Differential variational inequalities

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions. The work of J.-S. Pang is supported by the National Science Foundation under grants CCR-0098013 CCR-0353074, and DMS-0508986, by a Focused Research Group Grant DMS-0139715 to the Johns Hopkins University and DMS-0353016 to Rensselaer Polytechnic Institute, and by the Office of Naval Research under grant N00014-02-1-0286. The work of D. E. Stewart is supported by the National Science Foundation under a Focused Research Group grant DMS-0138708.     

Finite convergence of algorithms for nonlinear programs and variational inequalities

Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.The research of the first author was supported in part by the Office of Naval Research under Contract No. N00014-86-K-0173.The authors are indebted to Professors Olvi Mangasarian, Garth McCormick, Jong-Shi Pang, Hanif Sherali, and Hoang Tuy for helpful comments and suggestions and to two anonymous referees for constructive remarks and for bringing to their attention the results in Refs. 13 and 14.     

A pathsearch damped Newton method for computing general equilibria

Computable general equilibrium models and other types of variational inequalities play a key role in computational economics. This paper describes the design and implementation of a pathsearch damped Newton method for solving such problems. Our algorithm improves on the typical Newton method (which generates and solves a sequence of LCPs) in both speed and robustness. The underlying complementarity problem is reformulated as a normal map so that standard algorithmic enhancements of Newton's method for solving nonlinear equations can be easily applied. The solver is implemented as a GAMS subsystem, using an interface library developed for this purpose. Computational results obtained from a number of test problems arising in economics are given.This material is based on research supported by the National Science Foundation Grant CCR-9157632 and the Air Force Office of Scientific Research Grant F49620-94-1-0036.     

Derivation of linear beam equations using nonlinear continuum mechanics

Familiar linear elastic and viscoelastic beam equations (Euler-Bernoulli, Rayleigh, Kelvin-Voigt, Timoshenko, and Shear Diffusion) and boundary conditions are derived from a nonlinear theory of large motions rather than the usual variational techniques. Also included is a fairly detailed derivation of the nonlinear theory and a careful discussion of the hypotheses.This work has been partially supported by the Office of Naval Research under grant number N00014-88-K0417 and by the National Science Foundation under grant number DMS-8801412.     

A B-differentiable equation-based,globally and locally quadratically convergent algorithm for nonlinear programs,complementarity and variational inequality problems

This paper presents a globally convergent, locally quadratically convergent algorithm for solving general nonlinear programs, nonlinear complementarity and variational inequality problems. The algorithm is based on a unified formulation of these three mathematical programming problems as a certain system of B-differentiable equations, and is a modification of the damped Newton method described in Pang (1990) for solving such systems of nonsmooth equations. The algorithm resembles several existing methods for solving these classes of mathematical programs, but has some special features of its own; in particular, it possesses the combined advantage of fast quadratic rate of convergence of a basic Newton method and the desirable global convergence induced by one-dimensional Armijo line searches. In the context of a nonlinear program, the algorithm is of the sequential quadratic programming type with two distinct characteristics: (i) it makes no use of a penalty function; and (ii) it circumvents the Maratos effect. In the context of the variational inequality/complementarity problem, the algorithm provides a Newton-type descent method that is guaranteed globally convergent without requiring the F-differentiability assumption of the defining B-differentiable equations.This work was based on research supported by the National Science Foundation under Grant No. ECS-8717968.     

Solving geometric programs using GRG: Results and comparisons

This paper describes the performance of a general-purpose GRG code for nonlinear programming in solving geometric programs. The main conclusions drawn from the experiments reported are: (i) GRG competes well with special-purpose geometric programming codes in solving geometric programs; and (ii) standard time, as defined by Colville, is an inadequate means of compensating for different computing environments while comparing optimization algorithms.This research was partially supported by the Office of Naval Research under Contracts Nos. N00014-75-C-0267 and N00014-75-C-0865, the US Energy Research and Development Administration, Contract No. E(04-3)-326 PA-18, and the National Science Foundation, Grant No. DCR75-04544 at Stanford University; and by the Office of Naval Research under Contract No. N00014-75-C-0240, and the National Science Foundation, Grant No. SOC74-23808, at Case Western Reserve University.     

