Convexification of nonconvex functions and application to minimum and maximum principles for nonconvex sets
The authors discuss the convexification of nonconvex functions in topological linear spaces. So the well-known results are pointed out that the convex envelope \(C_{nv}(f)\) and the lower semicontinuous convex envelope \(\overline C_{nv}(f)\) of a function \(f: E\to\mathbb{R}\) can be described by the convex hull and the closed convex hull of the epigraph of \(f\). The concave envelope \(C_{nc}(f)\) and the upper semicontinuous concave envelope \(\overline C_{nc}(f)\) can be characterized in the same manner replacing the epigraph by the hypograph. Hence, it is \[ \begin{aligned} C_{nv}(f)(x) & =\inf\{\sum \lambda_if(x_i)\mid\lambda_i\geq 0,\;\sum\lambda_i=1,\;\sum\lambda_ix_i=x\},\\ C_{nc}(f)(x) & =\sup\{\sum \lambda_if(x_i)\mid\lambda_i\geq 0,\;\sum\lambda_i=1,\;\sum\lambda_ix_i= x\}\end{aligned} \] and \[ \begin{aligned} \overline{C}_{nv}(f)(x) &=\inf\{\alpha\mid \alpha=\lim_l\inf C_{nv}(f)(x_i),\;\{x_l\}\to x\},\\ \overline C_{nc}(f)(x) &=\sup\{\alpha\mid\alpha= \lim_l\sup C_{nc}(f)(x_i),\;\{x_l\}\to x\}.\end{aligned} \] The results are used for the proof of different minimum and maximum principles for semicontinuous functions. All the assertions are extensions of the known Stegall theorem to nonconvex sets and have the following form: If \(f\) is a lower/upper semicontinuous and lower/upper bounded function on a set \(A\subseteq E\) (or \(A\subseteq E^*\)) which is closed, bounded and admits an additional property (e.g. \(A\) is contained in a closed convex set with Radon-Nikodým property), then for any \(\varepsilon>0\) there exists a point \(u\in E^*\) \((u\in E)\) with \(|u|<\varepsilon\) such that \(f(\cdot)+ u(\cdot)\) attains a strong minimum/maximum on \(A\).
scientific article; zbMATH DE number 880527