M-convex function minimization by continuous relaxation approach: Proximity theorem and algorithm

Satoko Moriguchi, Akiyoshi Shioura, Nobuyuki Tsuchimura

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


The concept of M-convexity for functions in integer variables, introduced by Murota [Adv. Math., 124 (1996), pp. 272-311], plays a primary role in the theory of discrete convex analysis. In this paper, we consider the problem of minimizing an M-convex function, which is a natural generalization of the separable convex resource allocation problem under a submodular constraint and contains some classes of nonseparable convex function minimization on integer lattice points. We propose a new approach for M-convex function minimization based on continuous relaxation. By establishing proximity theorems we develop a new algorithm based on continuous relaxation. We apply the approach to some special cases of the separable convex quadratic resource allocation problem and the convex quadratic tree resource allocation problem to obtain faster algorithms.

Original languageEnglish
Pages (from-to)633-668
Number of pages36
JournalSIAM Journal on Optimization
Issue number3
Publication statusPublished - 2011


  • Convex function
  • Discrete optimization
  • Matroid
  • Submodular function

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science


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