A multilevel algorithm for solving the trust-region subproblem

We present a multilevel numerical algorithm for the exact solution of the Euclidean trust-region subproblem. This particular subproblem typically arises when optimizing a nonlinear (possibly non-convex) objective function whose variables are discretized continuous functions, in which case the different levels of discretization provide a natural multilevel context. The trust-region problem is considered at the highest level (corresponding to the finest discretization), but information on the problem curvature at lower levels is exploited for improved efficiency. The algorithm is inspired by the method described in [J.J. Moré and D.C. Sorensen, On the use of directions of negative curvature in a modified Newton method, Math. Program. 16(1) (1979), pp. 1–20], for which two different multilevel variants will be analysed. Some preliminary numerical comparisons are also presented.

[1]  William L. Briggs,et al.  A multigrid tutorial, Second Edition , 2000 .

[2]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

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

[4]  Charles A. Bouman,et al.  A general framework for nonlinear multigrid inversion , 2005, IEEE Transactions on Image Processing.

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

[6]  David M. author-Gay.  Computing Optimal Locally Constrained Steps , 1981 .

[7]  T. Steihaug.  The Conjugate Gradient Method and Trust Regions in Large Scale Optimization , 1983 .

[8]  Serge,et al.  SECOND-ORDER CONVERGENCE PROPERTIES OF TRUST-REGION METHODS USING INCOMPLETE CURVATURE INFORMATION, WITH AN APPLICATION TO MULTIGRID OPTIMIZATION * , 2006 .

[9]  Jacqueline Grennon.  , 2nd Ed. , 2002, The Journal of nervous and mental disease.

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

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

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

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

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

[15]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[16]  DONALD GOLDFARB,et al.  A LINE SEARCH MULTIGRID METHOD FOR LARGE-SCALE CONVEX OPTIMIZATION , 2007 .

[17]  Danny C. Sorensen,et al.  On the use of directions of negative curvature in a modified newton method , 1979, Math. Program..

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

[19]  M. D. Hebden,et al.  An algorithm for minimization using exact second derivatives , 1973 .

[20]  Richard Zippel,et al.  Proving Polynomial-Time for Sphere-Constrained Quadratic Programming , 1990 .

[21]  S. McCormick,et al.  A multilevel variational method for Au= Bu on composite Grids , 1989 .

[22]  D. Sorensen.  Newton's method with a model trust region modification , 1982 .

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