Numerical experiments with modern methods for large scale QP-problems

Abstract

We describe the outcome of numerical experiments using three approaches for solving convex QP-problems in standard form 1 2 x T Bx + b T x ! = min ; A T x \Gamma a = 0 ; (0.1) x 0 ; namely the unconstrained technique of Kanzow [14], the bound constrained technique of Friedlander, Mart'inez and Santos [8] and the author's bound constrained quadratic extended Lagrangian [23]. These three methods solve (0.1) by a single unconstrained respectively bound constrained minimization. For our test purposes a test generator has been written which generates problems of this type with free choice of the condition number of the reduced Hessian, condition number of matrix of gradients of binding constraints and number of binding constraints. The exact solution is randomly generated. As a minimizer the new limited-memory BFGS-method (for bound constrained problems) of Byrd, Lu, Nocedal and Zhu [2] has been chosen. This allows using exactly the same minimization technology with exactly the ..