On the Identification of Local Minimizers in Inertia-Controlling Methods for Quadratic Programming

Abstract

Abstract. The verification of a local minimizer of a general (i.e., nonconvex) quadratic program is in general an NP-hard problem. The difficulty concerns the optimality of certain points (which we call dead points) at which the first-order necessary conditions for optimality are satisfied, but strict complementarity does not hold. One important class of methods for solving general quadratic programming problems are called inertia-controlling quadratic programming (ICQP) methods. We derive a computational scheme for proceeding at a dead point that is appropriate for a general ICQP method.

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

“…The ability of a solver to reveal and modify the inertia of K is referred to as inertia control, and hence a solver that has this capability is referred to as inertia-controlling solver. The inertia control has been mainly investigated in the context of the factorization of the KKT matrix [30,32,42,51]. To illustrate the basic idea of inertia-controlling factorizations, we consider a LBL T factorizations of K, where, due to the Sylvester law of inertia, the matrix B has the same inertia as K. The inertia of B is readily available, because of the particular block-diagonal structure of this matrix.…”

“…Inequality (32) gives the possibility of relating the accuracy of the approximate solution u of the KKT system to the quality of the current IP iterate, thus allowing to reduce the computational cost of the solution of the system, and hence the overall cost of the IP method [3,14,52]. The idea is to use adaptive inner iteration stopping criteria that require low accuracy when the outer IP iterate is far from the optimal solution and to require higher accuracy as soon as the IP iterate approaches the solution.…”

Barrier Methods for Large-Scale Quadratic Programming

“…At such points, all conventional quadratic programming methods will find it difficult to proceed, since it can be shown that the problem of distinguishing a dead point that is not a minimizer is an NP-hard problem (see Forsgren, Gill and Murray [FGM89b] for a precise definition of a dead point and a computational scheme within the context of inertia-controlling methods for QP that will attempt to determine if a dead point is a local minimizer). We emphasize that this difficulty is inherent in the problem, and is independent of the solution method.…”

An algorithm for indefinite quadratic programming based on a partial Cholesky factorization

“…In constrat, the strategy mentioned above does not allow to leave the Kuhn-Tucker point; see for instance Gill et al [8]. Nevertheless, there is a suggested procedure to deal with these situations given by Forsgren et al [5].…”