Convex envelopes generated from finitely many compact convex sets (Q1942266)

The authors focus on the problem of constructing the convex envelope of a lower semicontinuous function \(\phi\) defined over a compact convex subset \(C\) of \(\mathbb{R}^n\). The convex envelope of \(\phi\) on \(C\) can be fully characterized by the set of extreme points of the convex hull of \(\mathrm{epi}_C\phi\); the projection of this set on \(C\) is the generating set of the convex envelope of \(\phi\) over \(C\). The authors present a convex NLP formula for the problem of constructing the convex envelope of an lsc function whose generating set is representable as the union of a finite number of compact convex sets; in this case, the envelope representation problem is significantly simplified. Their convexification argument is based on the concept of perspective transformation. The authors focus on functions of the form \(\phi(x,y)= f(x)g(y)\) (\(x\in \mathbb{R}^m,\) \(y\in \mathbb{R}^n\)) over a box, where \(f\) is nonnegative and convex and \(g\) is nonnegative and componentwise concave. In Section 3, they study the case where \(f\) is assumed to be a power, or to have exponential form, and they provide a characterization of the convex envelope of \(\phi\). In Section 4, analytical expressions for the convex envelopes of functions of the form \(\phi(x,y)= f(x)g(y)\), where \(f\) is nonnegative convex and \(g\) is componenwise concave are presented, under the additional assumption that \(g\) is univariate and bivariate in turn.