The scalar curvature equation on 2- and 3-spheres

In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.Research of first author supported in part by NSF grant 91-03949Research of second author supported by a NSF Postdoctoral Fellowship.Research of third author supported in part by NSF grant 91-02872 and the Ellentuck Fund.     

A least-squares-based method for a class of nonsmooth minimization problems with applications in plasticity

This paper introduces a globally convergent algorithm for solving a class of nonsmooth optimization problems, involving square roots of quadratic forms. The class includes in particular limit analysis problems in plasticity. The algorithm combines smoothing with successive approximation. The main computational effort in each iteration is solving a linear weighted least-squares problem. The convergence of the algorithm is proved and ana priori error estimate is obtained. Numerical results are presented for two limit analysis problems.The work of the first author was partially supported by NSF Grant DDM-89-96112. Parts of the work was done during his stay at the University of Bayreuth as a guest of the DFG. The work of the second author was supported in part by the Air Force Office of Scientific Research under contract AFOSR-88-0218 and by a National Science Foundation Grant ECS-8802239 at the University of Maryland, Baltimore County Campus.     

A second-order method for unconstrained optimization

This paper presents a quadratically converging algorithm for unconstrained minimization. All the accumulation points that it constructs satisfy second-order necessary conditions of optimality. Thus, it avoids second-order saddle andinflection points, an essential feature for a method to be used in minimizing the modified Lagrangians in multiplier methods.The work of the first author was supported by NSF RANN AEN 73-07732-A02 and JSEP Contract No. F44620-71-C-0087; the work of the second author was supported by NSF Grant No. GK-37672 and the ARO Contract No. DAHCO4-730C-0025.     

A primal-dual approximation algorithm for generalized steiner network problems

We present the first polynomial-time approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. Ifk is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primal-dual and shows the importance of this technique in designing approximation algorithms.Research supported by an NSF Graduate Fellowship, DARPA contracts N00014-91-J-1698 and N00014-92-J-1799, and AT&T Bell Laboratories.Research supported in part by Air Force contract F49620-92-J-0125 and DARPA contract N00014-92-J-1799.Part of this work was done while the author was visiting AT&T Bell Laboratories and Bellcore.     

On the convergence of primal-dual interior-point methods with wide neighborhoods

We study primal-dual interior-point methods for linear programs. After proposing a new primaldual potential function we describe a new potential reduction algorithm. We make connections between the new potential function and primal-dual interior-point algorithms with wide neighborhoods. Then we describe an algorithm that is a slightly modified version of existing primal-dual algorithms using wide neighborhoods. Assuming the optimal solution is non-degenerate, the algorithm is 1-step Q-quadratically convergent. We also study the degenerate case and show that the neighborhoods of the central path stay large as the iterates approach the optimal solutions.Research performed while the author was a Ph.D. student at Cornell University and was supported in part by the United States Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute of Cornell University, Contract DAAL03-91-C-0027 and also by NSF, AFOSR and ONR through NSF Grant DMS-8920550.     

Second order parallel algorithms for piecewise smooth displacement kernels

Second order parallel algorithms for Fredholm integral equations with piecewise smooth displacement kernels are derived. One is based on a difference scheme of Runge-Kutta type for an unusual partial differential equations for continuous functions of two variables. The other is based on the trapezoidal quadrature rule applied to a modified integral equations. It is found that the Runge-Kutta type algorithm exhibits certain advantages.The work of these authors was supported in part by the NSF Grant DMS-9007030The work of this author was supported in part by a grant from the National Science and Engineering Research Council of Canada     

On the convergence of the iteration sequence in primal-dual interior-point methods

