Stephen C. Billups

A Comparison of Large Scale Mixed Complementarity Problem Solvers

This paper provides a means for comparing various computercodes for solving large scale mixed complementarity problems. Wediscuss inadequacies in how solvers are currently compared, andpresent a testing environment that addresses these inadequacies. Thistesting environment consists of a library of test problems, along withGAMS and MATLAB interfaces that allow these problems to be easilyaccessed. The environment is intended for use as a tool byother researchers to better understand both their algorithms and theirimplementations, and to direct research toward problem classes thatare currently the most challenging. As an initial benchmark, eightdifferent algorithm implementations for large scale mixedcomplementarity problems are briefly described and tested with defaultparameter settings using the new testing environment.

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.

A Homotopy Based Algorithm for Mixed Complementarity Problems

This paper develops an algorithm for solving mixed complementarity problems that is based upon probability-one homotopy methods. After the complementarity problem is reformulated as a system of nonsmooth equations, a homotopy method is used to solve a sequence of smooth approximations to this system of equations. The global convergence properties of this approach are qualitatively different from those of other recent methods, which rely upon decrease of a merit function. This enables the algorithm to reliably solve certain classes of problems that prove troublesome for other methods. To improve efficiency, the homotopy algorithm is embedded in a generalized Newton method.

Derivative-Free Optimization of Expensive Functions with Computational Error Using Weighted Regression

We propose a derivative-free algorithm for optimizing computationally expensive functions with computational error. The algorithm is based on the trust region regression method by Conn, Scheinberg, and Vicente [A. R. Conn, K. Scheinberg, and L. N. Vicente, IMA J. Numer. Anal., 28 (2008), pp. 721--748] but uses weighted regression to obtain more accurate model functions at each trust region iteration. A heuristic weighting scheme is proposed that simultaneously handles (i) differing levels of uncertainty in function evaluations and (ii) errors induced by poor model fidelity. We also extend the theory of

Solving the canonical dual of box-and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm

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A Probability-One Homotopy Algorithm for Nonsmooth Equations and Mixed Complementarity Problems

-poisedness and strong

Stochastic derivative-free optimization using a trust region framework

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Identifying significant temporal variation in time course microarray data without replicates

-poisedness to weighted regression. We report computational results comparing interpolation, regression, and weighted regression methods on a collection of benchmark problems. Weighted regression appears to outperform interpolation and regression models on nondifferentiable functions and functions with deterministic noise.

Probability-one homotopy maps for mixed complementarity problems

This paper presents a massively parallel global deterministic direct search method (VTDIRECT) for solving nonconvex quadratic minimization problems with either box or±1 integer constraints. Using the canonical dual transformation, these well-known NP-hard problems can be reformulated as perfect dual stationary problems (with zero duality gap). Under certain conditions, these dual problems are equivalent to smooth concave maximization over a convex feasible space. Based on a perturbation method proposed by Gao, the integer programming problem is shown to be equivalent to a continuous unconstrained Lipschitzian global optimization problem. The parallel algorithm VTDIRECT is then applied to solve these dual problems to obtain global minimizers. Parallel performance results for several nonconvex quadratic integer programming problems are reported.

Convergence of an Infeasible Interior-Point Algorithm from Arbitrary Positive Starting Points

Convergence theory for a new probability-one homotopy algorithm for solving non-smooth equations is given. This algorithm is able to solve problems involving highly nonlinear equations, where the norm of the residual has nonglobal local minima. The algorithm is based on constructing homotopy mappings that are smooth in the interior of their domains. The algorithm is specialized to solve mixed complementarity problems (MCP) through the use of MCP functions and associated smoothers. This specialized algorithm includes an option to ensure that all iterates remain feasible. Easily satisfiable sufficient conditions are given to ensure that the homotopy zero curve remains feasible, and global convergence properties for the MCP algorithm are proved. Computational results on the MCPLIB test library demonstrate the effectiveness of the algorithm.