Trust region affine scaling algorithms for linearly constrained convex and concave programs

We study a trust region affine scaling algorithm for solving the linearly constrained convex or concave programming problem. Under primal nondegeneracy assumption, we prove that every accumulation point of the sequence generated by the algorithm satisfies the first order necessary condition for optimality of the problem. For a special class of convex or concave functions satisfying a certain invariance condition on their Hessians, it is shown that the sequences of iterates and objective function values generated by the algorithm convergeR-linearly andQ-linearly, respectively. Moreover, under primal nondegeneracy and for this class of objective functions, it is shown that the limit point of the sequence of iterates satisfies the first and second order necessary conditions for optimality of the problem. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.The work of these authors was based on research supported by the National Science Foundation under grant INT-9600343 and the Office of Naval Research under grants N00014-93-1-0234 and N00014-94-1-0340.     

A note on the existence of the Alizadeh-Haeberly-Overton direction for semidefinite programming

This note establishes a new sufficient condition for the existence and uniqueness of the Alizadeh-Haeberly-Overton direction for semidefinite programming. The work of these authors was based on research supported by the National Science Foundation under grants INT-9600343 and CCR-970048 and the Office of Naval Research under grant N00014-94-1-0340.     

Superlinear primal-dual affine scaling algorithms for LCP

We describe an interior-point algorithm for monotone linear complementarity problems in which primal-dual affine scaling is used to generate the search directions. The algorithm is shown to have global and superlinear convergence with Q-order up to (but not including) two. The technique is shown to be consistent with a potential-reduction algorithm, yielding the first potential-reduction algorithm that is both globally and superlinearly convergent.Corresponding author. The work of this author was based on research supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.The work of this author was based on research supported by the National Science Foundation under grant DDM-9109404 and the Office of Naval Research under grant N00014-93-1-0234. This work was done while the author was a faculty member of the Systems and Industrial Engineering Department at the University of Arizona.     

Local convergence of interior-point algorithms for degenerate monotone LCP

Most asymptotic convergence analysis of interior-point algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is removed.The work of this author was based on research supported by the National Science Foundation under grant DDM-9109404 and the Office of Naval Research under grant N00014-93-1-0234.The work of this author was based on research supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Numerical stability for solving nonlinear equations

Summary The concepts of the condition number, numerical stability and well-behavior for solving systems of nonlinear equationsF(x)=0 are introduced. Necessary and sufficient conditions for numerical stability and well-behavior of a stationary are given. We prove numerical stability and well-behavior of the Newton iteration for solving systems of equations and of some variants of secant iteration for solving a single equation under a natural assumption on the computed evaluation ofF. Furthermore we show that the Steffensen iteration is unstable and show how to modify it to have well-behavior and hence stability.This work was supported in part by the Office of Naval Research under Contract N 00014-67-0314-0010 NR 044-422 and by the National Science Foundation under Grant GJ 32111     

The dimension of cycle-free orders

The existence of a four-dimensional cycle-free order is proved. This answers a question of Ma and Spinrad. Two similar problems are also discussed.Research partially supported by Office of Naval Research grant N00014-90-J-1206Research partially supported by the National Science Foundation under grant DMS     

The principal pivoting method revisited

The principal pivoting method (PPM) for the linear complementarity problem (LCP) is shown to be applicable to the class of LCPs involving the newly identified class of sufficient matrices.Research partially supported by the National Science Foundation grant DMS-8800589, U.S. Department of Energy grant DE-FG03-87ER25028 and Office of Naval Research grant N00014-89-J-1659.Dedicated to George B. Dantzig on the occasion of his 75th birthday.     

