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SIAM J. on Optimization

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1999

Volume 9, Issue 4, pp. vii-1204

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Dedication

Michael L. Overton and Robert B. Schnabel

SIAM J. Optim. 9, pp. vii-viii ( pages)

Online Publication Date: July 31, 2006

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SIAM Journal on Optimization would not exist were it not for the vision, energy, and dedication of John E. Dennis, Jr. It was he who, in the late 1980s, recognized the need for a SIAM journal focusing broadly on optimization. He was inspired partly by the success of the SIAM conferences on optimization held regularly since 1984 and partly by the example of the Mathematical Programming Society, whose effectiveness in establishing optimization, especially its algorithmic aspects, as a discipline rested both on its international symposia and on its well-respected journal. John had long played an important role in SIAM, serving as an editor for SIAM Journal on Numerical Analysis from 1975 to 1981, as co-chair of the second SIAM Conference on Optimization in Houston in 1987, as a member of the SIAM Council from 1985 to 1990, as chair of the SIAM activity group on optimization from 1989 to 1992, and as an advocate for optimization as a subject that has a natural home in SIAM. When SIAM Journal on Optimization was established, he was a natural choice as Editor-in-Chief, serving from 1990 to 1994, when he stepped down to become the Chair of the Mathematical Programming Society. With the passage of time, it is clear that the establishment of the journal brought optimization to full status as one of the leading disciplinary areas within SIAM.
The two of us have known John since the late 1970s when we were graduate students, one of us as his advisee. He was very supportive to us as young scientists and by his support played a critical role throughout our careers, for which we are enormously grateful. John is particularly proud of the success of his more than thirty Ph.D. graduates at Cornell and Rice; seven of these former students are authors of papers in this special issue. John has always made a special point of showing interest in young scientists beginning their careers, whether or not they studied with him; in our view, nothing a senior scientist can do is more important than that.
John began his career in Utah as a functional analyst and only later turned to computational mathematics. He has authored or coauthored dozens of well known papers in optimization and applied mathematics. To single out one contribution is difficult, but he is particularly well known for his pioneering convergence analysis of quasi-Newton methods (also known as secant or variable metric methods) with C. G. Broyden and J. J. Moré, and his survey paper with Moré in SIAM Review (1977) became required reading for a generation of graduate students. In more recent years John's special interest has been multidisciplinary optimization, emphasizing industrial application of optimization, especially in the aeronautical and oil industries. Nothing could be closer to the central mission of SIAM.
John has long been an advocate of electronic publication, and we find it especially appropriate that, exploiting this medium, we are publishing this special issue of SIAM Journal on Optimization actually on his 60th birthday. We are honored to be able to dedicate this issue to John Dennis. Finally, we also salute his family: Ann, Jed and Katie, of whom he is so proud. Happy Birthday, John, and many happy returns!

Linear Programming in O([n3/ln n]L) Operations

K. M. Anstreicher

SIAM J. Optim. 9, pp. 803-812 (10 pages) | Cited 9 times

Online Publication Date: July 31, 2006

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We show that the complexity to solve linear programming problems, using standard linear algebra, can be reduced to O([n3/ln n]L) operations, where n is the number of variables in a standard-form problem with integer data of bit size L. Our technique combines partial updating with a preconditioned conjugate gradient method, in a scheme first suggested by Nesterov and Nemirovskii.

Free Material Design via Semidefinite Programming: The Multiload Case with Contact Conditions

A. Ben-Tal, M. Kocvara, A. Nemirovski, and J. Zowe

SIAM J. Optim. 9, pp. 813-832 (20 pages) | Cited 16 times

Online Publication Date: July 31, 2006

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Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material and the material itself can freely vary. The case of one single load has been discussed in several recent papers, and an efficient numerical approach was presented in [M. Kocvara, M. Zibulevsky, and J. Zow, RAIRO Modél. Math. Anal. Numér. 32 (1998), pp. 255--281]. We attack here the multiload situation (understood in the worst-case sense), which is of much more interest for applications but also significantly more challenging from both the theoretical and the numerical points of view. After a series of transformation steps we reach a problem formulation for which we can prove existence of a solution; a suitable discretization leads to a semidefinite programming problem for which modern polynomial time algorithms of interior point type are available. A number of numerical examples demonstrates the efficiency of our approach.

