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A New Approach to Solving Stochastic Optimal Control Problems

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  • Pablo T. Rodriguez-Gonzalez

    (Tecnologico Nacional de Mexico en Celaya, Departamento de Ingenieria Quimica, Av. Tecnologico y Garcia Cubas S/N, Celaya 38010, Guanajuato, Mexico)

  • Vicente Rico-Ramirez

    (Tecnologico Nacional de Mexico en Celaya, Departamento de Ingenieria Quimica, Av. Tecnologico y Garcia Cubas S/N, Celaya 38010, Guanajuato, Mexico)

  • Ramiro Rico-Martinez

    (Tecnologico Nacional de Mexico en Celaya, Departamento de Ingenieria Quimica, Av. Tecnologico y Garcia Cubas S/N, Celaya 38010, Guanajuato, Mexico)

  • Urmila M. Diwekar

    (Center for Uncertain Systems: Tools for Optimization and Management, Vishwamitra Research Institute, Crystal Lake, IL 60012, USA)

Abstract

A conventional approach to solving stochastic optimal control problems with time-dependent uncertainties involves the use of the stochastic maximum principle (SMP) technique. For large-scale problems, however, such an algorithm frequently leads to convergence complexities when solving the two-point boundary value problem resulting from the optimality conditions. An alternative approach consists of using continuous random variables to capture uncertainty through sampling-based methods embedded within an optimization strategy for the decision variables; such a technique may also fail due to the computational intensity involved in excessive model calculations for evaluating the objective function and its derivatives for each sample. This paper presents a new approach to solving stochastic optimal control problems with time-dependent uncertainties based on BONUS (Better Optimization algorithm for Nonlinear Uncertain Systems). The BONUS has been used successfully for non-linear programming problems with static uncertainties, but we show here that its scope can be extended to the case of optimal control problems with time-dependent uncertainties. A batch reactor for biodiesel production was used as a case study to illustrate the proposed approach. Results for a maximum profit problem indicate that the optimal objective function and the optimal profiles were better than those obtained by the maximum principle.

Suggested Citation

  • Pablo T. Rodriguez-Gonzalez & Vicente Rico-Ramirez & Ramiro Rico-Martinez & Urmila M. Diwekar, 2019. "A New Approach to Solving Stochastic Optimal Control Problems," Mathematics, MDPI, vol. 7(12), pages 1-13, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1207-:d:295815
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    References listed on IDEAS

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    1. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    2. Salazar, Juan M. & Diwekar, Urmila & Constantinescu, Emil & Zavala, Victor M., 2013. "Stochastic optimization approach to water management in cooling-constrained power plants," Applied Energy, Elsevier, vol. 112(C), pages 12-22.
    3. Lee, Adrian J. & Diwekar, Urmila M., 2012. "Optimal sensor placement in integrated gasification combined cycle power systems," Applied Energy, Elsevier, vol. 99(C), pages 255-264.
    4. Verma, Puneet & Sharma, M.P., 2016. "Review of process parameters for biodiesel production from different feedstocks," Renewable and Sustainable Energy Reviews, Elsevier, vol. 62(C), pages 1063-1071.
    5. Kemal Sahin & Urmila Diwekar, 2004. "Better Optimization of Nonlinear Uncertain Systems (BONUS): A New Algorithm for Stochastic Programming Using Reweighting through Kernel Density Estimation," Annals of Operations Research, Springer, vol. 132(1), pages 47-68, November.
    6. Shastri, Y. & Diwekar, U., 2008. "Optimal control of lake pH for mercury bioaccumulation control," Ecological Modelling, Elsevier, vol. 216(1), pages 1-17.
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    Cited by:

    1. Alena Vagaská & Miroslav Gombár & Ľuboslav Straka, 2022. "Selected Mathematical Optimization Methods for Solving Problems of Engineering Practice," Energies, MDPI, vol. 15(6), pages 1-22, March.
    2. Christelle Dleuna Nyoumbi & Antoine Tambue, 2023. "A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems," Computational Economics, Springer;Society for Computational Economics, vol. 61(1), pages 1-34, January.

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