Convex envelopes of bivariate functions through the solution of KKT systems (Q1756774)

The concept of the convex envelope of a function \(f\) over a compact region \(X\) in \(R^n\) is defined as the best convex underestimator of a non-convex function \(f\). It can be expressed as the supremum of all affine underestimators of \(f\) over \(X\). The author describes in the paper a variant of the approach to the problem of finding the convex envelope of bivariate functions over polytopes, which was published by \textit{M. Locatelli} and \textit{F. Schoen} [Math. Program. 144, No. 1--2 (A), 65--91 (2014; Zbl 1295.90055)]. The proposed procedure based on the solution of a KKT system simplifies the previously published method for finding convex envelopes published in the paper by Locatelli and Schoen in [loc. cit.]. The following two special cases are discussed: deriving the convex envelope of a bilinear function over general polytopes and deriving the convex envelope of the product of power functions over boxes.