A Global Convergence Analysis of an Algorithm for Large-Scale Nonlinear Optimization Problems

Paul T. Boggs, Anthony J. Kearsley, and Jon W. Tolle

SIAM J. Optim. 9, pp. 833-862 (30 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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In this paper we give a global convergence analysis of a basic version of an SQP algorithm described in [P. T. Boggs, A. J. Kearsley, and J. W. Tolle, SIAM J. Optim., 9 (1999), pp. 755--778] for the solution of large-scale nonlinear inequality-constrained optimization problems. Several procedures and options have been added to the basic algorithm to improve the practical performance; some of these are also analyzed. The important features of the algorithm include the use of a constrained merit function to assess the progress of the iterates and a sequence of approximate merit functions that are less expensive to evaluate. It also employs an interior point quadratic programming solver that can be terminated early to produce a truncated step.

Nonsymmetric Search Directions for Semidefinite Programming

Nathan Brixius, Florian A. Potra, and Rongqin Sheng

SIAM J. Optim. 9, pp. 863-876 (14 pages)

Online Publication Date: July 31, 2006

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Two nonsymmetric search directions for semidefinite programming, the XZ and ZX search directions, are proposed. They are derived from a nonsymmetric formulation of the semidefinite programming problem. The XZ direction corresponds to the direct linearization of the central path equation $XZ= \nu I,$ while the ZX direction corresponds to $ZX = \nu I$. The XZ and ZX directions are well defined if both X and Z are positive definite matrices, where X may be nonsymmetric. We present an algorithm using the XZ and ZX directions alternately following the Mehrotra predictor-corrector framework. Numerical results show that the XZ/ZX algorithm, in many cases, requires less CPU time than the XZ+ZX method of Alizadeh, Overton, and Haeberly [SIAM J. Optim., 8 (1998), pp. 746--768] while achieving similar accuracy.

An Interior Point Algorithm for Large-Scale Nonlinear Programming

Richard H. Byrd, Mary E. Hribar, and Jorge Nocedal

SIAM J. Optim. 9, pp. 877-900 (24 pages) | Cited 61 times

Online Publication Date: July 31, 2006

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The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primal-dual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests.

Nonlinear Optimization, Quadrature, and Interpolation

H. Cheng, V. Rokhlin, and N. Yarvin

SIAM J. Optim. 9, pp. 901-923 (23 pages) | Cited 16 times

Online Publication Date: July 31, 2006

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We present a nonlinear optimization procedure for the design of generalized Gaussian quadratures for a fairly broad class of functions. While some of the components of the algorithm have been published previously, we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the Newton-type process used to find the quadratures. The resulting procedure never failed when applied to Chebyshev systems (for which the existence and uniqueness of generalized Gaussian quadratures are well known); it also worked for many non-Chebyshev systems, for which the generalized Gaussian quadratures are not guaranteed to exist. The performance of the algorithm is illustrated with several numerical examples; some of the presented quadratures integrate efficiently large classes of singular functions.

Two-Step Algorithms for Nonlinear Optimization with Structured Applications

Andrew R. Conn, Luís N. Vicente, and Chandu Visweswariah

SIAM J. Optim. 9, pp. 924-947 (24 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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In this paper we propose extensions to trust-region algorithms in which the classical step is augmented with a second step that we insist yields a decrease in the value of the objective function. The classical convergence theory for trust-region algorithms is adapted to this class of two-step algorithms.
The algorithms can be applied to any problem with variable(s) whose contribution to the objective function is a known functional form. In the nonlinear programming package LANCELOT, they have been applied to update slack variables and variables introduced to solve minimax problems, leading to enhanced optimization efficiency. Extensive numerical results are presented to show the effectiveness of these techniques.

Where Best to Hold a Drum Fast

Steven J. Cox and Paul X. Uhlig

SIAM J. Optim. 9, pp. 948-964 (17 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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If we are allowed to fasten, say, one half of a drum's boundary, which half produces the lowest or highest bass note? The answer is a natural limit of solutions to a family of extremal Robin problems for the least eigenvalue of the Laplacian. We produce explicit extremizers when the drum is a disk while for general shapes we establish existence and necessary conditions, and we build and test a pair of numerical methods.

A Global Convergence Theory for Dennis, El-Alem, and Maciel's Class of Trust-Region Algorithms for Constrained Optimization without Assuming Regularity

Mahmoud El-Alem

SIAM J. Optim. 9, pp. 965-990 (26 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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This work presents a convergence theory for Dennis, El-Alem, and Maciel's class of trust-region-based algorithms for solving the smooth nonlinear programming problem with equality constraints. The results are proved under very mild conditions on the quasi-normal and tangential components of the trial steps. The Lagrange multiplier estimates and the Hessian estimates are assumed to be bounded. No regularity assumption is made. In particular, linear independence of the gradients of the constraints is not assumed. The theory proves global convergence for the class. In particular, it shows that a subsequence of the iteration sequence satisfies one of four types of Mayer--Bliss stationary conditions in the limit.

