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Non-polyhedral Extensions of the Frank and Wolfe Theorem

  • Juan Enrique Martínez-Legaz
  • Dominikus NollEmail author
  • Wilfredo Sosa
Chapter

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 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.

Keywords

Quadratic optimization Asymptotes Motzkin-sets Frank-and-Wolfe theorem 

AMS 2010 Subject Classification

49M20 65K10 90C30 

Notes

Acknowledgements

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 [16]. 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.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Juan Enrique Martínez-Legaz
    • 1
  • Dominikus Noll
    • 2
    Email author
  • Wilfredo Sosa
    • 3
  1. 1.Departament d’Economia i d’Història EconòmicaUniversitat Autònoma de Barcelona, and Barcelona Graduate School of Mathematics (BGSMath)BarcelonaSpain
  2. 2.Université de ToulouseInstitut de MathématiquesToulouseFrance
  3. 3.Programa de Pôs-Graduação em EconomiaUniversidade Católica de BrasíliaTaguatingaBrazil

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