Mathematical Programming

, Volume 90, Issue 1, pp 113–145 | Cite as

Logarithmic SUMT limits in convex programming

  • Garth P. McCormick
  • Christoph Witzgall


The limits of a class of primal and dual solution trajectories associated with the Sequential Unconstrained Minimization Technique (SUMT) are investigated for convex programming problems with non-unique optima. Logarithmic barrier terms are assumed. For linear programming problems, such limits – of both primal and dual trajectories – are strongly optimal, strictly complementary, and can be characterized as analytic centers of, loosely speaking, optimality regions. Examples are given, which show that those results do not hold in general for convex programming problems. If the latter are weakly analytic (Bank et al. [3]), primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability. That class of programming problems contains faithfully convex, linear, and convex quadratic programming problems as strict subsets. In the differential case, dual trajectory limits can be characterized similarly, albeit under different conditions, one of which suffices for strict complementarity.

Key words: analytic center – barrier function – convex programming – degeneracy – faithfully convex – interior point method – optimization – self-concordant – strictly complementary – strongly optimal – SUMT – weakly analytic – trajectory – Wolfe-dual program 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Garth P. McCormick
    • 1
  • Christoph Witzgall
    • 2
  1. 1.Emeritus, Department of Operations Research, SEAS, The George Washington University, Washington, D.C. 20052US
  2. 2.Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD 20899US

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