A stabilized SQP method: superlinear convergence

Stabilized sequential quadratic programming (sSQP) methods for nonlinear optimization generate a sequence of iterates with fast local convergence regardless of whether or not the active-constraint gradients are linearly dependent. This paper concerns the local convergence analysis of an sSQP method that uses a line search with a primal-dual augmented Lagrangian merit function to enforce global convergence. The method is provably well-defined and is based on solving a strictly convex quadratic programming subproblem at each iteration. It is shown that the method has superlinear local convergence under assumptions that are no stronger than those required by conventional stabilized SQP methods. The fast local convergence is obtained by allowing a small relaxation of the optimality conditions for the quadratic programming subproblem in the neighborhood of a solution. In the limit, the line search selects the unit step length, which implies that the method does not suffer from the Maratos effect. The analysis indicates that the method has the same strong first- and second-order global convergence properties that have been established for augmented Lagrangian methods, yet is able to transition seamlessly to sSQP with fast local convergence in the neighborhood of a solution. Numerical results on some degenerate problems are reported.

[1]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[2]  A. Forsgren.  Inertia-controlling factorizations for optimization algorithms , 2002 .

[3]  Alexey F. Izmailov,et al.  On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers , 2011, Math. Program..

[4]  Daniel P. Robinson.  Primal -dual methods for nonlinear optimization , 2007 .

[5]  Alexey F. Izmailov,et al.  Newton-Type Methods for Optimization Problems without Constraint Qualifications , 2004, SIAM J. Optim..

[6]  Alexey F. Izmailov,et al.  Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints , 2009, Comput. Optim. Appl..

[7]  A. F. Izmailov,et al.  On the influence of the critical lagrange multipliers on the convergence rate of the multiplier method , 2012 .

[8]  Francisco Facchinei,et al.  An LP-Newton method: nonsmooth equations, KKT systems, and nonisolated solutions , 2013, Mathematical Programming.

[9]  Stephen J. Wright.  Superlinear Convergence of a Stabilized SQP Method to a Degenerate Solution , 1998, Comput. Optim. Appl..

[10]  Alexey F. Izmailov,et al.  Stabilized SQP revisited , 2012, Math. Program..

[11]  Daniel P. Robinson,et al.  A primal-dual augmented Lagrangian , 2010, Computational Optimization and Applications.

[12]  Sven Leyffer,et al.  Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming , 2001, Comput. Optim. Appl..

[13]  William W. Hager,et al.  Stabilized Sequential Quadratic Programming , 1999, Comput. Optim. Appl..

[14]  Nicholas I. M. Gould,et al.  CUTEst: a Constrained and Unconstrained Testing Environment with safe threads for mathematical optimization , 2013, Computational Optimization and Applications.

[15]  A. F. Izmailov,et al.  Comments on: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it , 2015 .

[16]  A. F. Izmailov,et al.  Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it , 2015 .

[17]  Todd Munson,et al.  Benchmarking optimization software with COPS. , 2001 .

[18]  J. Pang,et al.  Computational optimization : a tribute to Olvi Mangasarian , 1999 .

[19]  A. F. Izmailov.  Solution sensitivity for Karush–Kuhn–Tucker systems with non-unique Lagrange multipliers , 2010 .

[20]  Stephen J. Wright.  Constraint identification and algorithm stabilization for degenerate nonlinear programs , 2000, Math. Program..

[21]  Anders Forsgren,et al.  Primal-Dual Interior Methods for Nonconvex Nonlinear Programming , 1998, SIAM J. Optim..

[22]  Andreas Fischer,et al.  Modified Wilson's Method for Nonlinear Programs with Nonunique Multipliers , 1999, Math. Oper. Res..

[23]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[24]  Francisco Facchinei,et al.  On the Accurate Identification of Active Constraints , 1998, SIAM J. Optim..

[25]  Frank E. Curtis,et al.  Flexible penalty functions for nonlinear constrained optimization , 2008 .

[26]  H. Kunzi,et al.  Lectu re Notes in Economics and Mathematical Systems , 1975 .

[27]  Daniel P. Robinson,et al.  A Globally Convergent Stabilized SQP Method , 2013, SIAM J. Optim..

[28]  Paulo J. S. Silva,et al.  Two New Weak Constraint Qualifications and Applications , 2012, SIAM J. Optim..

[29]  Daniel P. Robinson,et al.  A stabilized SQP method: global convergence , 2017 .

[30]  Jorge J. Moré,et al.  Benchmarking optimization software with COPS. , 2001 .

[31]  John E. Mitchell,et al.  An improved branch and bound algorithm for mixed integer nonlinear programs , 1994, Comput. Oper. Res..

[32]  J. M. Martínez,et al.  On second-order optimality conditions for nonlinear programming , 2007 .

[33]  Vyacheslav Kungurtsev,et al.  Second-Derivative Sequential Quadratic Programming Methods for Nonlinear Optimization , 2013 .

[34]  Francisco Facchinei,et al.  A family of Newton methods for nonsmooth constrained systems with nonisolated solutions , 2013, Math. Methods Oper. Res..

[35]  Lorenz T. Biegler,et al.  MPEC problem formulations and solution strategies with chemical engineering applications , 2008, Comput. Chem. Eng..

[36]  Stephen J. Wright.  An Algorithm for Degenerate Nonlinear Programming with Rapid Local Convergence , 2005, SIAM J. Optim..

[37]  Stephen J. Wright,et al.  Active Set Identification in Nonlinear Programming , 2006, SIAM J. Optim..

[38]  C. Floudas.  Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications , 1995 .

[39]  N. Maratos,et al.  Exact penalty function algorithms for finite dimensional and control optimization problems , 1978 .

[40]  Stephen J. Wright.  Modifying SQP for Degenerate Problems , 2002, SIAM J. Optim..

[41]  P. Gill,et al.  Sequential Quadratic Programming Methods , 2012 .

[42]  Stephen J. Wright,et al.  Numerical Behavior of a Stabilized SQP Method for Degenerate NLP Problems , 2002, COCOS.

[43]  Klaus Schittkowski,et al.  Test examples for nonlinear programming codes , 1980 .

[44]  Mikhail V. Solodov,et al.  Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems , 2010, Math. Program..