Global Convergence of Filter Methods for Nonlinear Programming

We present a general filter algorithm that allows a great deal of freedom in the step computation. Each iteration of the algorithm consists basically in computing a point which is not forbidden by the filter, from the current point. We prove its global convergence, assuming that the step must be efficient, in the sense that, near a feasible nonstationary point, the reduction of the objective function is “large.” We show that this condition is reasonable, by presenting two classical ways of performing the step which satisfy it. In the first one, the step is obtained by the inexact restoration method of Martinez and Pilotta. In the second, the step is computed by sequential quadratic programming.

[1]  Stefan Ulbrich.  On the superlinear local convergence of a filter-SQP method , 2004, Math. Program..

[2]  J. M. Martínez,et al.  Inexact-Restoration Algorithm for Constrained Optimization1 , 2000 .

[3]  Clóvis C. Gonzaga,et al.  A Globally Convergent Filter Method for Nonlinear Programming , 2003, SIAM J. Optim..

[4]  Stefan Ulbrich,et al.  A globally convergent primal-dual interior-point filter method for nonlinear programming , 2004, Math. Program..

[5]  Nicholas I. M. Gould,et al.  A Multidimensional Filter Algorithm for Nonlinear Equations and Nonlinear Least-Squares , 2004, SIAM J. Optim..

[6]  Nicholas I. M. Gould,et al.  FILTRANE, a Fortran 95 filter-trust-region package for solving nonlinear least-squares and nonlinear feasibility problems , 2007, TOMS.

[7]  Benar Fux Svaiter,et al.  A Practical Optimality Condition Without Constraint Qualifications for Nonlinear Programming , 2003 .

[8]  José Mario Martínez,et al.  Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters , 1999, Math. Program..

[9]  P. Toint,et al.  A brief history of filter methods. , 2007 .

[10]  Lorenz T. Biegler,et al.  Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence , 2005, SIAM J. Optim..

[11]  Claudia A. Sagastizábal,et al.  A bundle-filter method for nonsmooth convex constrained optimization , 2008, Math. Program..

[12]  Sven Leyffer,et al.  On the Global Convergence of a Filter--SQP Algorithm , 2002, SIAM J. Optim..

[13]  LeyfferSven,et al.  Global Convergence of a Trust-Region SQP-Filter Algorithm for General Nonlinear Programming , 2002 .

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

[15]  Lorenz T. Biegler,et al.  Line Search Filter Methods for Nonlinear Programming: Local Convergence , 2005, SIAM J. Optim..

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

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

[18]  Elizabeth W. Karas,et al.  Global convergence of slanting filter methods for nonlinear programming , 2008, Appl. Math. Comput..

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

[20]  E. Omojokun.  Trust region algorithms for optimization with nonlinear equality and inequality constraints , 1990 .

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

[22]  J. M. Martínez,et al.  Inexact-Restoration Method with Lagrangian Tangent Decrease and New Merit Function for Nonlinear Programming , 2001 .

[23]  Nicholas I. M. Gould,et al.  A Filter-Trust-Region Method for Unconstrained Optimization , 2005, SIAM J. Optim..

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

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

[26]  Claudia A. Sagastizábal,et al.  An Infeasible Bundle Method for Nonsmooth Convex Constrained Optimization without a Penalty Function or a Filter , 2005, SIAM J. Optim..

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