Existence theorems in vector optimization

In this paper, existence theorems for minimal, weakly minimal, and properly minimal elements of a subset of a partially-ordered, real linear space are presented.This paper was written when the author was a visitor at the Department of Mathematics, North Carolina State University, Raleigh, North Carolina.     

Diagonalized multiplier methods and quasi-Newton methods for constrained optimization

Two approaches to quasi-Newton methods for constrained optimization problems inR ⁿ are presented. These approaches are based on a class of Lagrange multiplier approximation formulas used by the author in his previous work on Newton's method for constrained problems. The first approach is set in the framework of a diagonalized multiplier method. From this point of view, a new update rule for the Lagrange multipliers which depends on the particular quasi-Newton method employed is given. This update rule, in contrast to most other update rules, does not require exact minimization of the intermediate unconstrained problem. In fact, the optimal convergence rate is attained in the extreme case when only one step of a quasi-Newton method is taken on this intermediate problem. The second approach transforms the constrained optimization problem into an unconstrained problem of the same dimension.The author would like to thank J. Moré and M. J. D. Powell for comments related to the material in Section 13. He also thanks J. Nocedal for the computer results in Tables 1–3 and M. Wright for the results in Table 4, which were obtained via one of her general programs. Discussions with M. R. Hestenes and A. Miele regarding their contributions to this area were very helpful. Many individuals, including J. E. Dennis, made useful general comments at various stages of this paper. Finally, the author is particularly thankful to R. Byrd, M. Heath, and R. McCord for reading the paper in detail and suggesting many improvements.This work was supported by the Energy Research and Development Administration, Contract No. E-(40-1)-5046, and was performed in part while the author was visiting the Department of Operations Research, Stanford University, Stanford, California.     

Path problems in skew-symmetric graphs

We study path problems in skew-symmetric graphs. These problems generalize the standard graph reachability and shortest path problems. We establish combinatorial solvability criteria and duality relations for the skew-symmetric path problems and use them to design efficient algorithms for these problems. The algorithms presented are competitive with the fastest algorithms for the standard problems.This research was done while the first author was at Stanford University Computer Science Department, supported in part by ONR Office of Naval Research Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation.This research was done while the second author was visiting Stanford University Computer Science Department and supported by the above mentioned NSF and Powell Foundation Grants.     

On accuracy and unconditional stability of linear multistep methods for second order differential equations

Linear multistep methods for solution of the equationy=f(t, y) are studied by means of the test equationy=–² y, with real. It is shown that the order of accuracy cannot exceed 2 for an unconditionally stable method.This work was supported by the NASA-Ames Research Center, Moffett Field, California, under Interchange No. NCA2-OR745-712, while the author was a visitor at the Computer Science Department, Stanford University, Stanford, California.     

A convergence theorem of Rosen’s gradient projection method

Since Rosen’s gradient projection method was published in 1960, a rigorous convergence proof of his method has remained an open question. A convergence theorem is given in this paper. Part of this author’s work was done while he studied at the Department of Mathematics, University of California at Santa Barbara, and was supported by the National Science Foundation under Grant No. MCS83-14977. Part of this author’s work was done while he visited the Computer Science Department, University of Minnesota, Minneapolis, and was supported by the National Science Foundation under Grant No. MCS81-01214.     

On a Newton-like method for constrained nonlinear minimization via slack variables

We show that Tapia's quasi-Newton diagonalized approach to constrained minimization can be formulated in such a way that no linear systems have to be solved of dimension larger than the natural ones or which present singularities. Numerical experiments indicate fast local convergence, but also substantial difficulties of global convergence.This work was supported by the National Science Council of Italy in the framework of the SOFTMAT project. Part of the work was done during visits at the Computer Science Department, Stanford University, and at the Numerical Optimization Centre, Hatfield Polytechnic; the author thanks Prof. G. Golub and Prof. L. Dixon for providing a stimulating atmosphere. Thanks are also due to Dr. M. Bertocchi, University of Bergamo, for collaboration in performing the numerical experiments.     

The value of the stochastic solution in stochastic linear programs with fixed recourse

Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.This paper is an extension of part of the author's dissertation in the Department of Operations Research, Stanford University, Stanford, California. The research was supported at Stanford by the Department of Energy under Contract DE-AC03-76SF00326, PA#DE-AT03-76ER72018, Office of Naval Research under Contract N00014-75-C-0267 and the National Science Foundation under Grants MCS76-81259, MCS-7926009 and ECS-8012974 (formerly ENG77-06761).     

Matrix factorizations in optimization of nonlinear functions subject to linear constraints

Several ways of implementing methods for solving nonlinear optimization problems involving linear inequality and equality constraints using numerically stable matrix factorizations are described. The methods considered all follow an active constraint set approach and include quadratic programming, variable metric, and modified Newton methods.Part of this work was performed while the author was a visitor at Stanford University. This research was supported in part by the National Science Foundation under Grant GJ 36472 and in part by the Atomic Energy Commission Contract No. AT(04-3)-326PA30.     

