Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods

Angelia Nedić, Asuman Ozdaglar
2009 SIAM Journal on Optimization  
In this paper, we study methods for generating approximate primal solutions as a by-product of subgradient methods applied to the Lagrangian dual of a primal convex (possibly nondifferentiable) constrained optimization problem. Our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods, such as the recent resource allocation problems in large-scale networks. For such problems, we propose and
more » ... yze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions. These algorithms use a constant stepsize in view of its simplicity and practical significance. We provide estimates on the primal infeasibility and primal sub-optimality of the generated approximate primal solutions. These estimates are given per iteration, thus providing a basis for analyzing the trade-offs between the desired level of error and the selection of the stepsize value. Our analysis relies on the Slater condition and the inherited boundedness properties of the dual problem under this condition. It also relies on the boundedness of subgradients, which is ensured by assuming the compactness of the constraint set. Keywords: subgradient methods, averaging, approximate primal solutions, convergence rate estimates. * We thank Robert Freund and Pablo Parrilo for useful comments and discussions. Also, we thank Vladimir Norkin and the two anonymous referees for valuable comments and suggestions.
doi:10.1137/070708111 fatcat:lwgywn3nkja7pkxdhgdvtzy6ji