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Efficient upper and lower bounds for global mixed-integer optimal control

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  • Sebastian Sager
  • Mathieu Claeys
  • Frédéric Messine

Abstract

We present a control problem for an electrical vehicle. Its motor can be operated in two discrete modes, leading either to acceleration and energy consumption, or to a recharging of the battery. Mathematically, this leads to a mixed-integer optimal control problem (MIOCP) with a discrete feasible set for the controls taking into account the electrical and mechanical dynamic equations. The combination of nonlinear dynamics and discrete decisions poses a challenge to established optimization and control methods, especially if global optimality is an issue. Probably for the first time, we present a complete analysis of the optimal solution of such a MIOCP: solution of the integer-relaxed problem both with a direct and an indirect approach, determination of integer controls by means of the sum up rounding strategy, and calculation of global lower bounds by means of the method of moments. As we decrease the control discretization grid and increase the relaxation order, the obtained series of upper and lower bounds converge for the electrical car problem, proving the asymptotic global optimality of the calculated chattering behavior. We stress that these bounds hold for the optimal control problem in function space, and not on an a priori given (typically coarse) control discretization grid, as in other approaches from the literature. This approach is generic and is an alternative to global optimal control based on probabilistic or branch-and-bound based techniques. The main advantage is a drastic reduction of computational time. The disadvantage is that only local solutions and certified lower bounds are provided with no possibility to reduce these gaps. For the instances of the electrical car problem, though, these gaps are very small. The main contribution of the paper is a survey and new combination of state-of-the-art methods for global mixed-integer optimal control and the in-depth analysis of an important, prototypical control problem. Despite the comparatively low dimension of the problem, the optimal solution structure of the relaxed problem exhibits a series of bang-bang, path-constrained, and sensitivity-seeking arcs. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Sebastian Sager & Mathieu Claeys & Frédéric Messine, 2015. "Efficient upper and lower bounds for global mixed-integer optimal control," Journal of Global Optimization, Springer, vol. 61(4), pages 721-743, April.
    • Handle: RePEc:spr:jglopt:v:61:y:2015:i:4:p:721-743
    DOI: 10.1007/s10898-014-0156-4
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    References listed on IDEAS

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    1. Sebastian Sager & Michael Jung & Christian Kirches, 2011. "Combinatorial integral approximation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 363-380, June.
    2. Eligius M. T. Hendrix & Boglárka G.-Tóth, 2010. "Nonlinear Programming algorithms," Springer Optimization and Its Applications, in: Introduction to Nonlinear and Global Optimization, chapter 5, pages 91-136, Springer.
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    Cited by:

    1. Dario Bauso & Quanyan Zhu & Tamer Başar, 2016. "Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 606-630, May.
    2. Ishan Bajaj & M. M. Faruque Hasan, 2020. "Global dynamic optimization using edge-concave underestimator," Journal of Global Optimization, Springer, vol. 77(3), pages 487-512, July.

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