, Volume 94, Issue 1, pp 41-69

The volume algorithm revisited: relation with bundle methods

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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.

Received: December 1999 / Accepted: September 2002 Published online: December 19, 2002
Key Words. volume algorithm – bundle methods – Steiner problems
Correspondence to: Claudia A. Sagastizábal, e-mail: sagastiz@impa.br