A truncated Newton method in an augmented Lagrangian framework for nonlinear programming

In this paper we propose a primal-dual algorithm for the solution of general nonlinear programming problems. The core of the method is a local algorithm which relies on a truncated procedure for the computation of a search direction, and is thus suitable for large scale problems. The truncated direction produces a sequence of points which locally converges to a KKT pair with superlinear convergence rate.The local algorithm is globalized by means of a suitable merit function which is able to measure and to enforce progress of the iterates towards a KKT pair, without deteriorating the local efficiency. In particular, we adopt the exact augmented Lagrangian function introduced in Pillo and Lucidi (SIAM J. Optim. 12:376–406, 2001), which allows us to guarantee the boundedness of the sequence produced by the algorithm and which has strong connections with the above mentioned truncated direction.The resulting overall algorithm is globally and superlinearly convergent under mild assumptions.

[1]  Robert J. Vanderbei,et al.  An Interior-Point Algorithm for Nonconvex Nonlinear Programming , 1999, Comput. Optim. Appl..

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

[3]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[4]  Jorge Nocedal,et al.  A trust region method based on interior point techniques for nonlinear programming , 2000, Math. Program..

[5]  Jorge Nocedal,et al.  An Interior Point Algorithm for Large-Scale Nonlinear Programming , 1999, SIAM J. Optim..

[6]  Nicholas I. M. Gould,et al.  Global Convergence of a Trust-Region SQP-Filter Algorithm for General Nonlinear Programming , 2002, SIAM J. Optim..

[7]  Hiroshi Yamashita,et al.  Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization , 1996, Math. Program..

[8]  L. Grippo,et al.  A class of nonmonotone stabilization methods in unconstrained optimization , 1991 .

[9]  L. Grippo,et al.  A truncated Newton method with nonmonotone line search for unconstrained optimization , 1989 .

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

[11]  Francisco Facchinei,et al.  Minimization of SC1 functions and the Maratos effect , 1995, Oper. Res. Lett..

[12]  J. Henry,et al.  System Modelling and Optimization: Proceedings of the 16th IFIP-TC7 Conference, Compiègne, France, July 5-9, 1993 , 1994, System Modelling and Optimization.

[13]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[14]  Nicholas I. M. Gould,et al.  CUTEr and SifDec: A constrained and unconstrained testing environment, revisited , 2003, TOMS.

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

[16]  Lorenz T. Biegler,et al.  Global and Local Convergence of Line Search Filter Methods for Nonlinear Programming , 2002 .

[17]  L. Grippo,et al.  An exact penalty function method with global convergence properties for nonlinear programming problems , 1986, Math. Program..

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

[19]  A. Mayne.  Parametric Optimization: Singularities, Pathfollowing and Jumps , 1990 .

[20]  I M GouldNicholas,et al.  CUTEr and SifDec , 2003 .

[21]  L. Grippo,et al.  Exact penalty functions in constrained optimization , 1989 .

[22]  L. Grippo,et al.  A class of continuously differentiable exact penalty function algorithms for nonlinear programming problems , 1984 .

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

[24]  Torkel Glad,et al.  A multiplier method with automatic limitation of penalty growth , 1979, Math. Program..

[25]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[26]  Michael L. Overton,et al.  A Primal-dual Interior Method for Nonconvex Nonlinear Programming , 1998 .

[27]  Ya-Xiang Yuan.  Advances in Nonlinear Programming , 1998 .

[28]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[29]  Laura Palagi,et al.  Convergence to Second-Order Stationary Points of a Primal-Dual Algorithm Model for Nonlinear Programming , 2005, Math. Oper. Res..

[30]  Stefano Lucidi,et al.  New Results on a Continuously Differentiable Exact Penalty Function , 1992, SIAM J. Optim..

[31]  Gianni Di Pillo,et al.  An Augmented Lagrangian Function with Improved Exactness Properties , 2002, SIAM J. Optim..

[32]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[33]  Richard H. Byrd,et al.  A Trust Region Algorithm for Nonlinearly Constrained Optimization , 1987 .