An augmented Lagrangian method exploiting an active-set strategy and second-order information

In this paper, we consider nonlinear optimization problems with nonlinear equality constraints and bound constraints on the variables. For the solution of such problems, many augmented Lagrangian methods have been defined in the literature. Here, we propose to modify one of these algorithms, namely ALGENCAN [2, 3], in such a way to incorporate second-order information into the augmented Lagrangian framework, using an active-set strategy. We show that the overall algorithm has the same convergence properties as ALGENCAN and an asymptotic quadratic convergence rate under suitable assumptions. The numerical results confirm that the proposed algorithm is a viable alternative to ALGENCAN with greater robustness.

[1]  José Mario Martínez,et al.  On Augmented Lagrangian Methods with General Lower-Level Constraints , 2007, SIAM J. Optim..

[2]  Gianni Di Pillo,et al.  An active set feasible method for large-scale minimization problems with bound constraints , 2012, Computational Optimization and Applications.

[3]  Immanuel M. Bomze,et al.  Active Set Complexity of the Away-Step Frank-Wolfe Algorithm , 2019, SIAM J. Optim..

[4]  A. Cristofari.  Active-set identification with complexity guarantees of an almost cyclic 2-coordinate descent method with Armijo line search , 2021 .

[5]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[6]  M. J. D. Powell,et al.  Nonlinear Programming—Sequential Unconstrained Minimization Techniques , 1969 .

[7]  José Mario Martínez,et al.  Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients , 2002, Comput. Optim. Appl..

[8]  Nicholas I. M. Gould,et al.  GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization , 2003, TOMS.

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

[10]  P. Toint,et al.  Lancelot: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A) , 1992 .

[11]  Iain S. Duff,et al.  MA57---a code for the solution of sparse symmetric definite and indefinite systems , 2004, TOMS.

[12]  E. G. Birgin,et al.  Complexity and performance of an Augmented Lagrangian algorithm , 2019, Optim. Methods Softw..

[13]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[14]  Stefano Lucidi,et al.  A Two-Stage Active-Set Algorithm for Bound-Constrained Optimization , 2016, Journal of Optimization Theory and Applications.

[15]  Mark W. Schmidt,et al.  “Active-set complexity” of proximal gradient: How long does it take to find the sparsity pattern? , 2017, Optim. Lett..

[16]  José Mario Martínez,et al.  Augmented Lagrangian methods under the constant positive linear dependence constraint qualification , 2007, Math. Program..

[17]  L. Palagi,et al.  A superlinearly convergent primal — dual algorithm model for constrained optimization problems with bounded variables , 2000 .

[18]  S. Lucidi,et al.  Quadratically and superlinearly convergent algorithms for the solution of inequality constrained minimization problems , 1995 .

[19]  Mikhail V. Solodov,et al.  Local Convergence of Exact and Inexact Augmented Lagrangian Methods under the Second-Order Sufficient Optimality Condition , 2012, SIAM J. Optim..

[20]  Zengxin Wei,et al.  On the Constant Positive Linear Dependence Condition and Its Application to SQP Methods , 1999, SIAM J. Optim..

[21]  R. Andreani,et al.  On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification , 2005 .

[22]  José Mario Martínez,et al.  Practical augmented Lagrangian methods for constrained optimization , 2014, Fundamentals of algorithms.

[23]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..