Non-polyhedral extensions of the Frank and Wolfe Theorem

Abstract:

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 larger classes of sets F with this Frank-and-Wolfe property will be identified. The existence of non-polyhedral Frank-and-Wolfe sets will be established, internal characterizations by way of asymptotic properties will be presented, and the stability of the Frank-and-Wolfe class under various operations will be discussed.

Juan Enrique Martinez-Legaz received his PhD degree from the Universitat de Barcelona in 1981. He is a full professor at the Universitat Autonoma de Barcelona since 1993. He was appointed EUROPT Fellow in 2011 and was awarded a doctorate honoris causa by the Universidad Nacional de Ingenieria (Lima, Peru) in 2013. He has supervised or co-supervised 7 PhD students at the Universitat de Barcelona (Spain), Universitat Autonoma de Barcelona (Spain), Università degli Studi di Bergamo (Italy), Universidade Federal do Parana (Brazil) and Pontificia Universidad Catolica del Peru (Peru). He has developed, as the main investigator, several research projects funded by Spanish agencies. Math>SciNet lists 151 papers published by him, which have received 845 citations by 412 authors. He was the Chairman of the Working Group on Generalized Convexity in the period 2000-2002 and the Editor-in-Chief of the journal Optimization during the years 2006-2012. He belongs to the editorial board of several journals, including Optimization, the Journal of Convex Analysis, the Journal of Global Optimization, TOP, the Journal of Optimization Theory and Applications and Minimax Theory and its Applications.