Numerical experience with a recursive trust-region method for multilevel nonlinear bound-constrained optimization

We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton et al. (A recursive trust-region method in infinity norm for bound-constrained nonlinear optimization, IMA J. Numer. Anal. 28(4) (2008), pp. 827–861) for bound-constrained nonlinear problems, and provide numerical experience on multilevel test problems. A suitable choice of the algorithm's parameters is identified on these problems, yielding a satisfactory compromise between reliability and efficiency. The resulting default algorithm is then compared with alternative optimization techniques such as mesh refinement and direct solution of the fine-level problem. It is also shown that its behaviour is similar to that of multigrid algorithms for linear systems.

[1]  Stephen G. Nash,et al.  Model Problems for the Multigrid Optimization of Systems Governed by Differential Equations , 2005, SIAM J. Sci. Comput..

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

[3]  Andrew R. Conn,et al.  Two-Step Algorithms for Nonlinear Optimization with Structured Applications , 1999, SIAM J. Optim..

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

[5]  Nicholas I. M. Gould,et al.  Sensitivity of trust-region algorithms to their parameters , 2005, 4OR.

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

[7]  Qiang Du,et al.  Uniform Convergence of a Nonlinear Energy-Based Multilevel Quantization Scheme , 2008, SIAM J. Numer. Anal..

[8]  S. Nash.  A multigrid approach to discretized optimization problems , 2000 .

[9]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[10]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.

[11]  Serge Gratton,et al.  Recursive Trust-Region Methods for Multiscale Nonlinear Optimization , 2008, SIAM J. Optim..

[12]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[13]  Zdenek Dostál,et al.  Projector preconditioning for partially bound‐constrained quadratic optimization , 2007, Numer. Linear Algebra Appl..

[14]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[15]  R. Kornhuber.  Monotone multigrid methods for elliptic variational inequalities I , 1994 .

[16]  Nicholas I. M. Gould,et al.  Lancelot: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A) , 1992 .

[17]  M. Powell,et al.  On the Estimation of Sparse Hessian Matrices , 1979 .

[18]  P. Toint,et al.  A recursive trust-region method in infinity norm for bound-constrained nonlinear optimization , 2007 .

[19]  Karl Kunisch,et al.  A globalization strategy for the multigrid solution of elliptic optimal control problems , 2006, Optim. Methods Softw..

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

[21]  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..

[22]  P. Gill,et al.  Combination trust-region line-search methods for unconstrained optimization , 1999 .

[23]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

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

[25]  E. Gelman,et al.  On multilevel iterative methods for optimization problems , 1990, Math. Program..

[26]  Serge Gratton,et al.  Numerical Experience with a Recursive Trust-Region Method for Multilevel Nonlinear Optimization , 2006 .

[27]  Jorge Nocedal,et al.  Combining Trust Region and Line Search Techniques , 1998 .

[28]  R. Kornhuber.  Monotone multigrid methods for elliptic variational inequalities II , 1996 .

[29]  P. Toint,et al.  A recursive ℓ∞-trust-region method for bound-constrained nonlinear optimization , 2008 .

[30]  P. Toint,et al.  Global convergence of a class of trust region algorithms for optimization with simple bounds , 1988 .

[31]  A. Bronstein,et al.  A MULTIGRID APPROACH FOR MULTI-DIMENSIONAL SCALING∗ , 2004 .