Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption
We present a local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$Q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>Q</mi> </math> </EquationSource> </InlineEquation>-superlinear convergence is established under the appropriate control of the penalty parameter values. For optimization problems, we demonstrate in addition local <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$Q$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>Q</mi> </math> </EquationSource> </InlineEquation>-linear convergence for sufficiently large fixed penalty parameters. Both exact and inexact versions of the method are considered. Contributions with respect to previous state-of-the-art analyses for equality-constrained problems consist in the extension to the variational setting, in using the weaker noncriticality assumption instead of the usual second-order sufficient optimality condition (SOSC), and in relaxing the smoothness requirements on the problem data. In the context of optimization problems, this gives the first local convergence results for the augmented Lagrangian method under the assumptions that do not include any constraint qualifications and are weaker than the SOSC. We also show that the analysis under the noncriticality assumption cannot be extended to the case with inequality constraints, unless the strict complementarity condition is added (this, however, still gives a new result). Copyright Springer Science+Business Media New York 2015
