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Multiplier Stabilization Applied to Two-Stage Stochastic Programs

Author

Listed:
  • Clara Lage

    (ENGIE
    IMPA)

  • Claudia Sagastizábal

    (IMECC - UNICAMP)

  • Mikhail Solodov

    (IMPA – Instituto de Matemática Pura e Aplicada)

Abstract

In many mathematical optimization applications, dual variables are an important output of the solving process, due to their role as price signals. When dual solutions are not unique, different solvers or different computers, even different runs in the same computer if the problem is stochastic, often end up with different optimal multipliers. From the perspective of a decision maker, this variability makes the price signals less reliable and, hence, less useful. We address this issue for a particular family of linear and quadratic programs by proposing a solution procedure that, among all possible optimal multipliers, systematically yields the one with the smallest norm. The approach, based on penalization techniques of nonlinear programming, amounts to a regularization in the dual of the original problem. As the penalty parameter tends to zero, convergence of the primal sequence and, more critically, of the dual is shown under natural assumptions. The methodology is illustrated on a battery of two-stage stochastic linear programs.

Suggested Citation

  • Clara Lage & Claudia Sagastizábal & Mikhail Solodov, 2019. "Multiplier Stabilization Applied to Two-Stage Stochastic Programs," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 158-178, October.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:1:d:10.1007_s10957-019-01550-7
    DOI: 10.1007/s10957-019-01550-7
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    References listed on IDEAS

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    1. Suvrajeet Sen & Yifan Liu, 2016. "Mitigating Uncertainty via Compromise Decisions in Two-Stage Stochastic Linear Programming: Variance Reduction," Operations Research, INFORMS, vol. 64(6), pages 1422-1437, December.
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    3. Wim Ackooij & Jérôme Malick, 2016. "Decomposition algorithm for large-scale two-stage unit-commitment," Annals of Operations Research, Springer, vol. 238(1), pages 587-613, March.
    4. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    5. Birge, John R. & Louveaux, Francois V., 1988. "A multicut algorithm for two-stage stochastic linear programs," European Journal of Operational Research, Elsevier, vol. 34(3), pages 384-392, March.
    6. Wim Ackooij & Jérôme Malick, 2016. "Decomposition algorithm for large-scale two-stage unit-commitment," Annals of Operations Research, Springer, vol. 238(1), pages 587-613, March.
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    Cited by:

    1. Guillaume Erbs & Clara Lage & Claudia Sagastizábal & Mikhail Solodov, 2023. "Increasing reliability of price signals in long term energy management problems," Computational Optimization and Applications, Springer, vol. 85(3), pages 787-820, July.

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