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.
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Acknowledgements
This research was supported by a CIFRE contract between ENGIE and Université de Paris Sorbonne, France. The first author is joint PhD candidate of IMPA—Instituto de Matemática Pura e Aplicada (Brazil) and Université de Paris Sorbonne. The second author’s research is also partially supported by CNPq Grant 303905/2015-8. Research of the third author is also supported in part by CNPq Grant 303724/2015-3, by FAPERJ Grant 203.052/2016, and by the Russian Foundation for Basic Research Grant 19-51-12003 NNIOa. The authors are grateful to W. de Oliveira for providing the MATLAB code for the test functions used in the performance profiles.
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Lage, C., Sagastizábal, C. & Solodov, M. Multiplier Stabilization Applied to Two-Stage Stochastic Programs. J Optim Theory Appl 183, 158–178 (2019). https://doi.org/10.1007/s10957-019-01550-7
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DOI: https://doi.org/10.1007/s10957-019-01550-7
Keywords
- Multiplier stability
- Dual regularization
- Penalty method
- Stochastic programming
- Two-stage stochastic programming
- Empirical approximations