A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization

ABSTRACT We present a modification of a primal-dual algorithm based on a mixed augmented Lagrangian and a log-barrier penalty function. The goal of this new feature is to quickly detect infeasibility. An additional parameter is introduced to balance the minimization of the objective function and the realization of the constraints. The global convergence of the modified algorithm is analysed under mild assumptions. We also show that under a suitable choice of the parameters along the iterations, the rate of convergence of the algorithm to an infeasible stationary point is superlinear. This is the first local convergence result for the class of interior point methods in the infeasible case. We finally report some numerical experiments to show that this new algorithm is quite efficient to detect infeasibility and does not deteriorate the overall behavior in the general case.

[1]  Riadh Omheni.  Méthodes primales-duales régularisées pour l'optimisation non linéaire avec contraintes , 2014 .

[2]  Hiroshi Yamashita,et al.  Quadratic Convergence of a Primal-Dual Interior Point Method for Degenerate Nonlinear Optimization Problems , 2005, Comput. Optim. Appl..

[3]  Hao Wang,et al.  A Sequential Quadratic Optimization Algorithm with Rapid Infeasibility Detection , 2014, SIAM J. Optim..

[4]  Riadh Omheni,et al.  A globally and quadratically convergent primal–dual augmented Lagrangian algorithm for equality constrained optimization , 2017, Optim. Methods Softw..

[5]  Dominique Orban,et al.  From global to local convergence of interior methods for nonlinear optimization , 2013, Optim. Methods Softw..

[6]  Jie Sun,et al.  A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs , 2018, Journal of Industrial & Management Optimization.

[7]  Stephen J. Wright,et al.  Local Convergence of a Primal-Dual Method for Degenerate Nonlinear Programming , 2002, Comput. Optim. Appl..

[8]  E. G. Birgin,et al.  Optimality properties of an Augmented Lagrangian method on infeasible problems , 2015, Comput. Optim. Appl..

[9]  Paul Armand,et al.  An Augmented Lagrangian Method for Equality Constrained Optimization with Rapid Infeasibility Detection Capabilities , 2018, J. Optim. Theory Appl..

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

[11]  José Mario Martínez,et al.  Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming , 2014, J. Glob. Optim..

[12]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

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

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

[15]  Riadh Omheni,et al.  A Mixed Logarithmic Barrier-Augmented Lagrangian Method for Nonlinear Optimization , 2017, J. Optim. Theory Appl..

[16]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[17]  Dominique Orban,et al.  Dynamic updates of the barrier parameter in primal-dual methods for nonlinear programming , 2008, Comput. Optim. Appl..

[18]  Paul Armand,et al.  Uniform boundedness of the inverse of a Jacobian matrix arising in regularized interior-point methods , 2013, Math. Program..

[19]  Vincent Pateloup,et al.  Study of a primal-dual algorithm for equality constrained minimization , 2011, Comput. Optim. Appl..

[20]  Jorge Nocedal,et al.  Knitro: An Integrated Package for Nonlinear Optimization , 2006 .

[21]  Paul Armand,et al.  A local convergence property of primal-dual methods for nonlinear programming , 2008, Math. Program..

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

[23]  Jorge Nocedal,et al.  An interior point method for nonlinear programming with infeasibility detection capabilities , 2014, Optim. Methods Softw..

[24]  Jorge Nocedal,et al.  Infeasibility Detection and SQP Methods for Nonlinear Optimization , 2010, SIAM J. Optim..

[25]  G. Debreu.  Definite and Semidefinite Quadratic Forms , 1952 .

[26]  Frank E. Curtis.  Noname manuscript No. (will be inserted by the editor) A Penalty-Interior-Point Algorithm for Nonlinear Constrained Optimization , 2011 .

[27]  José Mario Martínez,et al.  Handling infeasibility in a large-scale nonlinear optimization algorithm , 2012, Numerical Algorithms.

[28]  Jefferson G. Melo,et al.  Augmented Lagrangian methods for nonlinear programming with possible infeasibility , 2015, J. Glob. Optim..

[29]  Stephen J. Wright,et al.  Properties of the Log-Barrier Function on Degenerate Nonlinear Programs , 2002, Math. Oper. Res..