AUGMENTED PENALTY ALGORITHMS

We coin the term augmented penalty method to refer to an augmented Lag rangian-like method in which the penalty parameter is driven to zero i nstead of being kept bounded away from zero. The spirit of the algorit hm is thus that of a classical penalty in which an estimate of the Lag range multipliers is updated. For the classical augmented Lagrangian f unction (using the quadratic loss penalty term) applied to equality co nstrained optimization, Gould (1989 Siam J. Numer Anal. 26, 107) has o btained a two-step superlinear local convergence result using a consta nt Lagrange multiplier estimate. Dussault et al have generalized those results to inequality constrained optimization using other penalty te rms. On the other hand, convergence and rate of convergence results fo r augmented Lagrangian methods concern the convergence of the dual var iables, usually without relation to the actual effort required to obta in the approximate solutions of the unconstrained primal minimizations . In this paper we consider several updating rules for the Lagrange mu ltiplier estimates and we obtain rate of convergence results for both primal and dual variables, a two-step superlinear convergence of order alpha, with alpha < 2. In this result, each iteration uses the soluti on of a single primal-dual linear system. While this does not improve on the rate of convergence of simple penalty methods, it may alleviate some cancellation errors in internal computations.