Recently, numerous research efforts, most of them concerned with superlinear convergence of the duality gap sequence to zero in the Kojima—Mizuno—Yoshise primal-dual interior-point method for linear programming, have as a primary assumption the convergence of the iteration sequence. Yet, except for the case of nondegeneracy (uniqueness of solution), the convergence of the iteration sequence has been an important open question now for some time. In this work we demonstrate that for general problems, under slightly stronger assumptions than those needed for superlinear convergence of the duality gap sequence (except of course the assumption that the iteration sequence converges), the iteration sequence converges. Hence, we have not only established convergence of the iteration sequence for an important class of problems, but have demonstrated that the assumption that the iteration sequence converges is redundant in many of the above mentioned works.This research was supported in part by NSF Coop. Agr. No. CCR-8809615. A part of this research was performed in June, 1991 while the second and the third authors were at Rice University as visiting members of the Center for Research in Parallel Computation.Corresponding author. Research supported in part by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Research supported in part by NSF DMS-9102761 and DOE DE-FG05-91ER25100.Research supported in part by NSF DDM-8922636.     

Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights

The nonsymmetric Lanczos algorithm reduces a general matrix to tridiagonal by generating two sequences of vectors which satisfy a mutual bi-orthogonality property. The process can proceed as long as the two vectors generated at each stage are not mutually orthogonal, otherwise the process breaks down. In this paper, we propose a variant that does not break down by grouping the vectors into clusters and enforcing the bi-orthogonality property only between different clusters, but relaxing the property within clusters. We show how this variant of the matrix Lanczos algorithm applies directly to a problem of computing a set of orthogonal polynomials and associated indefinite weights with respect to an indefinite inner product, given the associated moments. We discuss the close relationship between the modified Lanczos algorithm and the modified Chebyshev algorithm. We further show the connection between this last problem and checksum-based error correction schemes for fault-tolerant computing.The research reported by this author was supported in part by NSF grant CCR-8813493.The research reported by this author was supported in part by ARO grant DAAL03-90-G-0105 and in part by NSF grant DCR-8412314.     

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.     

Computing the optimal commuting matrix pairs

A matrix can be modified by an additive perturbation so that it commutes with any given matrix. In this paper, we discuss several algorithms for computing the smallest perturbation in the Frobenius norm for a given matrix pair. The algorithms have applications in 2-D direction-of-arrival finding in array signal processing. The work of first author was supported in part by NSF grant CCR-9308399. The work of the second author was supported in part by China State Major Key Project for Basic Researches.     

Shortest paths algorithms: Theory and experimental evaluation

We conduct an extensive computational study of shortest paths algorithms, including some very recent algorithms. We also suggest new algorithms motivated by the experimental results and prove interesting theoretical results suggested by the experimental data. Our computational study is based on several natural problem classes which identify strengths and weaknesses of various algorithms. These problem classes and algorithm implementations form an environment for testing the performance of shortest paths algorithms. The interaction between the experimental evaluation of algorithm behavior and the theoretical analysis of algorithm performance plays an important role in our research. This work was done while Boris V. Cherkassky was visiting Stanford University Computer Science Department and supported by the NSF and Powell Foundation grants mentioned below. Andrew V. Goldberg was supported in part by ONR 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. Corresponding author. This work was done while Tomasz Radzik was a Postdoctoral Fellow at SORIE, Cornell University, and supported by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550, and by the Packard Fellowship of éva Tardos.     

Proximal minimization algorithm withD-functions

The original proximal minimization algorithm employs quadratic additive terms in the objectives of the subproblems. In this paper, we replace these quadratic additive terms by more generalD-functions which resemble (but are not strictly) distance functions. We characterize the properties of suchD-functions which, when used in the proximal minimization algorithm, preserve its overall convergence. The quadratic case as well as an entropy-oriented proximal minimization algorithm are obtained as special cases.The work of the first author was supported by NSF Grant CCR-8811135 and NIH Grant HL-28438, while visiting the Decision Sciences Department of the Wharton School and the Medical Image Processing Group at the Department of Radiology, both at the University of Pennsylvania, Philadelphia, Pennsylvania. The work of the second author was supported by NSF Grant CCR-91-04042 and AFOSR Grant 91-0168.The authors received valuable comments on the December 1989 and June 1990 versions of this paper, which led to considerable improvements of the results and their presentation. For this, they are indebted to D. Bertsekas, A. De Pierro, J. Eckstein, P. Eggermont, A. Iusem, M. Ferris, M. Teboulle, and three anonymous referees.     

