Stopping rules and backward error analysis for bound-constrained optimization

Termination criteria for the iterative solution of bound-constrained optimization problems are examined in the light of backward error analysis. It is shown that the problem of determining a suitable perturbation on the problem’s data corresponding to the definition of the backward error is analytically solvable under mild assumptions. Moreover, a link between existing termination criteria and this solution is clarified, indicating that some standard measures of criticality may be interpreted in the sense of backward error analysis. The backward error problem is finally considered from the multicriteria optimization point of view and some numerical illustration is provided.

[1]  Stefan M. Wild,et al.  Estimating Computational Noise , 2011, SIAM J. Sci. Comput..

[2]  N. Higham,et al.  Backward Error Bounds for Constrained Least Squares Problems , 1999 .

[3]  Guoliang Xue,et al.  The MINPACK-2 test problem collection , 1992 .

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  Nicholas I. M. Gould,et al.  Global Convergence of a Class of Trust Region Algorithms for Optimization Using Inexact Projections on Convex Constraints , 1993, SIAM J. Optim..

[6]  Jorge Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[7]  William W. Hager,et al.  A New Active Set Algorithm for Box Constrained Optimization , 2006, SIAM J. Optim..

[8]  Françoise Chaitin-Chatelin,et al.  Lectures on finite precision computations , 1996, Software, environments, tools.

[9]  Andrew J. Wathen,et al.  Stopping criteria for iterations in finite element methods , 2005, Numerische Mathematik.

[10]  J. J. Moré,et al.  On the identification of active constraints , 1988 .

[11]  Iain S. Duff,et al.  Stopping Criteria for Iterative Solvers , 1992, SIAM J. Matrix Anal. Appl..

[12]  Paul H. Calamai,et al.  Projected gradient methods for linearly constrained problems , 1987, Math. Program..

[13]  Gerardo Toraldo,et al.  Convergence properties of trust region methods for linear and convex constraints , 1990, Math. Program..

[14]  Mélodie Mouffe.  Multilevel optimization in infinity norm and associated stopping criteria. Optimisation multiniveaux en norme infinie et critères d'arrêt associés. , 2009 .

[15]  James Hardy Wilkinson,et al.  Error Analysis of Direct Methods of Matrix Inversion , 1961, JACM.

[16]  J. Burke.  On identification of active constraints II: the nonconvex case , 1990 .

[17]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[18]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .

[19]  R. Weiner.  Lecture Notes in Economics and Mathematical Systems , 1985 .

[20]  Todd Munson,et al.  Benchmarking optimization software with COPS 3.0. , 2001 .

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

[22]  J. L. Rigal,et al.  On the Compatibility of a Given Solution With the Data of a Linear System , 1967, JACM.