Convergence Properties of Minimization Algorithms for Convex Constraints Using a Structured Trust Region

In this paper, we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region. This development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others, a common occurrence in large-scale nonlinear applications. After describing the structured trust region mechanism, we prove global convergence for all algorithms in our class.

[1]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

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

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

[4]  Jorge J. Moré,et al.  The Levenberg-Marquardt algo-rithm: Implementation and theory , 1977 .

[5]  T. M. Williams,et al.  Practical Methods of Optimization. Vol. 1: Unconstrained Optimization , 1980 .

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

[7]  Ekkehard W. Sachs,et al.  Algorithmic methods in optimal control , 1981 .

[8]  Jorge J. Moré,et al.  Recent Developments in Algorithms and Software for Trust Region Methods , 1982, ISMP.

[9]  Andreas Griewank,et al.  On the unconstrained optimization of partially separable functions , 1982 .

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

[11]  M. J. D. Powell,et al.  On the global convergence of trust region algorithms for unconstrained minimization , 1984, Math. Program..

[12]  Andreas Griewank,et al.  Numerical experiments with partially separable optimization problems , 1984 .

[13]  Richard A. Tapia,et al.  A trust region strategy for nonlinear equality constrained op-timization , 1984 .

[14]  Andreas Griewank,et al.  On the existence of convex decompositions of partially separable functions , 1984, Math. Program..

[15]  A. Vardi.  A Trust Region Algorithm for Equality Constrained Minimization: Convergence Properties and Implementation , 1985 .

[16]  Richard H. Byrd,et al.  A Trust Region Algorithm for Nonlinearly Constrained Optimization , 1987 .

[17]  Philippe L. Toint,et al.  Global convergence of the partitioned BFGS algorithm for convex partially separable optimization , 1986, Math. Program..

[18]  P. Toint,et al.  On Large Scale Nonlinear Least Squares Calculations , 1987 .

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

[20]  J. Dunn.  On the convergence of projected gradient processes to singular critical points , 1987 .

[21]  M. Lescrenier,et al.  Partially separable optimization and parallel computing , 1988 .

[22]  P. Toint,et al.  Testing a class of methods for solving minimization problems with simple bounds on the variables , 1988 .

[23]  P. Toint.  Global Convergence of a a of Trust-Region Methods for Nonconvex Minimization in Hilbert Space , 1988 .

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

[25]  Jorge J. Moré,et al.  Trust regions and projected gradients , 1988 .

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

[27]  Daniel Tuyttens,et al.  On large scale nonlinear Network optimization , 1990, Math. Program..

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

[29]  M. El-Alem.  A global convergence theory for the Celis-Dennis-Tapia trust-region algorithm for constrained optimization , 1991 .

[30]  Ya-Xiang Yuan,et al.  A trust region algorithm for equality constrained optimization , 1990, Math. Program..

[31]  Richard H. Byrd,et al.  Parallel global optimization: numerical methods, dynamic scheduling methods, and application to molecular configuration , 1994 .

[32]  M. Lescrenier.  Convergence of trust region algorithms for optimization with bounds when strict complementarity does not hold , 1991 .

[33]  Andreas Griewank,et al.  The global convergence of partitioned BFGS on problems with convex decompositions and Lipschitzian gradients , 1991, Math. Program..

[34]  R. Carter.  On the global convergence of trust region algorithms using inexact gradient information , 1991 .

[35]  Gerardo Toraldo,et al.  On the Solution of Large Quadratic Programming Problems with Bound Constraints , 1991, SIAM J. Optim..

[36]  Daniel Tuyttens,et al.  LSNNO, a FORTRAN subroutine for solving large-scale nonlinear network optimization problems , 1992, TOMS.

[37]  P. Toint,et al.  Lancelot: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A) , 1992 .

[38]  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..

[39]  Donald Goldfarb,et al.  Partial-Update Newton Methods for Unary, Factorable, and Partially Separable Optimization , 1993, SIAM J. Optim..

[40]  P. Toint,et al.  Performance of a multifrontal scheme for partially separable optimization , 1994 .

[41]  Nicholas I. M. Gould,et al.  CUTE: constrained and unconstrained testing environment , 1995, TOMS.