Ron S. Dembo

Truncated-newtono algorithms for large-scale unconstrained optimization

We present an algorithm for large-scale unconstrained optimization based onNewtons method. In large-scale optimization, solving the Newton equations at each iteration can be expensive and may not be justified when far from a solution. Instead, an inaccurate solution to the Newton equations is computed using a conjugate gradient method. The resulting algorithm is shown to have strong convergence properties and has the unusual feature that the asymptotic convergence rate is a user specified parameter which can be set to anything between linear and quadratic convergence. Some numerical results on a 916 vriable test problem are given. Finally, we contrast the computational behavior of our algorithm with Newtons method and that of a nonlinear conjugate gradient algorithm.

On Reporting Computational Experiments with Mathematical Software

Many papers appearing in journals reporting computational experiments use computer generated evidence to compare or rank competing mathematical software techmques. Unfortunately, to date there have been no standards or gmdehnes indicating how computer experiments should be conducted or how the results should be presented. An initial attempt is made to rectify this situation, and a summary of unportant points which should be considered when writing or evaluating a paper m which computational results are reported is provided.

Reporting computational experiments in mathematical programming

This paper presents a set of recommended standards for the presentation of computational experiments in mathematical programming.

Nonlinear programming on generalized networks

We describe a specialization of the primal truncated Newton algorithm for solving nonlinear optimization problems on networks with gains. The algorithm and its implementation are able to capitalize on the special structure of the constraints. Extensive computational tests show that the algorithm is capable of solving very large problems. Testing of numerous tactical issues are described, including maximal basis, projected line search, and pivot strategies. Comparisons with NLPNET, a nonlinear network code, and MINOS, a general-purpose nonlinear programming code, are also included.

OR Practice-Large-Scale Nonlinear Network Models and Their Application

Nonlinear network applications arise in a variety of contexts: hydroelectric power system scheduling, financial planning, matrix estimation, air traffic control, and so on. This paper reviews applications of network optimization models with nonlinear objectives and, possibly, generalized arcs. Particular emphasis is placed on large-scale implementations. Specialized nonlinear network programming software is typically an order of magnitude faster than general purpose codes.

A smooth penalty function algorithm for network-structured problems

Abstract We discuss the design and implementation of an algorithm for the solution of large scale optimization problems with embedded network structures. The algorithm uses a linear-quadratic penalty (LQP) function to eliminate the side constraints and produces a differentiable, but non-separable, problem. A simplicial decomposition is subsequently used to decompose the problem into a sequence of linear network problems. Numerical issues and implementation details are also discussed. The algorithm is particularly suitable for vector architectures and was implemented on a CRAY Y-MP. We report very promising numerical results with a set of large linear multicommodity network flow problems drawn from a military planning application.

Computing equilibria on large multicommodity networks: An application of truncated quadratic programming algorithms

We present a general scheme for improving the asymptotic behavior of a given nonlinear programming algorithm without incurring a significant increase in storage overhead. To enhance the rate of convergence we compute search directions by partially solving a sequence of quadratic programming (QP) problems as suggested by Dembo [6]. The idea is illustrated on a class of extremely large nonlinear programming problems arising from traffic equilibrium calculations using both the Frank-Wolfe and PARTAN algorithms to partially solve the QP subproblems. Computational results indicate that the convergence rate of the underlying algorithm is indeed enhanced significantly when Frank-Wolfe is used to solve the QP subproblems but only marginally so in the case of PARTAN. It is conjectured, and supported by the theory [11], that with better algorithms for the QP subproblems the improvements due to the proposed framework would be more marked.

Dealing with degeneracy in reduced gradient algorithms

Many algorithms for linearly constrained optimization problems proceed by solving a sequence of subproblems. In these subproblems, the number of variables is implicitly reduced by using the linear constraints to express certain ‘basic’ variables in terms of other variables. Difficulties may arise, however, if degeneracy is present; that is, if one or more basic variables are at lower or upper bounds. In this situation, arbitrarily small movements along a feasible search direction in the reduced problem may result in infeasibilities for basic variables in the original problem. For such cases, the search direction is typically discarded, a new reduced problem is formed and a new search direction is computed. Such a process may be extremely costly, particularly in large-scale optimization where degeneracy is likely and good search directions can be expensive to compute. This paper is concerned with a practical method for ensuring that directions that are computed in the reduced space are actually feasible in the original problem. It is based on a generalization of the ‘maximal basis’ result first introduced by Dembo and Klincewicz for large nonlinear network optimization problems.

Second order algorithms for the posynomial geometric programming dual, part I: Analysis

In this paper we analyse algorithms for the geometric dual of posynomial programming problems, that make explicit use of second order information. Out of two possible approaches to the problem, it is shown that one is almost always superior. Interestingly enough, it is the second, inferior approach that has dominated the geometric programming literature.

Truncated policy iteration methods

Policy iteration methods are important but often computationally expensive approaches for solving certain stochastic optimization problems. Modified policy iteration methods have been proposed to reduce the storage and computational burden. The asymptotic speed-of-convergence of such methods is, however, not well understood. In this paper we show how modified policy iteration methods may be constructed to achieve a preassigned rate-of-convergence. Our analysis provides a framework for analyzing the local behavior of such methods and provides impetus for perhaps more computationally efficient procedures than currently exist.