Non-polyhedral Extensions of the Frank and Wolfe Theorem
In 1956 Marguerite Frank and Paul Wolfe proved that a quadratic function which is bounded below on a polyhedron P attains its infimum on P. In this work we search for larger classes of sets F with this Frank-and-Wolfe property. We establish the existence of non-polyhedral Frank-and-Wolfe sets, obtain internal characterizations by way of asymptotic properties, and investigate stability of the Frank-and-Wolfe class under various operations.
KeywordsQuadratic optimization Asymptotes Motzkin-sets Frank-and-Wolfe theorem
AMS 2010 Subject Classification49M20 65K10 90C30
Helpful discussions with B. Kummer (HU Berlin) and D. Klatte (Zürich) are gratefully acknowledged. We are indebted to Vera Roshchina (Australia) for having pointed out reference . J.E. Martínez-Legaz was supported by the MINECO of Spain, Grant MTM2014-59179-C2-2-P, and by the Severo Ochoa Programme for Centres of Excellence in R&D [SEV-2015-0563]. He is affiliated with MOVE (Markets, Organizations and Votes in Economics). D. Noll was supported by Fondation Mathématiques Jacques-Hadamard (FMJH) under PGMO Grant Robust Optimization for Control.
- 1.Andronov, V., Belousov, E., Shironin, V.: On solvability of the problem of polynomial programming (in Russian). Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika 4, 194–197 (1982). Translated as News of the Academy of Science of USSR, Dept. of Technical Sciences, Technical Cybernetics.Google Scholar
- 2.Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear parametric optimization. Birkhäuser, Basel-Boston-Stuttgart (1983)Google Scholar
- 3.Belousov, E.: Introduction to Convex Analysis and Integer Programming (in Russian). Moscow University Publisher (1977)Google Scholar
- 6.Collatz, L., Wetterling, W.: Optimization Problems. Springer Verlag (1975)Google Scholar
- 12.Kummer., B.: Globale Stabilität quadratischer Optimierungsprobleme. Wissenschaftliche Zeitschrift der Humboldt-Universität zu Berlin, Math.-Nat. R. XXVI(5), 565–569 (1977)Google Scholar
- 17.Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)Google Scholar
- 18.Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons (1986)Google Scholar