A Convex Optimization Toolbox

  • J. Frédéric BonnansEmail author
Part of the Universitext book series (UTX)


This chapter presents the duality theory for optimization problems, by both the minimax and perturbation approach, in a Banach space setting. Under some stability (qualification) hypotheses, it is shown that the dual problem has a nonempty and bounded set of solutions. This leads to the subdifferential calculus, which appears to be nothing but a partial subdifferential rule. Applications are provided to the infimal convolution, as well as recession and perspective functions. The relaxation of some nonconvex problems is analyzed thanks to the Shapley–Folkman theorem.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Inria-Saclay and Centre de Mathématiques AppliquéesÉcole PolytechniquePalaiseauFrance

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