Expressing Complementarity Problems in an Algebraic Modeling Language and Communicating Them to Solvers

Michael C. Ferris, Robert Fourer, and David M. Gay

SIAM J. Optim. 9, pp. 991-1009 (19 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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Diverse problems in optimization, engineering, and economics have natural formulations in terms of complementarity conditions, which state (in their simplest form) that either a certain nonnegative variable must be zero or a corresponding inequality must hold with equality, or both. A variety of algorithms has been devised for solving problems expressed in terms of complementarity conditions. It is thus attractive to consider extending algebraic modeling languages, which are widely used for sending ordinary equations and inequality constraints to solvers, so that they can express complementarity problems directly. We describe an extension to the AMPL modeling language that can express the most common complementarity conditions in a concise and flexible way, through the introduction of a single new "complements" operator. We present details of an efficient implementation that incorporates an augmented presolve phase to simplify complementarity problems, and that converts complementarity conditions to a canonical form convenient for solvers.

On the Complexity of Solving Feasible Linear Programs Specified with Approximate Data

Sharon Filipowski

SIAM J. Optim. 9, pp. 1010-1040 (31 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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The problem of solving linear programs specified with approximate data is considered. Algorithms are given for linear programs having both general inequality and nonnegativity constraints and for linear programs having only general inequality constraints.
Given approximate data for the actual (unknown) instance, the algorithms use knowledge that the instance in question is primal feasible to reduce the data precision necessary to give an approximation to the solution set of the actual instance when the actual instance has an optimal solution. In some cases, problem instances that would otherwise require perfect precision to solve can now be solved with approximate data because of the knowledge of primal feasibility.
The algorithms are computationally efficient. Furthermore, the algorithms require not much more data accuracy than the minimum amount necessary to give an approximate solution of a desired accuracy when the actual instance has an optimal solution. This work aids in the development of a computational complexity theory that uses approximate data and knowledge.

On Modified Factorizations for Large-Scale Linearly Constrained Optimization

Nicholas Ian Mark Gould

SIAM J. Optim. 9, pp. 1041-1063 (23 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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We consider the algebraic issues concerning the solution of general, large-scale, linearly constrained nonlinear optimization problems. Particular attention is given to suitable methods for solving the linear systems that occur at each iteration of such methods. The main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the null-space of the constraints while neither adversely affecting the convergence of Newton's method nor incurring a significant computational overhead. Numerical evidence to support the theoretical developments is provided.

A Trust Region Method for Parabolic Boundary Control Problems

C. T. Kelley and E. W. Sachs

SIAM J. Optim. 9, pp. 1064-1081 (18 pages)

Online Publication Date: July 31, 2006

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In this paper we develop a trust region algorithm for constrained parabolic boundary control problems. For the computation of a trust region step we propose an iterative scheme which is a projected form of the Steihaug trust region conjugate gradient method. To ensure the good local convergence properties in the terminal phase, a smoothing step at each iteration is added. This step and the projection require the modification of the standard trust region algorithm and its convergence proof. The algorithm has sup-norm convergence in the terminal phase and L2 convergence in the global phase. The results are illustrated for a parabolic boundary control problem.

Pattern Search Algorithms for Bound Constrained Minimization

Robert Michael Lewis and Virginia Torczon

SIAM J. Optim. 9, pp. 1082-1099 (18 pages) | Cited 82 times

Online Publication Date: July 31, 2006

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We present a convergence theory for pattern search methods for solving bound constrained nonlinear programs. The analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints. This analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems. We prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.

Newton's Method for Large Bound-Constrained Optimization Problems

Chih-Jen Lin and Jorge J. Moré

SIAM J. Optim. 9, pp. 1100-1127 (28 pages) | Cited 33 times

Online Publication Date: July 31, 2006

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We analyze a trust region version of Newton's method for bound-constrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints. We also show that the convergence theory leads to an efficient implementation for large bound-constrained problems.

