heuristics for the design of tree networks

We analyze the tour partitioning heuristics for the Capacitated Minimum Spanning Tree problem. Lower bounds for the worst-case performance ratios of these heuristics are obtained by using worst-case examples. We also generalize the heuristics to the multi-center case with the same worst-case bounds.The work of the first author was supported by a Dean Summer Research Grant from Owen Graduate School of Management, Vanderbilt University.Work done in part in the Department of Industrial Engineering and Operations Research at Columbia University.The work of the last two authors was supported in part by ONR contract N00014-90-J-1649, NSF contract DDM-8922712 and the Center for Telecommunications Research under NSF contract CDR 84-21402.     

Complete solution of a family of quartic Thue equations

This paper evolved from a visit of the first author to the University of Debrecen, Hungary, which was supported by the Austrian-Hungarian Science Cooperation project Nr. 10-U-3. Research of the second author was partially supported by Hungarian National Foundation for Scientific Research Grant 1641/90.     

Mean convergence of hermite interpolation revisited

The first author is supported by the Hungarian National Foundation for Scientific Research Grant No. 1910 and No. T7570, and the second author is supported by the National Science Foundation Grant No. 9302721. The work was done during the first author's visit in Eugene, Oregon in 1993, and was completed during the second author's visit to the Mathematisches Institut, University of Erlange-Nürnberg, supported by the Alexander von Humboldt Foundation.     

A penalty optimization technique for a class of regulator problems

This paper describes a function space algorithm for the solution of a class of linear-quadratic optimal control problems.The research of the first author was supported by Senate Research Grant No. 8.187.17, University of Ilorin, Ilorin, Kwara State, Nigeria.The authors thank two anonymous referees for their useful and challenging suggestions and comments which improved the quality of this paper. The authors are also indebted to I. Orisamolu for carrying out the computing work.     

Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result.The first author was partially supported by the Russian Foundation for Basic Research Grant 02-01-00334. The second author was partially supported by the Russian Foundation for Basic Research Grant 00-01-00869. The third author was partially supported by Fundacao para a Ciencia e Tecnologia and by INVOTAN Grant.     

Nonobtuse Tetrahedral Partitions that Refine Locally Towards Fichera-Like Corners

Linear tetrahedral finite elements whose dihedral angles are all nonobtuse guarantee the validity of the discrete maximum principle for a wide class of second order elliptic and parabolic problems. In this paper we present an algorithm which generates nonobtuse face-to-face tetrahedral partitions that refine locally towards a given Fichera-like corner of a particular polyhedral domain. The first author was supported by the Swedish Foundation for Strategic Research, the second author was supported by Grant No. 49051 of the Academy of Finland, the third author was supported by Grant No. A 1019201 of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AV0Z 10190503.     

Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited

Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.Dedicated to the Memory of Magnus R. Hestenes, 1906–1991This research was supported in part by NSF Cooperative Agreement CCR-88-09615 and was initiated while the first author was at Rice University as a Visiting Member of the Center for Research in Parallel Computation.The authors thank Yinyu Ye for constructive comments and discussions concerning this material.This author was supported in part by NSF Grant DMS-91-02761 and DOE Grant DE-FG05-91-ER25100.This author was supported in part by AFOSR Grant 89-0363, DOE Grant DE-FG05-86-ER25017, and ARO Grant 9DAAL03-90-G-0093.     

Sharp Minima for Multiobjective Optimization in Banach Spaces

We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.This research was supported by a Central Research Grant of The Hong Kong Polytechnic University (Grant No. G-T 507). Research of the first author was also supported by the National Natural Science Foundation of PR China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, China (Grant No. 2003A002M).     

Weakly self-injective semilattices

In this note we characterize weakly self-injective semilattices as Brouwerian semilattices which are compact in the residuated interval topology. We also characterize weakly self-injective semigroups which are semilattices of groups with trivial multiplication. Research of first author supported in part by a research grant from the Faculty Research Committee of Bowling Green State University. Research of second author supported in part by a postdoctoral fellowship in the Biomathematics Program at North Carolina State University under PHS Grant #GM-678 from NIGMS.