A classical result due to Frank and Wolfe (1956) says that a quadratic function $f$ attains its supremum on a nonempty polyhedron $M$ if $f$ is bounded from above on $M$. In this note, we present a stringent proof of the extension of this result to cubic optimization (known from Andronov, Belousov and Shironin (1982)). Further, we discuss related results. In particular, we bring back to attention Kummer's (1977) generalization of the Frank-Wolfe theorem to the case that $f$ is quadratic, but $M$ is the Minkowski sum of a compact set and a polyhedral cone.
Preprint, Institut füt Betriebswirtschaftslehre, Universität Zürich, May 2018 After revision: Published online 13 Jan 2019 in Optimization - A Journal of Mathematical Programming and Operations Research; DOI 10.1080/02331934.2019.1566327