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Optimization-Constrained Differential Equations with Active Set Changes

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  • Peter Stechlinski

    (University of Maine)

Abstract

Foundational theory is established for nonlinear differential equations with embedded nonlinear optimization problems exhibiting active set changes. Existence, uniqueness, and continuation of solutions are shown, followed by lexicographically smooth (implying Lipschitzian) parametric dependence. The sensitivity theory found here accurately characterizes sensitivity jumps resulting from active set changes via an auxiliary nonsmooth sensitivity system obtained by lexicographic directional differentiation. The results in this article hold under easily verifiable regularity conditions (linear independence of constraints and strong second-order sufficiency), which are shown to imply generalized differentiation index one of a nonsmooth differential-algebraic equation system obtained by replacing the optimization problem with its optimality conditions and recasting the complementarity conditions as nonsmooth algebraic equations. The theory in this article is computationally relevant, allowing for implementation of dynamic optimization strategies (i.e., open-loop optimal control), and recovers (and rigorously formalizes) classical results in the absence of active set changes. Along the way, contributions are made to the theory of piecewise differentiable functions.

Suggested Citation

  • Peter Stechlinski, 2020. "Optimization-Constrained Differential Equations with Active Set Changes," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 266-293, October.
    • Handle: RePEc:spr:joptap:v:187:y:2020:i:1:d:10.1007_s10957-020-01744-4
    DOI: 10.1007/s10957-020-01744-4
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2005. "Lexicographic differentiation of nonsmooth functions," LIDAM Reprints CORE 1817, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. A. Caboussat & C. Landry & J. Rappaz, 2010. "Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 141-156, October.
    3. Kamil A. Khan & Paul I. Barton, 2014. "Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 355-386, November.
    4. N. R. Amundson & A. Caboussat & J. W. He & J. H. Seinfeld, 2006. "Primal-Dual Interior-Point Method for an Optimization Problem Related to the Modeling of Atmospheric Organic Aerosols," Journal of Optimization Theory and Applications, Springer, vol. 130(3), pages 377-409, September.
    5. Peter G. Stechlinski & Paul I. Barton, 2016. "Generalized Derivatives of Differential–Algebraic Equations," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 1-26, October.
    Full references (including those not matched with items on IDEAS)

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