Line Search Filter Methods for Nonlinear Programming: Motivation and Global Convergence

Line search methods are proposed for nonlinear programming using Fletcher and Leyffer's filter method [Math. Program., 91 (2002), pp. 239--269], which replaces the traditional merit function. Their global convergence properties are analyzed. The presented framework is applied to active set sequential quadratic programming (SQP) and barrier interior point algorithms. Under mild assumptions it is shown that every limit point of the sequence of iterates generated by the algorithm is feasible, and that there exists at least one limit point that is a stationary point for the problem under consideration. A new alternative filter approach employing the Lagrangian function instead of the objective function with identical global convergence properties is briefly discussed.

[1]  M. Powell.  A method for nonlinear constraints in minimization problems , 1969 .

[2]  Shih-Ping Han.  A globally convergent method for nonlinear programming , 1975 .

[3]  K. Schittkowski.  The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function , 1982 .

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

[5]  P. Toint,et al.  A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds , 1991 .

[6]  T. Tsuchiya,et al.  On the formulation and theory of the Newton interior-point method for nonlinear programming , 1996 .

[7]  Jorge Nocedal,et al.  On the Local Behavior of an Interior Point Method for Nonlinear Programming , 1997 .

[8]  Hiroshi Yamashita.  A globally convergent primal-dual interior point method for constrained optimization , 1998 .

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

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

[11]  Sven Leyffer,et al.  A bundle filter method for nonsmooth nonlinear optimization , 1999 .

[12]  L. Vicente,et al.  A Globally Convergent Primal-Dual Interior-Point Filter Method for Nonconvex Nonlinear Programming , 2000 .

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

[14]  Lorenz T. Biegler,et al.  Failure of global convergence for a class of interior point methods for nonlinear programming , 2000, Math. Program..

[15]  Nicholas I. M. Gould,et al.  Superlinear Convergence of Primal-Dual Interior Point Algorithms for Nonlinear Programming , 2000, SIAM J. Optim..

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

[17]  J. Nocedal,et al.  Foundations of Computational Mathematics: Feasibility control in nonlinear optimization , 2001 .

[18]  Nicholas I. M. Gould,et al.  Componentwise fast convergence in the solution of full-rank systems of nonlinear equations , 2002, Math. Program..

[19]  Robert J. Vanderbei,et al.  Interior-Point Methods for Nonconvex Nonlinear Programming: Filter Methods and Merit Functions , 2002, Comput. Optim. Appl..

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

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

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

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

[24]  Andreas Wächter,et al.  A Primal-Dual Interior-Point Method for Nonlinear Programming with Strong Global and Local Convergence Properties , 2003, SIAM J. Optim..

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

[26]  Charles Audet,et al.  A Pattern Search Filter Method for Nonlinear Programming without Derivatives , 2001, SIAM J. Optim..

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

[28]  Hiroshi Yamashita,et al.  A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization , 2005, Math. Program..

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

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