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Mathematics of Computation

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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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Keywords: Numerical software evaluation, affine scale invariance, minimization algorithms, optimization algorithms
Article copyright: © Copyright 1979 American Mathematical Society