Mathematical Programming

, Volume 86, Issue 2, pp 283–312

Ergodic, primal convergence in dual subgradient schemes for convex programming

  • Torbjörn Larsson
  • Michael Patriksson
  • Ann-Brith Strömberg

DOI: 10.1007/s101070050090

Cite this article as:
Larsson, T., Patriksson, M. & Strömberg, A. Math. Program. (1999) 86: 283. doi:10.1007/s101070050090

Abstract.

Lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large, structured optimization problems. The dual subgradient scheme does not automatically produce primal feasible solutions; there is an abundance of techniques for computing such solutions (via penalty functions, tangential approximation schemes, or the solution of auxiliary primal programs), all of which require a fair amount of computational effort.

We consider a subgradient optimization scheme applied to a Lagrangean dual formulation of a convex program, and construct, at minor cost, an ergodic sequence of subproblem solutions which converges to the primal solution set. Numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained, compared to those generated in the basic subgradient scheme.

Key words: convex programming – Lagrangean duality – Lagrangean relaxation – subgradient optimization – ergodic convergence – primal convergence – traffic equilibrium assignment – road pricing

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Torbjörn Larsson
    • 1
  • Michael Patriksson
    • 2
  • Ann-Brith Strömberg
    • 2
  1. 1.Division of Optimization, Department of Mathematics, Linköping University, S-581 83 Linköping, SwedenSE
  2. 2.Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden, Fax: +46 31 16 19 73, e-mail: anstr@math.chalmers.seSE