This paper draws from our experience in developing the AMPL modeling language, to show where numerical issues have been crucial to modeling language design and where modeling language advances have strongly influenced the design of solvers. CitationProceedings of the 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, D.F. Griffiths and G.A. Watson, eds., University of … Read more
In this paper we consider the question of solving equilibrium problems—formulated as complementarity problems and, more generally, mathematical programs with equilibrium constraints (MPEC’s)—as nonlinear programs, using an interior-point approach. These problems pose theoretical difficulties for nonlinear solvers, including interior-point methods. We examine the use of penalty methods to get around these difficulties, present an example … Read more
The homogeneous model for linear programs is an elegant means of obtaining the solution or certificate of infeasibility and has importance regardless of the method used for solving the problem, interior-point methods or other methods. In 1999, Andersen and Ye generalized this model to monotone complementarity problems (CPs) and showed that most of the desirable … Read more
The relationship between the mathematical program with linear complementarity constraints (MPCC) and its inequality relaxation is studied. A new sequential quadratic programming (SQP) method is presented for solving the MPCC based on this relationship. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived … Read more
We discuss in this paper continuity and differentiability properties of solutions of parameterized variational inequalities (generalized equations). To this end we use an approach of formulating variational inequalities in a form of optimization problems and applying a general theory of perturbation analysis of parameterized optimization problems. CitationSchool of Industrial and Systems Engineering, Georgia Institute of … Read more
Interior point methods for nonlinear programs (NLPs) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primal-dual algorithm has been adapted with a modified step calculation. The algorithm is shown to … Read more
SIAM’s SIAG/Opt Newsletter special issue on Large Scale Nonconvex Optimization. Guest editors Sven Leyffer and Jorge Nocedal, with contributions by Gould, Sachs, Biegler, Waechter, Leyffer, Bussieck and Pruessner. CitationSIAG/Opt Views-and-News, Volume 14 Number 1, April 2003. http://fewcal.uvt.nl/sturm/siagopt/ArticleDownload View PDF
We present a new formulation of the truss topology problem that results in unique design and unique displacements of the optimal truss. This is reached by adding an upper level to the original optimization problem and formulating the new problem as an MPCC (Mathematical Program with Complementarity Constraints). We derive optimality conditions for this problem … Read more
We consider a class of optimization problems with a generalized equation among the constraints. This class covers several problem types like MPEC (Mathematical Programs with Equilibrium Constraints) and MPCC (Mathematical Programs with Complementarity Constraints). We briefly review techniques used for numerical solution of these problems: penalty methods, nonlinear programming (NLP) techniques and Implicit Programming approach … Read more
A class of matrix valued functions defined by singular values of nonsymmetric matrices is shown to have many properties analogous to matrix valued functions defined by eigenvalues of symmetric matrices. In particular, the (smoothed) matrix valued Fischer-Burmeister function is proved to be strongly semismooth everywhere. This result is also used to show the strong semismoothness … Read more