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Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method. (English) Zbl 1065.90060

Summary: We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a self-contained convergence analysis that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in-depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts.

We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.

MSC:
90C25Convex programming
49J52Nonsmooth analysis (other weak concepts of optimality)
90-02Research monographs (optimization)
90C57Polyhedral combinatorics, branch-and-bound, branch-and-cut
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