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.

Abstract: In this paper we suggest a single bench mark problem family for use in evaluating unconstrained minimization algorithms or routines. In essence, this problem consists of measuring, for each algorithm, the rate at which it descends an unlimited helical valley. The periodic nature of the problem allows us to exploit affine scale invariance properties of the algorithm. As a result, the capacity of the algorithm to minimize a wide range of helical valleys of various scales may be summarized by calculating a single valued function ${g_Q}({X_1})$. The measurement of this function is not difficult, and the result provides information of a simple, general character for use in decisions about choice of algorithm.