Solving Large Scale Unconstrained Minimization Problems by a New ODE Numerical Integration Method

ABSTRACT

In reference [1], for large scale nonlinear equations , a new ODE solving method was given. This paper is a continuous work. Here has gradient structure i.e. , is a scalar function. The eigenvalues of the Jacobian of ; or the Hessian of , are all real number. So the new method is very suitable for this structure. For quadratic function the convergence was proved and the spectral radius of iteration matrix was given and compared with traditional method. Examples show for large scale problems (dimension ) the new method is very efficient.

In reference [1], for large scale nonlinear equations , a new ODE solving method was given. This paper is a continuous work. Here has gradient structure i.e. , is a scalar function. The eigenvalues of the Jacobian of ; or the Hessian of , are all real number. So the new method is very suitable for this structure. For quadratic function the convergence was proved and the spectral radius of iteration matrix was given and compared with traditional method. Examples show for large scale problems (dimension ) the new method is very efficient.

KEYWORDS

Unconstrained Minimization Problem, Gradient Equations, Quadratic Model, Spectral Radius, ODE Numerical Integration

Unconstrained Minimization Problem, Gradient Equations, Quadratic Model, Spectral Radius, ODE Numerical Integration

Cite this paper

nullT. Han, X. Luo and Y. Han, "Solving Large Scale Unconstrained Minimization Problems by a New ODE Numerical Integration Method,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 527-532. doi: 10.4236/am.2011.25069.

nullT. Han, X. Luo and Y. Han, "Solving Large Scale Unconstrained Minimization Problems by a New ODE Numerical Integration Method,"

References

[1] T. M. Han and Y. H. Han, “Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method,” Applied Mathematics, Vol. 1, No. 3, 2010, pp. 222-229. doi: 10.4236/am.2010.13027

[2] P. Deuflhard, “Newton Methods for Nonlinear Problems Affine Invariance and Adaptive Algorithms,” Springer- Verlag, Berlin, 2004.

[3] D. J. Higham, “Trust Region Algorithms and Timestep Selection,” SIAM Journal on Numerical Analysis, Vol. 37, No. 1, 1999, pp. 194-210. doi:10.1137/S0036142998335972

[4] D. M. Young, “Convergence Properties of the Symmetric and Unsymmetric Successive Overrelaxation Methods and Related Methods,” Mathematics of Computation, Vol. 24, No. 112, 1970, pp. 793-807. doi:10.1090/S0025-5718-1970-0281331-4

[5] Y. Saad, “Iterative Methods for Sparse Linear Systems,” PWS Publishing Company, Boston, 1996.

[6] A. Miele and J. W. Cantrell, “Study on a Memory Gradient Method for the Minimization of Functions,” Journal of Optimization Theory and Applications, Vol. 3, No. 6, 1969, pp. 459-470. doi:10.1007/BF00929359

[7] A. R. Conn, N. I. M. Gould and P. L. Toint, “Testing a Class of Methods for Solving Minimization Problems with Simple Bounds on the Variables,” Mathematics of Computation, Vol. 50, No. 182, 1988, pp. 399-430. doi:10.1090/S0025-5718-1988-0929544-3

[8] M. A. Branch, T. F. Coleman and Y. Y. Li, “A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems,” SIAM Journal on Scientific Computing, Vol. 21, No. 1,1999, pp. 1-23. doi:10.1137/S1064827595289108

[1] T. M. Han and Y. H. Han, “Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method,” Applied Mathematics, Vol. 1, No. 3, 2010, pp. 222-229. doi: 10.4236/am.2010.13027

[2] P. Deuflhard, “Newton Methods for Nonlinear Problems Affine Invariance and Adaptive Algorithms,” Springer- Verlag, Berlin, 2004.

[3] D. J. Higham, “Trust Region Algorithms and Timestep Selection,” SIAM Journal on Numerical Analysis, Vol. 37, No. 1, 1999, pp. 194-210. doi:10.1137/S0036142998335972

[4] D. M. Young, “Convergence Properties of the Symmetric and Unsymmetric Successive Overrelaxation Methods and Related Methods,” Mathematics of Computation, Vol. 24, No. 112, 1970, pp. 793-807. doi:10.1090/S0025-5718-1970-0281331-4

[5] Y. Saad, “Iterative Methods for Sparse Linear Systems,” PWS Publishing Company, Boston, 1996.

[6] A. Miele and J. W. Cantrell, “Study on a Memory Gradient Method for the Minimization of Functions,” Journal of Optimization Theory and Applications, Vol. 3, No. 6, 1969, pp. 459-470. doi:10.1007/BF00929359

[7] A. R. Conn, N. I. M. Gould and P. L. Toint, “Testing a Class of Methods for Solving Minimization Problems with Simple Bounds on the Variables,” Mathematics of Computation, Vol. 50, No. 182, 1988, pp. 399-430. doi:10.1090/S0025-5718-1988-0929544-3

[8] M. A. Branch, T. F. Coleman and Y. Y. Li, “A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems,” SIAM Journal on Scientific Computing, Vol. 21, No. 1,1999, pp. 1-23. doi:10.1137/S1064827595289108