A unifying polyhedral approxim ation framework for convex optimization

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

We propose a unifying framework for polyhedral approximation in convex optimization. It subsumes classical methods, such as cutting plane and simplicial decomposition, but also includes new methods and new versions/extensions of old methods, such as a simplicial decomposition method for nondifferentiable optimization and a new piecewise linear approximation method for convex single commodity network flow problems. Our framework is based on an extended form of monotropic programming, a broadly applicable model, which includes as special cases Fenchel duality and Rockafellar's monotropic programming, and is characterized by an elegant and symmetric duality theory. Our algorithm combines flexibly outer and inner linearization of the cost function. The linearization is progressively refined by using primal and dual differentiation, and the roles of outer and inner linearization are reversed in a mathematically equivalent dual algorithm. We provide convergence results for the general case where outer and inner linearization are combined in the same algorithm.

Original languageEnglish (US)
Pages (from-to)333-360
Number of pages28
JournalSIAM Journal on Optimization
Volume21
Issue number1
DOIs
StatePublished - 2011
Externally publishedYes

Keywords

  • Convex optimization
  • Cutting plane
  • Monotropic programming
  • Polyhedral approximation
  • Simplicial decomposition

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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