A study of auction mechanisms for multilateral procurement based on subgradient and bundle methods

Abstract The use of iterative auctions is very common in procurement processes, where the market-maker often does not have access to complete and truthful information about the bidders’ private valuations of the resources on sale. The literature on the design of iterative mechanisms for combinatorial auctions has addressed only the most basic cases and has been dominated by primal-dual approaches. In this paper, we consider a general production/consumption exchange of interdependent goods, for which we investigate iterative auction mechanisms based on mathematical programming dual decomposition methods. We focus on Lagrangian relaxation and the solution of the Lagrangian dual through subgradient algorithms and the bundle method. A case study of a simulated wood chip market is used to evaluate numerically the efficiency of the mechanisms.

[1]  Léon Walras.  Éléments d'économie politique pure , 1889 .

[2]  Dimitri P. Bertsekas,et al.  Auction algorithms for network flow problems: A tutorial introduction , 1992, Comput. Optim. Appl..

[3]  Samir Elhedhli,et al.  Nondifferentiable Optimization , 2009, Encyclopedia of Optimization.

[4]  Sven de Vries,et al.  Combinatorial Auctions: A Survey , 2003, INFORMS J. Comput..

[5]  Michael P. Wellman,et al.  Market-oriented programming: some early lessons , 1996 .

[6]  A. Mas-Colell,et al.  Microeconomic Theory , 1995 .

[7]  P. Wolfe,et al.  A METHOD OF CONJUGATE SUBGRADIENTS FOR , 1975 .

[8]  David C. Parkes,et al.  Ascending Price Vickrey Auctions for General Valuations , 2005, J. Econ. Theory.

[9]  Anthony M. Kwasnica,et al.  A New and Improved Design for Multiobject Iterative Auctions , 2005, Manag. Sci..

[10]  Antonio Frangioni,et al.  Generalized Bundle Methods , 2002, SIAM J. Optim..

[11]  P. Wolfe.  Note on a method of conjugate subgradients for minimizing nondifferentiable functions , 1974 .

[12]  Krzysztof C. Kiwiel,et al.  Proximity control in bundle methods for convex nondifferentiable minimization , 1990, Math. Program..

[13]  Lawrence M. Ausubel.  An efficient dynamic auction for heterogeneous commodities , 2006 .

[14]  Michael P. Wellman,et al.  The WALRAS Algorithm: A Convergent Distributed Implementation of General Equilibrium Outcomes , 1998 .

[15]  Antonio Frangioni,et al.  Convergence Analysis of Deflected Conditional Approximate Subgradient Methods , 2009, SIAM J. Optim..

[16]  Claude Lemaréchal,et al.  Bundle Methods in Stochastic Optimal Power Management: A Disaggregated Approach Using Preconditioners , 2001, Comput. Optim. Appl..

[17]  S. Clearwater.  Market-based control: a paradigm for distributed resource allocation , 1996 .

[18]  Antonio Frangioni,et al.  A stabilized structured Dantzig–Wolfe decomposition method , 2012, Mathematical Programming.

[19]  Tuomas Sandholm,et al.  An Implementation of the Contract Net Protocol Based on Marginal Cost Calculations , 1993, AAAI.

[20]  Michael P. Wellman.  A Market-Oriented Programming Environment and its Application to Distributed Multicommodity Flow Problems , 1993, J. Artif. Intell. Res..

[21]  Harvey J. Everett.  Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources , 1963 .

[22]  V. F. Dem'yanov,et al.  Nondifferentiable Optimization , 1985 .

[23]  Debasis Mishra,et al.  Efficient Iterative Combinatorial Auctions , 2011 .

[24]  A. M. Geoffrion.  Duality in Nonlinear Programming: A Simplified Applications-Oriented Development , 1971 .

[25]  Rahul Simha,et al.  A Microeconomic Approach to Optimal Resource Allocation in Distributed Computer Systems , 1989, IEEE Trans. Computers.

[26]  Michael P. Wellman,et al.  A Market-Based Approach to Allocating QoS for Multimedia Applications , 1996 .

[27]  Claude Lemaréchal,et al.  Lagrangian Relaxation , 2000, Computational Combinatorial Optimization.

[28]  Benjamin F. Hobbs,et al.  Efficient market-clearing prices in markets with nonconvexities , 2005, Eur. J. Oper. Res..

[29]  P. Camerini,et al.  On improving relaxation methods by modified gradient techniques , 1975 .

[30]  J. Ledyard,et al.  A NEW AND IMPROVED DESIGN FOR MULTI-OBJECT ITERATIVE AUCTIONS , 1999 .

[31]  Michael P. Wellman,et al.  Auction Protocols for Decentralized Scheduling , 2001, Games Econ. Behav..

[32]  D. Gale,et al.  Multi-Item Auctions , 1986, Journal of Political Economy.

[33]  Maddalena Nonato,et al.  Applying Bundle Methods to the Optimization of Polyhedral Functions: An Applications-Oriented Development , 1995 .

[34]  David C. Parkes,et al.  iBundle: an efficient ascending price bundle auction , 1999, EC '99.

[35]  Jochem Zowe,et al.  A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results , 1992, SIAM J. Optim..

[36]  Yan Chen,et al.  Multi-object auctions with package bidding: An experimental comparison of Vickrey and iBEA , 2010, Games Econ. Behav..

[37]  Krzysztof C. Kiwiel,et al.  Convergence of Approximate and Incremental Subgradient Methods for Convex Optimization , 2003, SIAM J. Optim..

[38]  Sushil Bikhchandani,et al.  The Package Assignment Model , 2002, J. Econ. Theory.

[39]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

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

[41]  Ennio Stacchetti,et al.  The English Auction with Differentiated Commodities , 2000, J. Econ. Theory.

[42]  Fredrik Ygge,et al.  Market-Oriented Programming and its Application to Power Load Management , 1998 .

[43]  S. Bikhchandani,et al.  Competitive Equilibrium in an Exchange Economy with Indivisibilities , 1997 .

[44]  Dimitri P. Bertsekas,et al.  A new algorithm for the assignment problem , 1981, Math. Program..

[45]  Michel Gendreau,et al.  Série Scientifique Scientific Series Design for Optimized Multi-lateral Multi- Commodity Markets Design for Optimized Multi-lateral Multi- Commodity Markets , 2022 .

[46]  Hans Akkermans,et al.  Resource-Oriented Multicommodity Market Algorithms , 2000, Autonomous Agents and Multi-Agent Systems.