Computational methods for large-scale quadratic programming

• Main Author: Morales-Perez, Jose Luis
• Published: Imperial College London 1993
• Subjects: 519.7
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.505913

For theoretical and practical reasons, quadratic programming problems have attracted the interest of the mathematical programming community. They naturally arise from applications and as subproblems in other numerical techniques. However most existing techniques, designed for solving small and dense problems, tend to be prohibitively expensive when applied directly to solve large-scale problems. In this work we explore methods suitable for solving large-scale sparse convex quadratic programming problems. An interior-point primal-dual algorithmic framework and its computational implementation are presented in the first part of this work. Primal and dual updates are computed at each step by iteratively solving the linear systems posed by the classical method of barriers using a preconditioned Krylov-subspace method. Several variants are suggested by a Taylor approximation of the central path. A truncated Newton strategy has been implemented in order to achieve a significant reduction in the CPU time. In the second part, sparse implementations for Lemke's algorithm and a row-action algorithm based on diagonal approximations of the Hessian, are suggested. Lemke's algorithm implementation is based on updating the sparse LU factorization of a matrix representing the basis at the current step. The implementation of the row-action algorithm relies on the efficient solution of single-constrained diagonal subproblems. In order to compare the relative merits of our implementations, numerical experimentation is conducted on two sets of problems that use randomly generated Hessian matrices and constraints taken from a subset of the netlib problems. Several aspects are studied: the use of iterative linear algebra for solving the linear systems of equations posed by the interior-point variants, the impact on the computational resources (memory and CPU) when different approaches are used to solve large scale problems, and finally, the effectiveness of a second order correction and the truncated Newton strategy implemented in the interior-point methods.