Han's method without solving QP

Abstract

Using an appropriate active set strategy Han's method for solving a general differentiable nonlinear programming problem can be modified such that only equality constrained quadratic subproblems (i.e. linear equations) have to be solved in order to find a descent direction of the Zangwill-Pietrzykowski penalty function of the problem. The present paper gives a complete description of a correponding algorithm along with proofs of convergence and rate of convergence results. A crucial assumption concerns the linear independence of the gradients of active constraints which makes return to the feasible set possible whenever desirable. Under this assumption every cluster point of the sequence generated is a Kuhn-Tucker point. If the sequence comes near to a Kuhn-Tucker point where second order sufficiency holds with strict complementary slackness, convergence is Q-superlinear in the primal and the dual variables, if one uses consistent approximations of the Hessian of the Lagrangian. If second order derivatives are used, convergence is second order of course.