Log-nonexpansive functions: terminology and references - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T23:34:09Z http://mathoverflow.net/feeds/question/107042 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107042/log-nonexpansive-functions-terminology-and-references Log-nonexpansive functions: terminology and references S. Sra 2012-09-12T20:49:13Z 2012-09-13T06:57:46Z <p>During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important.</p> <blockquote> <p>(<strong>Defn.</strong>). Let $h: (0,\infty) \to (0,\infty)$ be strictly positive, continuous function. We say $h$ is <em>log-nonexpansive</em> if \begin{equation*} |\log h(s)-\log h(t)| \le |\log s - \log t|,\qquad\text{for all}\ \ s, t > 0. \end{equation*}</p> </blockquote> <p>This definition is just an alternate way of saying that $h$ is non-expansive under the <em>hyperbolic distance</em> $d(x,y) := |\log x - \log y|$. </p> <blockquote> <p>My question is whether there exists a classification of such log-nonexpansive functions, or at least some sufficient conditions that ensure this non-expansivity?</p> </blockquote> <p>I will also be grateful for references to material where such log-nonexpansivity arises.</p> http://mathoverflow.net/questions/107042/log-nonexpansive-functions-terminology-and-references/107067#107067 Answer by Dirk for Log-nonexpansive functions: terminology and references Dirk 2012-09-13T06:57:46Z 2012-09-13T06:57:46Z <p>To expand Robert Israel's comment: The class of log-nonexpansinve functions on $]0,\infty[$ is the image of the nonexpansive functions of $\mathbb{R}$ under conjugation with $\exp:\mathbb{R}\to ]0,\infty[$, that is, every log-nonexpansive function $h:]0,\infty[\to]0,\infty[$ is obtained by a nonexpansive function $g$ via $h = \exp\circ g\circ \exp^{-1}$.</p> <p>Since nonexpansive functions are central in optimization of real-valued functions, it seems natural, that an important class of functions for optimization of positive valued functions is obtained by conjugation with a bijection from $\mathbb{R}$ to $]0,\infty[$.</p> <p>I wonder, if other bijections from $\mathbb{R}$ to $]0,\infty[$ would also work in your context (although $\exp$ seems very well suited).</p>