Lagrangian Relaxation via Ballstep Subgradient Methods

Authors:
Krzysztof C. Kiwiel;Torbjörn Larsson;P. O. Lindberg
Affiliations:
Systems Research Institute, Newelska 6, 01-447 Warsaw, Poland;Department of Mathematics, Linköping University, S-58183 Linköping, Sweden;Department of Mathematics, Linköping University, S-58183 Linköping, Sweden
Venue:
Mathematics of Operations Research
Year:
2007

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Abstract

We exhibit useful properties of ballstep subgradient methods for convex optimization using level controls for estimating the optimal value. Augmented with simple averaging schemes, they asymptotically find objective and constraint subgradients involved in optimality conditions. When applied to Lagrangian relaxation of convex programs, they find both primal and dual solutions, and have practicable stopping criteria. Up until now, similar results have only been known for proximal bundle methods, and for subgradient methods with divergent series stepsizes, whose convergence can be slow. Encouraging numerical results are presented for large-scale nonlinear multicommodity network flow problems.