Mathematical Programming

, Volume 94, Issue 1, pp 41–69

The volume algorithm revisited: relation with bundle methods

Authors

  • L. Bahiense
    • Universidade Federal do Rio de Janeiro, COPPE-Sistemas e Computacão, P.O.Box 68511, Rio de Janeiro, RJ 21945-970, Brazil, e-mail: bahiense@impa.br, maculan@cos.ufrj.br
  • N. Maculan
    • Universidade Federal do Rio de Janeiro, COPPE-Sistemas e Computacão, P.O.Box 68511, Rio de Janeiro, RJ 21945-970, Brazil, e-mail: bahiense@impa.br, maculan@cos.ufrj.br
  • C. Sagastizábal
    • IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro RJ 22460-320, Brazil. On leave from INRIA- Rocquencourt, BP 105, 78153 Le Chesnay, France

DOI: 10.1007/s10107-002-0357-3

Cite this article as:
Bahiense, L., Maculan, N. & Sagastizábal, C. Math. Program., Ser. A (2002) 94: 41. doi:10.1007/s10107-002-0357-3

Abstract.

 We revise the Volume Algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a way similar to the serious/null steps philosophy of bundle methods, VA produces green, yellow or red steps. In order to give convergence results, we introduce in VA a precise measure for the improvement needed to declare a green or serious step. This addition yields a revised formulation (RVA) that is halfway between VA and a specific bundle method, that we call BVA. We analyze the convergence properties of both RVA and BVA. Finally, we compare the performance of the modified algorithms versus VA on a set of Rectilinear Steiner problems of various sizes and increasing complexity, derived from real world VLSI design instances.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002