Probability-one homotopies in computational science

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that, under mild assumptions, are globally convergent for a wide range of problems in science and engineering. Convergence theory, robust numerical algorithms, and production quality mathematical software exist for general nonlinear systems of equations, and special cases such as Brouwer fixed point problems, polynomial systems, and nonlinear constrained optimization. Using a sample of challenging scientific problems as motivation, some pertinent homotopy theory and algorithms are presented. The problems considered are analog circuit simulation (for nonlinear systems), reconfigurable space trusses (for polynomial systems), and fuel-optimal orbital rendezvous (for nonlinear constrained optimization). The mathematical software packages HOMPACK90 and POLSYS_PLP are also briefly described.     

Scalar labelings for homotopy paths

Two scalar labelings are introduced for obtaining approximate solutions to systems of nonlinear equations by simplicial approximation. Under reasonable assumptions the new scalar-labeling algorithms are shown to follow, in a limiting sense, homotopy paths which can also be tracked by piecewise linear vector labeling algorithms. Though the new algorithms eliminate the need to pivot on a system of linear equations, the question of relative computational efficiency is unresolved.The work of this author was supported in part by NSF Grant No. MCS-77-15509 and by ARO Grant No. DAAG-29-78-G-0160.The work of this author was supported in part by ONR Grant No. N00014-75-C-0495 and NSF Grant No. 81058.     

Simple bounds for solutions of monotone complementarity problems and convex programs

For a solvable monotone complementarity problem we show that each feasible point which is not a solution of the problem provides simple numerical bounds for some or all components of all solution vectors. Consequently for a solvable differentiable convex program each primal-dual feasible point which is not optimal provides simple bounds for some or all components of all primal-dual solution vectors. We also give an existence result and simple bounds for solutions of monotone compementarity problems satisfying a new, distributed constraint qualification. This result carries over to a simple existence and boundedness result for differentiable convex programs satisfying a similar constraint qualification.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grants MCS-8200632 and MCS-8102684.     

Locally unique solutions of quadratic programs,linear and nonlinear complementarity problems

It is shown that McCormick's second order sufficient optimality conditions are also necessary for a solution to a quadratic program to be locally unique and hence these conditions completely characterize a locally unique solution of any quadratic program. This result is then used to give characterizations of a locally unique solution to the linear complementarity problem. Sufficient conditions are also given for local uniqueness of solutions of the nonlinear complementarity problem.Research supported by National Science Foundation Grant MCS74-20584 A02.     

QPCOMP: A quadratic programming based solver for mixed complementarity problems

QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel [14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudo-monotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm. This material is based on research supported by National Science Foundation Grant CCR-9157632, Department of Energy Grant DE-FG03-94ER61915, and the Air Force Office of Scientific Research Grant F49620-94-1-0036.     

A smoothing inexact Newton method for nonlinear complementarity problems

In this article, we propose a new smoothing inexact Newton algorithm for solving nonlinear complementarity problems (NCP) base on the smoothed Fischer-Burmeister function. In each iteration, the corresponding linear system is solved only approximately. The global convergence and local superlinear convergence are established without strict complementarity assumption at the NCP solution. Preliminary numerical results indicate that the method is effective for large-scale NCP.     

Interior-point methods for nonlinear complementarity problems

We present a potential reduction interior-point algorithm for monotone nonlinear complementarity problems. At each iteration, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. For problems satisfying a scaled Lipschitz condition, this requirement is satisfied by the approximate solution obtained by applying one Newton step to that nonlinear system. We discuss the global and local convergence rates of the algorithm, convergence toward a maximal complementarity solution, a criterion for switching from the interior-point algorithm to a pure Newton method, and the complexity of the resulting hybrid algorithm.This research was supported in part by NSF Grant DDM-89-22636.The authors would like to thank Rongqin Sheng and three anonymous referees for their comments leading to a better presentation of the results.     

Solution of monotone complementarity problems with locally Lipschitzian functions

The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsφ(x)=0 or as the problem of minimizing the merit functionΘ=1/2∥Φ∥ ₂ ² , we extend results which hold for sufficiently smooth functionsF to the nonsmooth case. In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesΘ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.     

Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems

In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented. Supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

A universal constant for the convergence of Newton's method and an application to the classical homotopy method

We give a new theorem concerning the convergence of Newton's method to compute an approximate zero of a system of equations. In this result, the constanth ₀=0.162434... appears, which plays a fundamental role in the localization of good initial points for the Newton iteration. We apply it to the determination of an appropriate discretization of the time interval in the classical homotopy method.     

A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems

A new algorithm for the solation of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10 000 variables are presented. Partially supported by Agenzia Spaziale Italiana, Roma, Italy.     

An algorithm for a class of nonlinear complementarity problems with non-Lipschitzian functions

In this paper, we focus on solving a class of nonlinear complementarity problems with non-Lipschitzian functions. We first introduce a generalized class of smoothing functions for the plus function. By combining it with Robinson's normal equation, we reformulate the complementarity problem as a family of parameterized smoothing equations. Then, a smoothing Newton method combined with a new nonmonotone line search scheme is employed to compute a solution of the smoothing equations. The global and local superlinear convergence of the proposed method is proved under mild assumptions. Preliminary numerical results obtained applying the proposed approach to nonlinear complementarity problems arising in free boundary problems are reported. They show that the smoothing function and the nonmonotone line search scheme proposed in this paper are effective.     

Cycling in linear complementarity problems

A bound for the minimum length of a cycle in Lemke's Algorithm is derived. An example illustrates that this bound is sharp, and that the fewest number of variables is seven.     

Iterative methods for variational and complementarity problems

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.This research was based on work supported by the National Science Foundation under grant ECS-7926320.     

Nonlinear relaxation methods for solving symmetric linear complementarity problems

A family of iterative algorithms is presented for the solution of the symmetric linear complementarity problem,

Newton's method for linear complementarity problems

This paper presents an iterative, Newton-type method for solving a class of linear complementarity problems. This class was discovered by Mangasarian who had established that these problems can be solved as linear programs. Cottle and Pang characterized solutions of the problems in terms of least elements of certain polyhedral sets. The algorithms developed in this paper are shown to converge to the least element solutions. Some applications and computational results are also discussed.     

Computing integral solutions of complementarity problems

In this paper an algorithm is proposed to find an integral solution of (nonlinear) complementarity problems. The algorithm starts with a nonnegative integral point and generates a unique sequence of adjacent integral simplices of varying dimension. Conditions are stated under which the algorithm terminates with a simplex, one of whose vertices is an integral solution of the complementarity problem under consideration.     

Consistent aggregation of linear complementarity problems

Aggregating linear complementarity problems under a general definition of constrained consistency leads to the possibility of consistent aggregation of linear and quadratic programming models and bimatrix games. Under this formulation, consistent aggregation of dual variables is also achieved. Furthermore, the existence of multiple sets of aggregation operators is discussed and illustrated with a numerical example. Constrained consistency can also be interpreted as a disaggregation rule. This aspect of the problem may be important for implementing macro (economic) policies by means of micro (economic) agents.Giannini Foundation Paper No. 548.     

An LP-based successive overrelaxation method for linear complementarity problems

A sparsity preserving LP-based SOR method for solving classes of linear complementarity problems including the case where the given matrix is positive-semidefinite is proposed. The LP subproblems need be solved only approximately by a SOR method. Heuristic enhancement is discussed. Numerical results for a special class of problems are presented, which show that the heuristic enhancement is very effective and the resulting program can solve problems of more than 100 variables in a few seconds even on a personal computer.This research was sponsored by the Air Force Office of Scientific Research, Grant No. AFOSR-86-0124. Part of this material is based on work supported by the National Science Foundation under Grant No. MCS-82-00632.The author is grateful to Dr. R. De Leone for his helpful and constructive comments on this paper.     

A class of nonlinear complementarity problems for multifunctions

Given a continuous mapF:R ⁿ R ⁿ and a lower semicontinuous positively homogeneous convex functionh:R ⁿ R, the nonlinear complementarity problem considered here is to findxR ₊ ⁿ andyh(x), the subdifferential ofh atx, such thatF(x)+y0 andx ^T (F(x)+y)=0. Some existence theorems for the above problem are given under certain conditions on the mapF. An application to quasidifferentiable convex programming is also shown.The authors are grateful to Professor O. L. Mangasarian and the referee for their substantive suggestions.