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On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

In: Facets of Combinatorial Optimization

Author

Listed:
  • Michael N. Jung

    (Ruprecht-Karls-Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen)

  • Christian Kirches

    (Ruprecht-Karls-Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen)

  • Sebastian Sager

    (Otto-von-Guericke-Universität Magdeburg, Institut für Mathematische Optimierung)

Abstract

Logical implications appear in a number of important mixed-integer nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, and vanishing constraints. In addition to the tightness of the respective relaxations, we also address the issue of constraint qualification and the behavior of computational methods for some formulations. As a benchmark, we formulate a truck cruise control problem with logical implications resulting from gear-choice specific constraints. We provide this benchmark problem in AMPL format along with different realistic scenarios. Computational results for this benchmark are used to investigate feasibility gaps, integer feasibility gaps, quality of local solutions, and well-behavedness of the presented reformulations of the benchmark problem. Vanishing constraints give the most satisfactory results.

Suggested Citation

  • Michael N. Jung & Christian Kirches & Sebastian Sager, 2013. "On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control," Springer Books, in: Michael Jünger & Gerhard Reinelt (ed.), Facets of Combinatorial Optimization, edition 127, pages 387-417, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-38189-8_16
    DOI: 10.1007/978-3-642-38189-8_16
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