New Theoretical Results on Recursive Quadratic Programming Algorithms

Abstract

Recursive quadratic programming is a family of techniques developed by Bartholomew-Biggs and other authors for solving nonlinear programming problems. The first-order optimality conditions for a local minimizer of the augmented Lagrangian are transformed into a nonlinear system where both primal and dual variables appear explicitly. The inner iteration of the algorithm is a Newton-like procedure that updates simultaneously primal variables and Lagrange multipliers. In this way, as observed by Gould, the implementation of the Newton method becomes stable, in spite of the possibility of having large penalization parameters. In this paper, the inner iteration is analyzed from a different point of view. Namely, the size of the convergence region and the speed of convergence of the inner process are considered and it is shown that, in some sense, both are independent of the penalization parameter when an adequate version of the Newton method is used. In other words, classical Newton-like iterations are improved, not only in relation to stability of the linear algebra involved, but also with regard to the ovearll convergence of the nonlinear process. Some numerical experiments suggset that, in fact, practical efficiency of the methods is related to these theoretical results.