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Cutting-Plane Algorithms for Non-smooth Convex Optimization Over Simple Domains

In: Methods of Nonsmooth Optimization in Stochastic Programming

Author

Listed:
  • Wim Stefanus Ackooij

    (Électricité de France (EDF R&D))

  • Welington Luis Oliveira

    (Mines Paris - PSL)

Abstract

This chapter deals with the problem of minimizing a non-smooth convex function f over a “simple” feasible set X ⊂ ℝ n $$X \subset \mathbb {R}^n$$ . By “simple”, we mean that minimizing a linear or quadratic function over X can be efficiently performed by specialized algorithms. This is the case if X is a polyhedron, a ball, a semidefinite domain, or even, in some cases, a mixed-integer set. We start by laying the groundwork for the subsequent sections with some central concepts and definitions such as cutting-plane approximations and oracles (black-boxes). Next, we present implementable algorithms, such as Kelley’s cutting-plane method and a few enhanced versions of the latter. No further information on the objective function is necessary. For instance, f may only be assessed by a first-order oracle. This is a significant difference with the algorithms of Chap. 9 , which require full knowledge of f (e.g. its algebraic expression and thus potential “confidential” information).

Suggested Citation

  • Wim Stefanus Ackooij & Welington Luis Oliveira, 2025. "Cutting-Plane Algorithms for Non-smooth Convex Optimization Over Simple Domains," International Series in Operations Research & Management Science, in: Methods of Nonsmooth Optimization in Stochastic Programming, chapter 0, pages 295-319, Springer.
    • Handle: RePEc:spr:isochp:978-3-031-84837-7_10
    DOI: 10.1007/978-3-031-84837-7_10
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