Constrained Dogleg methods for nonlinear systems with simple bounds

We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problems.

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

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

[3]  William W. Hager,et al.  An affine-scaling interior-point CBB method for box-constrained optimization , 2009, Math. Program..

[4]  Carl Tim Kelley,et al.  A pointwise quasi-Newton method for integral equations , 1988 .

[5]  Benedetta Morini,et al.  A Gauss–Newton method for solving bound-constrained underdetermined nonlinear systems , 2009, Optim. Methods Softw..

[6]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[7]  Christian Kanzow,et al.  An interior-point affine-scaling trust-region method for semismooth equations with box constraints , 2007, Comput. Optim. Appl..

[8]  Christian Kanzow,et al.  On Affine-Scaling Interior-Point Newton Methods for Nonlinear Minimization with Bound Constraints , 2006, Comput. Optim. Appl..

[9]  A. Morgan,et al.  A methodology for solving chemical equilibrium systems , 1987 .

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

[11]  S. Dirkse,et al.  Mcplib: a collection of nonlinear mixed complementarity problems , 1995 .

[12]  A. Stavrakoudis,et al.  On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods , 2010 .

[13]  Tao Wang,et al.  An interior point potential reduction method for constrained equations , 1996, Math. Program..

[14]  Homer F. Walker,et al.  Inexact Newton Dogleg Methods , 2005, SIAM J. Numer. Anal..

[15]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..

[16]  Jorge J. Moré,et al.  Algorithm 566: FORTRAN Subroutines for Testing Unconstrained Optimization Software [C5], [E4] , 1981, TOMS.

[17]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[18]  Benedetta Morini,et al.  Trust-region quadratic methods for nonlinear systems of mixed equalities and inequalities , 2009 .

[19]  Stefan Ulbrich,et al.  Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption , 1999, Math. Program..

[20]  Stefania Bellavia,et al.  STRSCNE: A Scaled Trust-Region Solver for Constrained Nonlinear Equations , 2004, Comput. Optim. Appl..

[21]  Stefania Bellavia,et al.  An affine scaling trust-region approach to bound-constrained nonlinear systems , 2003 .

[22]  Detong Zhu,et al.  An affine scaling trust-region algorithm with interior backtracking technique for solving bound-constrained nonlinear systems , 2005 .

[23]  Nataša Krejić,et al.  An interior-point method for solving box-constrained underdetermined nonlinear systems , 2005 .

[24]  Stefania Bellavia,et al.  Subspace Trust-Region Methods for Large Bound-Constrained Nonlinear Equations , 2006, SIAM J. Numer. Anal..

[25]  Klaus Schittkowski,et al.  Test examples for nonlinear programming codes , 1980 .

[26]  Andreas Griewank,et al.  Yury G. Evtushenko – a tribute , 2005, Optim. Methods Softw..

[27]  Stefania Bellavia,et al.  An interior global method for nonlinear systems with simple bounds , 2005, Optim. Methods Softw..

[28]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[29]  G. Dulikravich,et al.  Generalized nonlinear minimal residual (GNLMR) method for iterative algorithms , 1986 .

[30]  Alexander P. Morgan,et al.  Chemical equilibrium systems as numerical test problems , 1990, TOMS.