Hestenes Method for Symmetric Indefinite Systems in Interior-Point Method

Abstract

This paper deals with the analysis and the solution of the Karush- Kuhn-Tucker (KKT) system that arises at each iteration of an Interior-Point (IP)method for minimizing a nonlinear function subject to equality and inequality constraints. This system is generally large and sparse and it can be reduced so that the coefficient matrix is still sparse, symmetric and indefinite, with size equal to the number of the primal variables and of the equality constraints. Instead of transforming this reduced system to a quasidefinite form by regularization techniques used in available codes on IP methods, under standard assumptions on the nonlinear problem, the system can be viewed as the optimality Lagrange conditions for a linear equality constrained quadratic programming problem, so that Hestenes multipliers’ method can be applied. Numerical experiments on elliptic control problems with boundary and distributed control show the effectiveness of Hestenes scheme as inner solver for IP methods