Modifying the inertia of matrices arising in optimization

Abstract

Applications in constrained optimization (and other areas) produce symmetric matrices with a natural block 2 × 2 structure. An optimality condition leads to the problem of perturbing the (1,1) block of the matrix to achieve a specific inertia. We derive a perturbation of minimal norm, for any unitarily invariant norm, that increases the number of nonnegative eigenvalues by a given amount, and we show how it can be computed efficiently given a factorization of the original matrix. We also consider an alternative way to satisfy the optimality condition based on a projection approach. Theoretical tools developed here include an extension of Ostrowski's theorem on congruences and some lemmas on inertias of block 2 × 2 symmetric matrices.