Some numerical results using a sparse matrix updating formula in unconstrained optimization

Toint, Ph. L.

doi:10.1090/S0025-5718-1978-0483452-7

Some numerical results using a sparse matrix updating formula in unconstrained optimization
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by Ph. L. Toint PDF

Math. Comp. 32 (1978), 839-851 Request permission

Abstract:

This paper presents a numerical comparison between algorithms for unconstrained optimization that take account of sparsity in the second derivative matrix of the objective function. Some of the methods included in the comparison use difference approximation schemes to evaluate the second derivative matrix and others use an approximation to it which is updated regularly using the changes in the gradient. These results show what method to use in what circumstances and also suggest interesting future developments.

References

J. Inst. Math. Appl.

Variable Metric Method for Minimization

J. E. Dennis Jr. and Jorge J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev. 19 (1977), no. 1, 46–89. MR 445812, DOI 10.1137/1019005
R. Fletcher and M. J. D. Powell, A rapidly convergent descent method for minimization, Comput. J. 6 (1963/64), 163–168. MR 152116, DOI 10.1093/comjnl/6.2.163

An Algorithm for Minimization Using Exact Second Derivatives

H. Y. Huang, Unified approach to quadratically convergent algorithms for function minimization, J. Optim. Theory Appl. 5 (1970), 405–423. MR 288939, DOI 10.1007/BF00927440

Methods in Computational Physics

M. J. D. Powell, A new algorithm for unconstrained optimization, Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1970, pp. 31–65. MR 0272162

A Fortran Subroutine for Unconstrained Minimization, Requiring First Derivatives of the Objective Function

M. J. D. Powell, Convergence properties of a class of minimization algorithms, Nonlinear programming, 2 (Proc. Special Interest Group on Math. Programming Sympos., Univ. Wisconsin, Madison, Wis., 1974) Academic Press, New York, 1974, pp. 1–27. MR 0386270
M. J. D. Powell, A view of unconstrained optimization, Optimization in action (Proc. Conf., Univ. Bristol, Bristol, 1975) Academic Press, London, 1976, pp. 117–152. MR 0431680

Two Fortran Subroutines for Direct Solution of Linear Equations, whose Matrix is Sparse, Symmetric and Positive Definite

L. K. Schubert, Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian, Math. Comp. 24 (1970), 27–30. MR 258276, DOI 10.1090/S0025-5718-1970-0258276-9
Ph. L. Toint, On sparse and symmetric matrix updating subject to a linear equation, Math. Comp. 31 (1977), no. 140, 954–961. MR 455338, DOI 10.1090/S0025-5718-1977-0455338-4