On the relation between quadratic termination and convergence properties of minimization algorithms

Summary It is shown that the theory developed in part I of this paper [22] can be applied to some well-known minimization algorithms with the quadratic termination property to prove theirn-step quadratic convergence. In particular, some conjugate gradient methods, the rank-1-methods of Pearson and McCormick (see Pearson [18]) and the large class of rank-2-methods described by Oren and Luenberger [16, 17] are investigated.This work was supported in part at Stanford University, Stanford, California, under Energy Research and Development Administration, Contract E(04-3) 326 PA No. 30, and National Science Foundation Grant DCR 71-01996 A04 and in part by the Deutsche Forschungsgemeinschaft     

On theoretical and numerical aspects of the bang-bang-principle

Summary In the first part of this paper we are dealing with theoretical statements and conditions which finally lead to bang-bang-principles. A careful analysis of these theorems is used for the development of a numerical method. This method consists of two stages: During the first iterations the number and approximate location of the switching points of the optimal control are determined. In the second phase a rapidly convergent algorithm determines the exact location. We apply this method successfully to a parabolic boundary control problem and give an extensive discussion of numerical results.The work of the second author on this paper was partially done during his stay at North Carolina State University, Graduate Program in Operations Research and Department of Mathematics, Raleigh, USA     

Head injury criteria

The results of a study of the mechanical responses of the head will be presented. Driving point impedance responses of a variety of subhuman primates and unembalmed cadavers have been measured and used to develop a lumped parameter model of the head. The response of these heads to impacts (front and side) have been used to further evaluate model parameters and to develop injury criteria. The model indicates decreasing tolerance with impulse duration for pulses of approximately 10 times the first resonance period; a quasistatic response is indicated for impulses longer than 10 times the first resonance period. The results are compared with other current head injury criteria. Applications of the model to helmet design and restraint systems are suggested. It is intended that this information form part of a crash test device performance specification that will result in an analogue head with human-like responses to be used in the prediction of the injury-reducing potential of head protection devices.Duke University, Durham, North Carolina, U.S.A. Published in Mekhanika Polimerov, No. 3, pp. 465–477, May–June, 1976.     

Consistency and asymptotic normality of profilekernel and backfitting estimators in semiparametric reproductive dispersion nonlinear models

Semiparametric reproductive dispersion nonlinear model (SRDNM) is an extension of nonlinear reproductive dispersion models and semiparametric nonlinear regression models, and includes semiparametric nonlinear model and semiparametric generalized linear model as its special cases. Based on the local kernel estimate of nonparametric component, profile-kernel and backfitting estimators of parameters of interest are proposed in SRDNM, and theoretical comparison of both estimators is also investigated in this paper. Under some regularity conditions, strong consistency and asymptotic normality of two estimators are proved. It is shown that the backfitting method produces a larger asymptotic variance than that for the profile-kernel method. A simulation study and a real example are used to illustrate the proposed methodologies. This work was supported by National Natural Science Foundation of China (Grant Nos. 10561008, 10761011), Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. Y200805073), PhD Special Scientific Research Foundation of Chinese University (Grant No. 20060673002) and Program for New Century Excellent Talents in University (Grant No. NCET-07-0737)     

Lossless Condensers,Unbalanced Expanders,And Extractors

Trevisan showed that many pseudorandom generator constructions give rise to constructions of explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(logn) additional random bits that maps n bits strings to poly(logK) bit strings, such that any source with support size K is mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date. By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough min-entropies our extractors can output all of the randomness with only O(logn) bits. We also obtain a new disperser that works for every entropy loss, uses an O(logn) bit seed, and has only O(logn) entropy loss. This is the best disperser construction to date, and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced bipartite graph with strong expansion properties. * Much of this work was done while the author was in the Computer Science Division, University of California, Berkeley, and supported in part by a David and Lucile Packard Fellowship for Science and Engineering and NSF NYI Grant No. CCR-9457799. The work was also supported in part by an Alon fellowship and by the Israel Science Foundation. † Much of this work was done while the author was a graduate student in the Computer Science Division, University of California, Berkeley. Supported in part by NSF Grants CCR-9820897, CCF-0346991, and an Alfred P. Sloan Research Fellowship. ‡ Much of this work was done while the author was on leave at the Computer Science Division, University of California, Berkeley. Supported in part by a David and Lucile Packard Fellowship for Science and Engineering, NSF Grants CCR-9912428 and CCR-0310960, NSF NYI Grant CCR-9457799, and an Alfred P. Sloan Research Fellowship.     

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.     

