On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers

It has been previously demonstrated that in the case when a Lagrange multiplier associated to a given solution is not unique, Newton iterations [e.g., those of sequential quadratic programming (SQP)] have a tendency to converge to special multipliers, called critical multipliers (when such critical multipliers exist). This fact is of importance because critical multipliers violate the second-order sufficient optimality conditions, and this was shown to be the reason for slow convergence typically observed for problems with degenerate constraints (convergence to noncritical multipliers results in superlinear rate despite degeneracy). Some theoretical and numerical validation of this phenomenon can be found in Izmailov and Solodov (Comput Optim Appl 42:231–264, 2009;Math Program 117:271–304, 2009). However, previous studies concerned the basic forms of Newton iterations. The question remained whether the attraction phenomenon still persists for relevant modifications, as well as in professional implementations. In this paper, we answer this question in the affirmative by presenting numerical results for the well known MINOS and SNOPT software The research of A. F. Izmailov is supported by the Russian Foundation for Basic Research Grants 07-01-00270, 07-01-00416 and 09-01-90001-Bel, and by RF President’s Grant NS-693.2008.1 for the support of leading scientific schools. M. V. Solodov is supported in part by CNPq Grants 301508/2005-4 and 471267/2007-4, by PRONEX–Optimization, and by FAPERJ. A. F. Izmailov Department of Operations Research, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskiye Gory, GSP-2, 119992 Moscow, Russia e-mail: [email protected] M. V. Solodov (B) IMPA, Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil e-mail: [email protected]