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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The affine scale invariance of minimization algorithms


Author: J. N. Lyness
Journal: Math. Comp. 33 (1979), 265-287
MSC: Primary 65K05; Secondary 90C30
MathSciNet review: 514823
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Abstract: Let $ f(x)$ be a general objective function and let $ \bar f(x) = h + mf(Ax + d)$. An analytic estimation of the minimum of one would resemble an analytic estimation of the other in all nontrivial respects. However, the use of a minimization algorithm on either might or might not lead to apparently unrelated sequences of calculations.

This paper is devoted to providing a general theory for the affine scale invariance of algorithms. Key elements in this theory are groups of transformations T whose elements relate $ \bar f(x)$ and $ f(x)$ given above. The statement that a specified algorithm is scale invariant with respect to a specified group T is defined. The scale invariance properties of several well-known algorithms are discussed.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1979-0514823-9
PII: S 0025-5718(1979)0514823-9
Keywords: Numerical software evaluation, affine scale invariance, minimization algorithms, optimization algorithms
Article copyright: © Copyright 1979 American Mathematical Society



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