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.
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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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Martínez-Legaz, J.E., Noll, D., Sosa, W. (2019). Non-polyhedral Extensions of the Frank and Wolfe Theorem. In: Bauschke, H., Burachik, R., Luke, D. (eds) Splitting Algorithms, Modern Operator Theory, and Applications. Springer, Cham. https://doi.org/10.1007/978-3-030-25939-6_12
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