Dr. Nicolas Hadjisavvas
Department of Product and Systems Design Engineering
University of the Aegean, Greece

A function $f : R^n \rightarrow R$ is called quasiconvex if for all $x,y$ in $R^n$ and all $0$$<$$t$$<$$1$, $f(tx + (1-t) y) \leq \max\{f(x),f(y)\}$. Quasiconvex functions may have local minima which are not global minima, and this may create theoretical and computational difficulties in optimization. The question at the origin of this work is the following: Is it always possible to write a quasiconvex function $f(x)$ as a composition $h(g(x))$, where $h$ is nonincreasing and $g$ has the property that all local minima are global minima? If this is the case, then studying the points of global minimum of $f$ would be equivalent to the study of points of the minimum of $g$, which in general is easier. We answer this question by the affirmative, under a very weak assumption: the function $f$ should have a property that is weaker even to lower semicontinuity.