Existence, uniqueness, and convergence of the regularized primal-dual central path

  • Authors:

  • Jordi Castro;Jordi Cuesta

  • Affiliations:

  • Department of Statistics and Operations Research, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Catalonia, Spain;Operations Research unit, Department of Chemical Engineering, Universitat Rovira i Virgili, Tarragona, Catalonia, Spain

  • Venue:
  • Operations Research Letters
  • Year:
  • 2010

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Abstract

In a recent work [J. Castro, J. Cuesta, Quadratic regularizations in an interior-point method for primal block-angular problems, Mathematical Programming, in press (doi:10.1007/s10107-010-0341-2)] the authors improved one of the most efficient interior-point approaches for some classes of block-angular problems. This was achieved by adding a quadratic regularization to the logarithmic barrier. This regularized barrier was shown to be self-concordant, thus fitting the general structural optimization interior-point framework. In practice, however, most codes implement primal-dual path-following algorithms. This short paper shows that the primal-dual regularized central path is well defined, i.e., it exists, it is unique, and it converges to a strictly complementary primal-dual solution.