Unconstrained optimization using tensor methods.


Chow, Ta-Tung, Eskow, Elizabeth, and Schnabel, Robert B. (1994). Algorithm 739. A software package for unconstrained optimization using tensor methods. ACM Transactions on Mathematical Software, 20(4), 518-530.

Chow, T., Eskow, E, and Schnabel, R. B. (1990). A software package for unconstrained optimization using tensor methods, Technical Report CU-CS-492-90, Department of Computer Science, University of Colorado, December 1990.

Schnabel, Robert B., and Chow, T. (1991). Tensor methods for unconstrained optimization using second derivatives. SIAM Journal of Optimization, 1, 293-315.


This paper describes a software package for finding the unconstrained minimizer of a nonlinear function of n variables. The package is intended for problems where n is not too large—say, n 100—so that the cost of storing one n × n matrix, and factoring it at each iteration, is acceptable. The software allows the user to choose between a recently developed “tensor method” for unconstrained optimization and an analogous standard method based on a quadratic model. The tensor method bases each iteration on a specially constructed fourth-order model of the objective function not significantly more expensive to form, store, or solve than the standard quadratic model. In our experience, the tensor method requires significantly fewer iterations and function evaluations to solve most unconstrained optimization problems than standard methods based on quadratic models, and also solves a somewhat wider range of problems. For these reasons, it may be a useful addition to numerical software libraries. [Chow, Eskow, Schnabel, 1994, p. 518.]

The software can be called with an interface where the user supplies only the function, number of variables, and starting point; default choices are made for all other input parameters. It is especially useful for problems in which the Hessian matrix at the solution is singular. An alternative interface allows the user to specify any input parameters that are different from the defaults. [NEOS Guide,]


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