DP Dual subgradient method with averaging for optimal resource allocation

Abstract A dual subgradient method is proposed for solving convex optimization problems with linear constraints. As novelty, the recovering of primal solutions can be avoided. Instead, the optimal convergence rate for the whole sequence of primal-dual iterates is obtained. This is due to the primal-dual averaging strategies which are incorporated into the iterative scheme. We apply our dual subgradient method with averaging to optimal resource allocation within a multi-agent environment. The proposed dual subgradient method naturally corresponds to a distributed process of production/price adjustments and effectively leads to a market equilibrium.

[1]  E. Friedman,et al.  The complexity of resource allocation and price mechanisms under bounded rationality , 1995 .

[2]  T. Larsson,et al.  A Lagrangean Relaxation Scheme for Structured Linear Programs With Application To Multicommodity Network Flows , 1997 .

[3]  Hanif D. Sherali,et al.  Recovery of primal solutions when using subgradient optimization methods to solve Lagrangian duals of linear programs , 1996, Oper. Res. Lett..

[4]  Paul Milgrom,et al.  Putting Auction Theory to Work , 2004 .

[5]  Michael Patriksson,et al.  Primal convergence from dual subgradient methods for convex optimization , 2014, Mathematical Programming.

[6]  Asuman E. Ozdaglar,et al.  Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods , 2008, SIAM J. Optim..

[7]  Francisco Facchinei,et al.  Generalized Nash Equilibrium Problems , 2010, Ann. Oper. Res..

[8]  D. E. Campbell.  Resource Allocation Mechanisms , 1987 .

[9]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

[10]  Andrzej Ruszczynski,et al.  A merit function approach to the subgradient method with averaging , 2008, Optim. Methods Softw..

[11]  Andreas Fischer,et al.  GENERALIZED NASH EQUILIBRIUM PROBLEMS - RECENT ADVANCES AND CHALLENGES , 2014 .

[12]  Hanif D. Sherali,et al.  A class of convergent primal-dual subgradient algorithms for decomposable convex programs , 1986, Math. Program..

[13]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[14]  Marc Teboulle,et al.  An $O(1/k)$ Gradient Method for Network Resource Allocation Problems , 2014, IEEE Transactions on Control of Network Systems.

[15]  Yu. Nesterov,et al.  Quasi-monotone Subgradient Methods for Nonsmooth Convex Minimization , 2015, J. Optim. Theory Appl..

[16]  A. Kakhbod.  Resource Allocation in Decentralized Systems with Strategic Agents: An Implementation Theory Approach , 2013 .

[17]  Michael Patriksson,et al.  Ergodic, primal convergence in dual subgradient schemes for convex programming , 1999, Mathematical programming.