Algebraic Core and Convex Calculus Without Topology
Research of Boris S. Mordukhovich was partly supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research grant # DMS-1716057, and by the Australian Research Council under Discovery Project DP-190100555. Research of Nguyen Mau Nam was partly supported by the USA National Science Foundation under grant DMS-1716057.
In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. Functions of this type play a crucial role in many aspects of convex optimization and its applications.
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Van Cuong, D., Mordukhovich, B. S., Mau Nam, N., & Cartmell, A. (2020). Algebraic core and convex calculus without topology. Optimization, 1–27. https://doi.org/10.1080/02331934.2020.1800700