Combining stabilized SQP with the augmented Lagrangian algorithm

For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advantage of the well-known robust behavior of the augmented Lagrangian algorithm, including on problems with degenerate constraints, and at the same time try to reduce the overall algorithm locally to sSQP (which gives fast local convergence rate under weak assumptions). Specifically, the algorithm first verifies whether the primal-dual sSQP step (with unit stepsize) makes good progress towards decreasing the violation of optimality conditions for the original problem, and if so, makes this step. Otherwise, the primal part of the sSQP direction is used for linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate fixed for the time being. The overall algorithm has reasonable global convergence guarantees, and inherits strong local convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alternatives are reported. Research of the first author is supported by the Russian Foundation for Basic Research Grant 14-01-00113 and by CNPq Grant PVE 401119/2014-9 (Brazil). The second author is supported in part by CNPq Grant 302637/2011-7, by PRONEX–Optimization, and by FAPERJ. The third author is supported by the Russian Foundation for Basic Research Grant 14-01-00113. A. F. Izmailov, E. I. Uskov IO Department, VMK Faculty, Lomonosov Moscow State University (MSU), Uchebniy Korpus 2, Leninskiye Gory, 119991 Moscow, Russia E-mail: [email protected] E. I. Uskov E-mail: [email protected] M. V. Solodov IMPA – Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil E-mail: [email protected] 2 Izmailov, Solodov, and Uskov