Computing projections with LSQR

LSQR uses the Golub-Kahan bidiagonalization process to solve sparse least-squares problems with and without regularization. In some cases, projections of the right-hand side vector are required, rather than the least-squares solution itself. We show that projections may be obtained from the bidiagonalization as linear combinations of (the-oretically) orthogonal vectors. Even the least-squares solution may be obtained from orthogonal vectors, perhaps more accurately than the usual LSQR solution. (However, LSQR has proved equally good in all examples so far.) Presented at the Cornelius Lanczos International Centenary Conference, North Carolina State University, Raleigh, NC, December 1993. Partially supported by Department of Energy grant DE-FG03-92ER25117, National Science Foundation grants DMI-9204208 and DMI-9500668, and Office of Naval Research grants N00014-90-J-1242 and N00014-96-1-0274.     

Large-scale linearly constrained optimization

An algorithm for solving large-scale nonlinear programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.This research was supported by the U.S. Office of Naval Research (Contract N00014-75-C-0267), the National Science Foundation (Grants MCS71-03341 A04, DCR75-04544), the U.S. Energy Research and Development Administration (Contract E(04-3)-326 PA #18), the Victoria University of Wellington, New Zealand, and the Department of Scientific and Industrial Research Wellington, New Zealand.     

The customer response times in the processor sharing queue are associated

The customer response times in the egalitarian processor sharing queue are shown to be associated random variables under renewal inputs and general independent service times assumptions.The work by this author was supported in part by the National Science Foundation under grant ASC 88-8802764 and by the Office of Naval Research under grant ONR N00014-87-K-0796.     

Optimization with unary functions

Most nonlinear programming problems consist of functions which are sums of unary functions of linear functions. Advantage can be taken of this form to calculate second and higher order derivatives easily and at little cost. Using these, high order optimization techniques such as Halley's method can be utilized to accelerate the rate of convergence to the solution. These higher order derivatives can also be used to compute second order sensitivity information. These techniques are applied to the solution of the classical chemical equilibrium problem.Supported by National Science Foundation grant ECS-8709795, co-funded by the U.S. Air Force Office of Scientific Research and by the Office of Naval Research grant N00014-86-K0052.Supported by National Science Foundation grant ECS-8709795, co-funded by the U.S. Air Force Office of Scientific Research.     

On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method

Interest in linear programming has been intensified recently by Karmarkar’s publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrier-function methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a “projected Newton barrier” method. This method is shown to be equivalent to Karmarkar’s projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several non-trivial test problems, and the implications for future developments in linear programming are discussed. The research of the Stanford authors was supported by the U.S. Department of Energy Contract DE-AA03-76SF00326, PA No. DE-AS03-76ER72018; National Science Foundation Grants DCR-8413211 and ECS-8312142; the Office of Naval Research Contract N00014-85-K-0343; and the U.S. Army Research Office Contract DAAG29-84-K-0156. The research of J.A. Tomlin was supported by Ketron, Inc. and the Office of Naval Research Contract N00014-85-C-0338.     

On the solution of large,structured linear complementarity problems: The block partitioned case

This paper delineates the underlying theory of an efficient method for solving a class of specially-structured linear complementarity problems of potentially very large size. We propose a block-iterative method in which the subproblems are linear complementarity problems. Problems of the type considered here arise in the process of making discrete approximations to differential equations in the presence of special side conditions. This problem source is exemplified by the free boundary problem for finite-length journal bearings. Some of the authors' computational experience with the method is presented.Research partially supported by the Office of Naval Research under contract N-00014-67-A-0112-0011; U.S. Atomic Energy Commission Contract AT (04-3)-326 PA No. 18.Research partially supported by U.S. Atomic Energy Commission Contract AT(04-3)-326 PA No. 30; and National Science Foundation Grant GJ 35135X.     

On the convergence of a basic iterative method for the implicit complementarity problem

In Part 1 of this study (Ref. 1), we have defined the implicit complementarity problem and investigated its existence and uniqueness of solution. In the present paper, we establish a convergence theory for a certain iterative algorithm to solve the implicit complementarity problem. We also demonstrate how the algorithm includes as special cases many existing iterative methods for solving a linear complementarity problem.This research was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract No. N00014-75-C-0621-NR-047-048 with the Office of Naval Research.