Stochastic partial differential equations in Hölder spaces

Summary We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.The work of the first author was supported in part by the NSF grant DMS-91-01360     

On the formulation and theory of the Newton interior-point method for nonlinear programming

In this work, we first study in detail the formulation of the primal-dual interior-point method for linear programming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.The first two authors were supported in part by NSF Cooperative Agreement No. CCR-8809615, by Grants AFOSR 89-0363, DOE DEFG05-86ER25017, ARO 9DAAL03-90-G-0093, and the REDI Foundation. The fourth author was supported in part by NSF DMS-9102761 and DOE DE-FG02-93ER25171. The authors would like to thank Sandra Santos for painstakingly proofreading an earlier verion of this paper.     

A generalized hypergreedy algorithm for weighted perfect matching

We give a generalization of the hypergreedy algorithm for minimum weight perfect matching on a complete edge weighted graph whose weights satisfy the triangle inequality. With a modified version of this algorithm we obtain a logn-approximate perfect matching heuristic for points in the Euclidean plane, inO(n log² n) time.This research was supported in part by the DIMACS Grant No. NSF-STC88-09648.This research was supported in part by the NSF under Grant No. CCR 88-07518.     

On computing the center of a convex quadratically constrained set

We propose a method for finding analytic center of a convex feasible region whose boundaries are defined by quadratic functions. The algorithm starts from an arbitrary initial point and approaches to the desired center by simultaneously reducing infeasibility or slackness of all constraints. A partial Newton step is taken at each iteration.Research supported in part by the ONR under grant N00014-87-K-0214 and by the NSF under grant CCR-8810107.Research supported in part by the NSF under grant ECS-8721709.     

A primal-dual interior-point method for linear programming based on a weighted barrier function

One motivation for the standard primal-dual direction used in interior-point methods is that it can be obtained by solving a least-squares problem. In this paper, we propose a primal-dual interior-point method derived through a modified least-squares problem. The direction used is equivalent to the Newton direction for a weighted barrier function method with the weights determined by the current primal-dual iterate. We demonstrate that the Newton direction for the usual, unweighted barrier function method can be derived through a weighted modified least-squares problem. The algorithm requires a polynomial number of iterations. It enjoys quadratic convergence if the optimal vertex is nondegenerate.The research of the second author was supported in part by ONR Grants N00014-90-J-1714 and N00014-94-1-0391.     

Multiplicative commutators in division rings

In this paper we study division rings in which the multiplicative commutators are periodic or periodic relative to the center. This work was supported by NSF grant MCS-76-06683 at the University of Chicago. This paper was written while the author was a vistor at the Institute for Advanced Studies, The Hebrew University of Jerusalem.     

A superquadratic infeasible-interior-point method for linear complementarity problems

We consider a modification of a path-following infeasible-interior-point algorithm described by Wright. In the new algorithm, we attempt to improve each major iterate by reusing the coefficient matrix factors from the latest step. We show that the modified algorithm has similar theoretical global convergence properties to those of the earlier algorithm while its asymptotic convergence rate can be made superquadratic by an appropriate parameter choice. The work of this author was based on research supported by the Office of Scientific Computing, US Department of Energy, under Contract W-31-109-Eng-38. The work of this author was based on research supported in part by the US Department of Energy under Grant DE-FG02-93ER25171.     

A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPs

 The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, we proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints. Received: June 22, 2001 / Accepted: January 20, 2002 Published online: December 9, 2002 RID="†" ID="†"Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. RID="⋆" ID="⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203426 RID="⋆⋆" ID="⋆⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203113 RID="⋆⋆⋆" ID="⋆⋆⋆"This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339. Key Words. semidefinite program – semidefinite relaxation – nonlinear programming – interior-point methods – limited memory quasi-Newton methods. Mathematics Subject Classification (1991): 90C06, 90C27, 90C30.