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Large-Scale Convex Optimization Via Saddle Point Computation

Author

Listed:
  • Markku Kallio

    (International Institute for Applied Systems Analysis, Laxenburg, Austria; Helsinki School of Economics, Helsinki, Finland)

  • Charles H. Rosa

    (International Institute for Applied Systems Analysis, Laxenburg, Austria; Argonne National Laboratory, Argonne, IL 60439, USA)

Abstract

This article proposes large-scale convex optimization problems to be solved via saddle points of the standard Lagrangian. A recent approach for saddle point computation is specialized, by way of a specific perturbation technique and unique scaling method, to convex optimization problems with differentiable objective and constraint functions. In each iteration the update directions for primal and dual variables are determined by gradients of the Lagrangian. These gradients are evaluated at perturbed points that are generated from current points via auxiliary mappings. The resulting algorithm suits massively parallel computing, though in this article we consider only a serial implementation. We test a version of our code embedded within GAMS on 16 nonlinear problems, which are mainly large. These models arise from multistage optimization of economic systems. For larger problems with adequate precision requirements, our implementation appears faster than MINOS.

Suggested Citation

  • Markku Kallio & Charles H. Rosa, 1999. "Large-Scale Convex Optimization Via Saddle Point Computation," Operations Research, INFORMS, vol. 47(1), pages 93-101, February.
    • Handle: RePEc:inm:oropre:v:47:y:1999:i:1:p:93-101
    DOI: 10.1287/opre.47.1.93
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    References listed on IDEAS

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    1. Sen, Suvrajeet & Yakowitz, Diana, 1986. "A primal-dual subgradient method for time staged capacity expansion planning," European Journal of Operational Research, Elsevier, vol. 27(3), pages 301-312, December.
    2. Alan Manne & Richard Richels, 1992. "Buying Greenhouse Insurance: The Economic Costs of CO2 Emission Limits," MIT Press Books, The MIT Press, edition 1, volume 1, number 026213280x, April.
    3. Markku Kallio & Seppo Salo, 1994. "Tatonnement Procedures for Linearly Constrained Convex Optimization," Management Science, INFORMS, vol. 40(6), pages 788-797, June.
    4. Alan Manne & Richard Richels, 1995. "The Greenhouse Debate: Economic Efficiency, Burden Sharing and Hedging Strategies," The Energy Journal, , vol. 16(4), pages 1-37, October.
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    Cited by:

    1. Larsson, Torbjorn & Patriksson, Michael & Stromberg, Ann-Brith, 2003. "On the convergence of conditional [var epsilon]-subgradient methods for convex programs and convex-concave saddle-point problems," European Journal of Operational Research, Elsevier, vol. 151(3), pages 461-473, December.
    2. A. Nedić & A. Ozdaglar, 2009. "Subgradient Methods for Saddle-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 205-228, July.

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