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# Function Minimization Algorithm

IP.com Disclosure Number: IPCOM000080129D
Original Publication Date: 1973-Jan-01
Included in the Prior Art Database: 2005-Feb-27
Document File: 2 page(s) / 49K

IBM

## Related People

Gaston, CA: AUTHOR

## Abstract

Described is an iterative process of fitting a hyperparaboloid to a set of reference points to predict the function minimum, and periodically refining the semimajor axes of the paraboloid, so it will converge towards a good approximation of the defined function in the vicinity of its minimum. At any time in the iterative process a number of reference points (possibly widely scattered) are used, and a hyperparaboloid is fitted to those points. The minimum of the hyperparaboloid is taken as a new reference point, and the worst of the old points may be discarded. (Image Omitted)

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Function Minimization Algorithm

Described is an iterative process of fitting a hyperparaboloid to a set of reference points to predict the function minimum, and periodically refining the semimajor axes of the paraboloid, so it will converge towards a good approximation of the defined function in the vicinity of its minimum. At any time in the iterative process a number of reference points (possibly widely scattered) are used, and a hyperparaboloid is fitted to those points. The minimum of the hyperparaboloid is taken as a new reference point, and the worst of the old points may be discarded.

(Image Omitted)

If the a(j) are assumed known, this set of equations can be solved for the X(j) (corresponding to fitting a hyperparaboloid of known shape to a set of points).

If the X(j) are assumed known, this set of equations can be solved for the a(j) (corresponding to determining the best shape of hyperparaboloid to fit a set of points including the center, or minimum, point).

Iteration between these two types of solutions yields a tightly clustered set of reference points defining a hyperparaboloid, which is accurately centered over a minimum of the defined function.

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