Convergence of Trust Region Algorithms for Optimization with Bounds when Strict Complementarity Does Not Hold

Abstract

Conn, Gould, and Toint [SIAM J. Numer. Anal., 25 (1988), pp. 433–460; 26 (1989), p. 764] have proposed a class of trust region algorithms for minimizing nonlinear functions whose variables are subjected to simple bound constraints. In their convergence analysis, they show that if the strict complementarily condition holds, the considered algorithms reduce to an unconstrained calculation after finitely many iterations, allowing fast asymptotic rates of convergence. This paper analyses the behaviour of these iterative processes in the case where the strict complementarily condition is violated. It is proved that inexact Newton methods lead to superlinear or quadratic rates of convergence, even if the set of active bounds at the solution is not entirely detected. Practical criteria for stopping the inner iterations of the algorithms are deduced, ensuring these rates of convergence.