INERTIA-REVEALING PRECONDITIONING FOR LARGE-SCALE NONCONVEX CONSTRAINED OPTIMIZATION

Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is guaranteed to be a descent direction only if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints; otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general they do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete LBLT factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale three-dimensional (3D) PDE-constrained optimal control problems, and on 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with 150(3) state variables and 150(2) control variables, both subject to bound constraints.