Stabilized SQP revisited

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91–124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernández and Solodov (Math Program 125:47–73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover, Research of the first author is supported by the Russian Foundation for Basic Research Grant 10-01-00251, and by RF President’s Grant NS-693.2008.1 for the support of leading scientific schools. The second author is supported in part by CNPq Grants 300513/2008-9 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]