We consider the class of weakly convex sets with respect to a quasiball in a Banach space. This class generalizes the classes of sets with positive reach, proximal smooth sets and prox-regular sets. We prove the well-posedness of the closest points problem of two sets, one of which is weakly convex with respect to a quasiball M, and the other one is a summand of the quasiball −rM, where r ∈ (0, 1). We show that if a quasiball B is a summand of a quasiball M, then a set that is weakly convex with respect to the quasiball M is also weakly convex with respect to the quasiball B. We consider the class of weakly convex functions with respect to a given convex continuous function γ that consists of functions whose epigraphs are weakly convex sets with respect to the epigraph of γ. We obtain a sufficient condition for the well-posedness of the infimal convolution problem, and also a sufficient condition for the existence, uniqueness, and continuous dependence on parameters of the minimizer.
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Ivanov, G.E., Lopushanski, M.S. Well-Posedness of Approximation and Optimization Problems for Weakly Convex Sets and Functions. J Math Sci 209, 66–87 (2015). https://doi.org/10.1007/s10958-015-2485-3
- Banach Space
- Weakly Convex
- Convex Continuous Function
- Subgradient Optimization
- Convex Lower Semicontinuous Function