An active set algorithm for nonlinear programming using linear programming and equality constrain

This paper describes an active set algorithm for large scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza The step computation is performed in two stages In the rst stage a linear program is solved to estimate the active set at the solution The linear program is obtained by making a linear approximation to the penalty function inside a trust region In the second stage an equality constrained quadratic program EQP is solved involving only those constraints that are active at the solution of the linear program The EQP incorporates a trust region constraint and is solved inexactly by means of a projected conjugate gradient method Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr test set Department of Computer Science University of Colorado Boulder CO richard cs colorado edu This author was supported by Air Force O ce of Scienti c Research grant F Army Research O ce Grant DAAG and National Science Foundation grant INT Computational Science and Engineering Department Rutherford Appleton Laboratory Chilton Ox fordshire OX Qx England EU n gould rl ac uk This author was supported in part by the EPSRC grant GR R Department of Electrical and Computer Engineering Northwestern University Evanston IL USA These authors were supported by National Science Foundation grant CCR and Depart ment of Energy grant DE FG ER A

[1]  M. Powell.  A New Algorithm for Unconstrained Optimization , 1970 .

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

[3]  Michael A. Saunders,et al.  Large-scale linearly constrained optimization , 1978, Math. Program..

[4]  Philip E. Gill,et al.  Practical optimization , 1981 .

[5]  Michael A. Saunders,et al.  A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints , 1982 .

[6]  Michael A. Saunders,et al.  MINOS 5. 0 user's guide , 1983 .

[7]  Richard A. Tapia,et al.  Practical Methods of Optimization, Volume 2: Constrained Optimization (R. Fletcher) , 1984 .

[8]  Ya-Xiang Yuan,et al.  Conditions for convergence of trust region algorithms for nonsmooth optimization , 1985, Math. Program..

[9]  Roger Fletcher,et al.  Nonlinear programming and nonsmooth optimization by successive linear programming , 1989, Math. Program..

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

[11]  Nicholas I. M. Gould,et al.  CUTE: constrained and unconstrained testing environment , 1995, TOMS.

[12]  Jorge Nocedal,et al.  Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization , 1997, TOMS.

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

[14]  Nicholas I. M. Gould,et al.  Solving the Trust-Region Subproblem using the Lanczos Method , 1999, SIAM J. Optim..

[15]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[16]  Nicholas I. M. Gould,et al.  On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization , 2001, SIAM J. Sci. Comput..

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

[18]  J. Nocedal,et al.  Algorithms for large-scale nonlinear optimization , 2002 .

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

[20]  Roger Fletcher,et al.  On the global convergence of an SLP–filter algorithm that takes EQP steps , 2003, Math. Program..