Inertia-controlling factorizations for optimization algorithms

Abstract

When solving smooth optimization problems by sequential quadratic programming methods or interior methods using exact second-derivative information, often a linear system of equations involving a large sparse symmetric indefinite matrix has to be solved at each iteration. This matrix is often referred to as the barrier KKT matrix or the KKT matrix, depending on if an interior approach is used or not. For simplicity, we refer to it as the KKT matrix below. The inertia of the KKT matrix reveals if the problem is locally convex at the current iterate. In the convex case, the KKT matrix always has the correct inertia, and in this situation an off-the shelf sparse factorization routine can be used, where the pivot selection is based on sparsity and numerical stability. In the nonconvex case, it is of interest to find out the inertia during the factorization process so that the matrix may be modified a posteriori. This can be done by imposing an additional restriction on the pivot selection, based on controlling the inertia of the submatrix factorized so far. We describe the issues of an inertia-controlling factorization and describe our experiments with a particular sparse implementation.