Numerical Experience with a Recursive Trust-Region Method for Multilevel Nonlinear Optimization

We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton, Sartenaer and Toint (2004), and provide significant numerical experience on multilevel test problems. A suitable choice of the algorithm’s parameters is identified on these problems, yielding a very satisfactory compromise between reliability and efficiency. The resulting default algorithm is then compared to alternative optimization techniques such as mesh refinement and direct solution of the fine-level problem. The sensibility of the default variant with respect to most important algorithmic parameters is finally investigated.

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

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

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

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

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

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

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

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

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

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

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

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

[13]  Philippe L. Toint,et al.  Towards an efficient sparsity exploiting newton method for minimization , 1981 .

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

[15]  Irad Yavneh,et al.  A Multilevel Nonlinear Method , 2006, SIAM J. Sci. Comput..

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

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

[18]  M. Powell.  A New Algorithm for Unconstrained Optimization , 1970 .

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