A variable-metric method for function minimization derived from invariancy to nonlinear scaling

The effect of nonlinearly scaling the objective function on the variable-metric method is investigated, and Broyden's update is modified so that a property of invariancy to the scaling is satisfied. A new three-parameter class of updates is generated, and criteria for an optimal choice of the parameters are given. Numerical experiments compare the performance of a number of algorithms of the resulting class.The author is indebted to Professor S. S. Oren, Economic Engineering Department, Stanford University, Stanford, California, for stimulating discussions during the development of this paper. He also recognizes the financial support by the National Research Council of Italy (CNR) for his stay at Stanford University.     

The truncatedSVD as a method for regularization

The truncated singular value decomposition (SVD) is considered as a method for regularization of ill-posed linear least squares problems. In particular, the truncated SVD solution is compared with the usual regularized solution. Necessary conditions are defined in which the two methods will yield similar results. This investigation suggests the truncated SVD as a favorable alternative to standard-form regularization in cases of ill-conditioned matrices with well-determined numerical rank.This work was carried out while the author visited the Dept. of Computer Science, Stanford University, California, U.S.A., and was supported in part by National Science Foundation Grant Number DCR 8412314, by a Fulbright Supplementary Grant, and by the Danish Space Board.     

Duality in inexact fractional programming with set-inclusive constraints

This paper extends the fractional programming problem with set-inclusive constraints studied earlier by replacing every coefficient vector in the objective function with a convex set. A dual is formulated, and well-known duality results are established. A numerical example illustrates the dual strategy to obtain the value of the initial problem.The research of the first author was conducted while he was on sabbatical at the Department of Operations Research, Stanford University, Stanford, California. The financial assistance of the International Council for Exchange of Scholars is gratefully acknowledged. The author is grateful to the Department of Operations Research at Stanford for the use of its research facilities.     

Graph Theoretic and Spectral Analysis of Enron Email Data

Analysis of social networks to identify communities and model their evolution has been an active area of recent research. This paper analyzes the Enron email data set to discover structures within the organization. The analysis is based on constructing an email graph and studying its properties with both graph theoretical and spectral analysis techniques. The graph theoretical analysis includes the computation of several graph metrics such as degree distribution, average distance ratio, clustering coefficient and compactness over the email graph. The spectral analysis shows that the email adjacency matrix has a rank-2 approximation. It is shown that preprocessing of data has significant impact on the results, thus a standard form is needed for establishing a benchmark data. Anurat Chapanond is currently a Ph.D. student in Computer Science, RPI. Anurat graduated B. Eng. degree in Computer Engineering from Chiangmai University (Thailand) in 1997, M. S. in Computer Science from Columbia University in 2002. His research interest is in web data mining analyses and algorithms. M.S. Krishnamoorthy received the B.E. degree (with honors) from Madras University in 1969, the M. Tech degree in Electrical Engineering from the Indian Institute of Technology, Kanpur, in 1971, and the Ph. D. degree in Computer Science, also from the Indian Institute of Technology, in 1976. From 1976 to 1979, he was an Assistant Professor of Computer Science at the Indian Institute of Technology, Kanpur. From 1979 to 1985, he was an Assistant Professor of Computer Science at Rensselaer Polytechnic Institute, Troy, NY, and since, 1985, he has been an Associate Professor of Computer Science at Rensselaer. Dr. Krishnamoorthy's research interests are in the design and analysis of combinatorial and algebraic algorithms, visualization algorithms and programming environments. Bulent Yener is an Associate Professor in the Department of Computer Science and Co-Director of Pervasive Computing and Networking Center at Rensselaer Polytechnic Institute in Troy, New York. He is also a member of Griffiss Institute of Information Assurance. Dr. Yener received MS. and Ph.D. degrees in Computer Science, both from Columbia University, in 1987 and 1994, respectively. Before joining to RPI, he was a Member of Technical Staff at the Bell Laboratories in Murray Hill, New Jersey. His current research interests include bioinformatics, medical informtatics, routing problems in wireless networks, security and information assurance, intelligence and security informatics. He has served on the Technical Program Committee of leading IEEE conferences and workshops. Currently He is an associate editor of ACM/Kluwer Winet journal and the IEEE Network Magazine. Dr. Yener is a Senior Member of the IEEE Computer Society.     

The fixed point property for small sets

There exist exactly eleven (up to isomorphism and duality) ordered sets of size 10 with the fixed point property and containing no irreducible elements.The great part of the work presented here has been done when the author was visiting Ivan Rival at the University of Ottawa, Department of Computer Science.     

On surfaces with a rectilinear geodesic circle

Summary We say that a surface has a “rectilinear geodesic circle“ if it contains a straight line segment AB and a point F whose geodesic distance from a variable point of AB is a constant. The problem of generating such surfaces is solved here by constructing families of cones which behave locally (along a curce) like solutions, and then taking their envelopes. This method generates a class of solutions which depend on an arbitrary curve. This is an excerpt from a thesis presented in partial fulfillment of the Ph. D. Requirements of the Mathematics Department of Stanford University. It was supported in part by the Ballistics Research Laboratory of the Ordnance Department, U. S. Army, under Contract No. DA-04-200-ORD-294.