Polyhedral Boundary Projection

O. L. Mangasarian

SIAM J. Optim. 9, pp. 1128-1134 (7 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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We consider the problem of projecting a point in a polyhedral set onto the boundary of the set using an arbitrary norm for the projection. Two types of polyhedral sets, one defined by a convex combination of k points in Rn and the second by the intersection of m closed half-spaces in Rn, lead to disparate optimization problems for finding such a projection. The first case leads to a mathematical program with a linear objective function and constraints that are linear inequalities except for a single nonconvex cylindrical constraint. Interestingly, for the 1-norm, this nonconvex problem can be solved by solving 2n linear programs. The second polyhedral set leads to a much simpler problem of determining the minimum of m easily evaluated numbers. These disparate mathematical complexities parallel known ones for the related problem of finding the largest ball, with radius measured by an arbitrary norm, that can be inscribed in the polyhedral set. For a polyhedral set of the first type this problem is NP-hard for the 2-norm and the $\infty$-norm [R. M. Freund and J. B. Orlin, Math. Programming, 33 (1985), pp. 139--145] and solvable by a single linear program for the 1-norm [P. Gritzmann and V. Klee, Math. Programming, 59 (1993), pp. 163--213], while for the second type this problem leads to a single linear program even for a general norm [P. Gritzmann and V. Klee, Discrete Comput. Geom., 7 (1992), pp. 255--280].

A Revised Modified Cholesky Factorization Algorithm

Robert B. Schnabel and Elizabeth Eskow

SIAM J. Optim. 9, pp. 1135-1148 (14 pages) | Cited 11 times

Online Publication Date: July 31, 2006

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A modified Cholesky factorization algorithm introduced originally by Gill and Murray and refined by Gill, Murray, and Wright is used extensively in optimization algorithms. Since its introduction in 1990, a different modified Cholesky factorization of Schnabel and Eskow has also gained widespread usage. Compared with the Gill--Murray--Wright algorithm, the Schnabel--Eskow algorithm has a smaller a priori bound on the perturbation, added to ensure positive definiteness, and some computational advantages, especially for large problems. Users of the Schnabel--Eskow algorithm, however, have reported cases from two different contexts where it makes a far larger modification to the original matrix than is necessary and than is made by the Gill--Murray--Wright method. This paper reports on a simple modification to the Schnabel--Eskow algorithm that appears to correct all the known computational difficulties with the method, without harming its theoretical properties or its computational behavior in any other cases. In new computational tests, the modifications to the original matrix made by the new algorithm appear virtually always to be smaller than those made by the Gill--Murray--Wright algorithm, sometimes by significant amounts. The perturbed matrix is allowed to be more ill-conditioned with the new algorithm, but this seems to be appropriate in the known contexts where the underlying problem is ill-conditioned.

Parallel Adaptive GMRES Implementations for Homotopy Methods

Maria Sosonkina, Donald C. S. Allison, and Layne T. Watson

SIAM J. Optim. 9, pp. 1149-1158 (10 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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The success of probability-one homotopy methods in solving large-scale optimization problems and nonlinear systems of equations on parallel architectures may be significantly enhanced by the accurate parallel solution of large sparse nonsymmetric linear systems. Iterative solution techniques, such as GMRES(k), favor parallel implementations. However, their straightforward parallelization usually leads to a poor parallel performance because of global communication incurred by processors. One variation of GMRES(k) considered here is to adapt the restart value k for any given problem and use Householder reflections in the orthogonalization phase, coupled with graph-based matrix partitioning, to achieve high accuracy and reduce the communication overhead. This particular GMRES implementation is tailored to the uniquely stringent requirements imposed on a linear system solver by probability-one homotopy algorithms: occasionally unusually high accuracy, ability to adapt to problems of widely varying difficulty, and parallelism.

Modified Cholesky Factorizations in Interior-Point Algorithms for Linear Programming

Stephen J. Wright

SIAM J. Optim. 9, pp. 1159-1191 (33 pages) | Cited 9 times

Online Publication Date: July 31, 2006

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We investigate a modified Cholesky algorithm typical of those used in most interior-point codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky algorithms (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use alternative factorizations; and they perform robustly on most practical problems, yielding good interior-point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill conditioned. We investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the potential limitations of this approach.

The Quasi-Cauchy Relation and Diagonal Updating

M. Zhu, J. L. Nazareth, and H. Wolkowicz

SIAM J. Optim. 9, pp. 1192-1204 (13 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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The quasi-Cauchy (QC) relation is the weak quasi-Newton relation of Dennis and Wolkowicz [SIAM J. Numer. Anal., 30 (1993), pp. 1291--1314] with the added restriction that full matrices are replaced by diagonal matrices. This relation is justified and explored and, in particular, two basic variational techniques for updating diagonal matrices that satisfy it are formulated.
For purposes of illustration, a numerical experiment is described where a diagonal updated matrix with hereditary positive definiteness is used to precondition Cauchy's steepest-descent direction. The resulting QC algorithm is shown to be significantly accelerated.
In the concluding section, the following topics are briefly discussed: additional variational principles, use of diagonal updates within other optimization algorithms together with some further numerical experience (summarized in an appendix), and an interesting connection between QC-diagonal updating and trust-region techniques.
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