A classical theorem due to Frank and Wolfe (1956) says that a quadratic function $f$ in $n$ real variables attains its supremum on a nonempty polyhedral set $M$ in $\real^n$ if $f$ is bounded from above on $M$.
In 1982, Andronov, Belousov and Shironin published the extension of this result to the case of cubic polynomials $f$. However, their original proof is rather compactly written, in particular, various arguments are missing or show some lack of clarity. On the other hand, the idea of this proof is very nice and elegant, and the gaps are not difficult to fill. To top it all, one finds sometimes in the literature even the (unverified) claim that the result is not true. So we think it is desirable to give a stringent proof of the Frank-Wolfe theorem for cubic optimization problems. This will be done in this talk.
Moreover, we bring back to attention two other generalizations of the Frank-Wolfe theorem. Kummer (1977) extended 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. Recently, this extension was rediscovered by Martinez-Legaz, Noll and Sosa (2018), but the authors were not aware of Kummer's paper. Since there is a new interest in this kind of result, we sketch the idea of proof. Also in 1977, Belousov (1977) generalized the Frank-Wolfe theorem to the minimization of convex polynomials over a constraint set defined by finitely many convex polynomial inequalities, see also Belousov, Klatte (2002) for a proof of this theorem and some consequences. We remind of this result as a tribute to Evgeny Beluosov who passed away in February 2018.
The talk is based on the recent short paper Klatte (2018).
Personal website of Diethard Klatte