Abstract
Directional convexity generalizes the concept of classical convexity. We investigate aC-convexity generated by the intersections of C-semispaces that efficiently approximates directional convexity. We consider the following optimization problem in case of the direction set of aC-convexity being infinite. Given a compact aC-convex set A, maximize a linear form L subject to A. We prove that there exists an aC-extreme solution of the problem. A Krein-Milman type theorem has been proved for aC-convexity. We show that the aC-convex hull of a finite point set represents the union of a finite set of polytopes in case of the direction set being finite.
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References
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Naidenko, V. (2003). Optimization on Directionally Convex Sets. In: Leopold-Wildburger, U., Rendl, F., Wäscher, G. (eds) Operations Research Proceedings 2002. Operations Research Proceedings 2002, vol 2002. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55537-4_57
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DOI: https://doi.org/10.1007/978-3-642-55537-4_57
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00387-8
Online ISBN: 978-3-642-55537-4
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