Presolving for Semide nite Programs Without Constraint Quali cations

Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identiies possible infeasibility and unboundedness. In semideenite programming, identically zero variables corresponds to lack of a constraint qualiication which can result in both theoretical and numerical diiculties. A nonzero duality gap can exist which nulliies the elegant and powerful duality theory. Small perturbations can result in infeasibility and/or large perturbations in solutions. Such problems fall into the class of ill-posed problems. It is interesting to note that classes of problems where constraint qualiications fail arise from semideenite programming relaxations of hard combinatorial problems. We look at several such classes and present two approaches to nd regularized solutions. Some preliminary numerical results are included.