Mathematical Programming

, Volume 93, Issue 1, pp 129–171 | Cite as

Self-regular functions and new search directions for linear and semidefinite optimization

  • Jiming Peng
  • Cornelis Roos
  • Tamás Terlaky

Abstract.

In this paper, we introduce the notion of a self-regular function. Such a function is strongly convex and smooth coercive on its domain, the positive real axis. We show that any such function induces a so-called self-regular proximity function and a corresponding search direction for primal-dual path-following interior-point methods (IPMs) for solving linear optimization (LO) problems. It is proved that the new large-update IPMs enjoy a polynomial ?(n\(\frac{q+1}{2q}\)log\(\frac{n}{\varepsilon}\)) iteration bound, where q≥1 is the so-called barrier degree of the kernel function underlying the algorithm. The constant hidden in the ?-symbol depends on q and the growth degree p≥1 of the kernel function. When choosing the kernel function appropriately the new large-update IPMs have a polynomial ?(\(\sqrt{n}\)lognlog\(\frac{n}{\varepsilon}\)) iteration bound, thus improving the currently best known bound for large-update methods by almost a factor \(\sqrt{n}\). Our unified analysis provides also the ?(\(\sqrt{n}\)log\(\frac{n}{\varepsilon}\)) best known iteration bound of small-update IPMs. At each iteration, we need to solve only one linear system. An extension of the above results to semidefinite optimization (SDO) is also presented.

Key words: linear optimization – semidefinite optimization – interior-point method – primal-dual method – self-regularity – polynomial complexity 
Mathematics Subject Classification (1991): 90C05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jiming Peng
    • 1
  • Cornelis Roos
    • 2
  • Tamás Terlaky
    • 3
  1. 1.Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada, L8S 4L7, e-mail: pengj@mcmaster.caCA
  2. 2.Faculty of Information Technology and Systems, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands, e-mail: C.Roos@its.tudelft.nlNL
  3. 3.Advanced Optimization Laboratory, Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada, L8S 4L7, e-mail: terlaky@mcmaster.caCA

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