On a Frank-Wolfe type theorem in cubic optimization

A classical result due to Frank and Wolfe [An algorithm for quadratic programming. Naval Res Log Quart. 1956;3:95–110] 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 [On solvability of the problem of polynomial programming (In Russian). Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika. 1982;4:194–197. Translation appeared in News of the Academy of Science of USSR, Dept. of Technical Sciences, Technical Cybernetics.]). Further, we discuss related results. In particular, we bring back to attention Kummer's [Globale Stabilität quadratischer Optimierungsprobleme. Wissenschaftliche Zeitschrift der Humboldt- Universität zu Berlin, Math-Nat R. 1977;XXVI(5):565–569] 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.