UC San Diego

Computational methods are considered for finding a point satisfying the second-order necessary conditions for a general (possibly nonconvex) quadratic program (QP). A framework for the formulation and analysis of feasible- point active-set methods is proposed for a generic QP. This framework is defined by reformulating and extending an inertia-controlling method for general QP that was first proposed by Fletcher and subsequently modified by Gould. This reformulation defines a class of methods in which a primal-dual search pair is the solution of a "KKT system'' of equations associated with an equality- constrained QP subproblem defined in terms of a "working set'' of linearly independent constraints. It is shown that, under certain circumstances, the solution of this KKT system may be updated using a simple recurrence relation, thereby giving a significant reduction in the number of systems that need to be solved. The use of inertia control guarantees that the KKT systems remain nonsingular throughout, thereby allowing the utilization of third-party linear algebra software. The algorithm is suitable for indefinite problems, making it an ideal QP solver for stand-alone applications and for use within a sequential quadratic programming method using exact second derivatives. The proposed framework is applied to primal and dual quadratic problems, as well as to single-phase problems that combine the feasibility and optimality phases of the active-set method, producing a range of formats that are suitable for a variety of applications. The algorithm is implemented in the Fortran code icQP. Its performance is evaluated using different symmetric and unsymmetric linear solvers on a set of convex and nonconvex problems. Results are presented that compare the performance of icQP with the convex QP solver SQOPT on a large set of convex problems