Facial Reduction Algorithms for Conic Optimization Problems

Hayato Waki, Masakazu Muramatsu

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

In the conic optimization problems, it is well-known that a positive duality gap may occur, and that solving such a problem is numerically difficult or unstable. For such a case, we propose a facial reduction algorithm to find a primal-dual pair of conic optimization problems having the zero duality gap and the optimal value equal to one of the original primal or dual problems. The conic expansion approach is also known as a method to find such a primal-dual pair, and in this paper we clarify the relationship between our facial reduction algorithm and the conic expansion approach. Our analysis shows that, although they can be regarded as dual to each other, our facial reduction algorithm has ability to produce a finer sequence of faces of the cone including the feasible region. A simple proof of the convergence of our facial reduction algorithm for the conic optimization is presented. We also observe that our facial reduction algorithm has a practical impact by showing numerical experiments for graph partition problems; our facial reduction algorithm in fact enhances the numerical stability in those problems.

Original languageEnglish
Pages (from-to)188-215
Number of pages28
JournalJournal of Optimization Theory and Applications
Volume158
Issue number1
DOIs
Publication statusPublished - Jul 1 2013

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Conic Optimization
Optimization Problem
Primal-dual
Zero Duality Gap
Graph Partition
Duality Gap
Feasible region
Convergence of numerical methods
Numerical Stability
Dual Problem
Optimization problem
Cones
Cone
Unstable
Numerical Experiment
Face

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

Cite this

Facial Reduction Algorithms for Conic Optimization Problems. / Waki, Hayato; Muramatsu, Masakazu.

In: Journal of Optimization Theory and Applications, Vol. 158, No. 1, 01.07.2013, p. 188-215.

Research output: Contribution to journalArticle

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