Abstract
An important field of application of non-smooth optimization refers to decomposition of large-scale or complex problems by Lagrangian duality. In this setting, the dual problem consists in maximizing a concave non-smooth function that is defined as the sum of sub-functions. The evaluation of each sub-function requires solving a specific optimization sub-problem, with specific computational complexity. Typically, some sub-functions are hard to evaluate, while others are practically straightforward. When applying a bundle method to maximize this type of dual functions, the computational burden of solving sub-problems is preponderant in the whole iterative process. We propose to take full advantage of such separable structure by making a dual bundle iteration after having evaluated only a subset of the dual sub-functions, instead of all of them. This type of incremental approach has already been applied for subgradient algorithms. In this work we use instead a specialized variant of bundle methods and show that such an approach is related to bundle methods with inexact linearizations. We analyze the convergence properties of two incremental-like bundle methods. We apply the incremental approach to a generation planning problem over an horizon of one to three years. This is a large scale stochastic program, unsolvable by a direct frontal approach. For a real-life application on the French power mix, we obtain encouraging numerical results, achieving a significant improvement in speed without losing accuracy.
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References
Solodov, M., Zavriev, S.: Error stability properties of generalized gradient-type algorithms. J. Optim. Theory Appl. 98, 663–680 (1998)
Nedič, A., Bertsekas, D.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12(1), 109–138 (2001)
Kiwiel, K.: Convergence of approximate and incremental subgradient methods for convex optimization. SIAM J. Optim. 14(3), 807–840 (2003)
Gaudioso, M., Giallombardo, G., Miglionico, G.: An incremental method for solving convex finite min-max problems. Math. Oper. Res. 31(1), 173–187 (2006)
Kiwiel, K.: A proximal bundle method with approximate subgradient linearizations. SIAM J. Optim. 16(4), 1007–1023 (2006)
Kiwiel, K.: An algorithm for nonsmooth convex minimization with errors. Math. Comput. 45, 173–180 (1985)
Kiwiel, K.: Approximations in proximal bundle methods and decomposition of convex programs. J. Optim. Theory Appl. 84(3), 529–548 (1995)
Hintermüller, M.: A proximal bundle method based on approximate subgradients. Comput. Optim. Appl. 20(3), 245–266 (2001)
Solodov, M.: On approximations with finite precision in bundle methods for nonsmooth optimization. J. Optim. Theory Appl. 119(1), 151–165 (2003)
Kiwiel, K., Lemaréchal, C.: An inexact bundle variant suited to column generation. Math. Program. Ser. A (2007)
Heitsch, H., Römisch, W., Strugarek, C.: Stability of multistage stochastic programs. SIAM J. Optim. 17(2), 511–525 (2006)
Bacaud, L., Lemaréchal, C., Renaud, A., Sagastizábal, C.: Bundle methods in stochastic optimal power management: a disaggregate approach using preconditioners. Comp. Opt. Appl. 20(3), 227–244 (2001)
Lemaréchal, C., Sagastizábal, C.: Variable metric bundle methods: from conceptual to implementable forms. Math. Program. 76(3), 393–410 (1997)
Bonnans, J., Gilbert, J., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. Springer, Berlin (2006)
Hiriart-Urruty, J., Lemaréchal, C.: Convex Analysis and Minimization Algorithms 2. Springer, Berlin (1996)
Frangioni, A.: Generalized bundle methods. SIAM J. Optim. 13(1), 117–156 (2002)
Feltenmark, S., Kiwiel, K.: Dual applications of proximal bundle methods, including Lagrangian relaxation of nonconvex problems. SIAM J. Optim. 10(3), 697–721 (2000)
Avellà–Fluvià, M., Boukir, K., Martinetto, P.: Handling a CO2 reservoir in mid term generation scheduling. In: Proceedings 15th Power System Computation Conference (2005)
Mifflin, R., Sagastizábal, C.: A \(\mathcal{VU}\)-algorithm for convex minimization. Math. Program. 104(2–3), 583–608 (2005)
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The first author research was supported by a PhD grant from Electricité de France R&D, France. The work of the second author was partially supported by a research contract with EDF, CNPq Grant No. 303540-03/6, PRONEX-Optimization, and FAPERJ.
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Emiel, G., Sagastizábal, C. Incremental-like bundle methods with application to energy planning. Comput Optim Appl 46, 305–332 (2010). https://doi.org/10.1007/s10589-009-9288-8
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DOI: https://doi.org/10.1007/s10589-009-9288-8