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Global Solution of Optimization Problems with Parameter-Embedded Linear Dynamic Systems

Author

Listed:
  • A. B. Singer

    (Massachusetts Institute of Technology)

  • P. I. Barton

    (Massachusetts Institute of Technology)

Abstract

This paper develops a theory for the global solution of nonconvex optimization problems with parameter-embedded linear dynamic systems. A quite general problem formulation is introduced and a solution is shown to exists. A convexity theory for integrals is then developed to construct convex relaxations for utilization in a branch-and-bound framework to calculate a global minimum. Interval analysis is employed to generate bounds on the state variables implied by the bounds on the embedded parameters. These bounds, along with basic integration theory, are used to prove convergence of the branch-and-bound algorithm to the global minimum of the optimization problem. The implementation of the algorithm is then considered and several numerical case studies are examined thoroughly

Suggested Citation

  • A. B. Singer & P. I. Barton, 2004. "Global Solution of Optimization Problems with Parameter-Embedded Linear Dynamic Systems," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 613-646, June.
    • Handle: RePEc:spr:joptap:v:121:y:2004:i:3:d:10.1023_b:jota.0000037606.79050.a7
    DOI: 10.1023/B:JOTA.0000037606.79050.a7
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    References listed on IDEAS

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    1. Carver, M.B., 1978. "Efficient integration over discontinuities in ordinary differential equation simulations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 20(3), pages 190-196.
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    Cited by:

    1. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
    2. Mario Villanueva & Boris Houska & Benoît Chachuat, 2015. "Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs," Journal of Global Optimization, Springer, vol. 62(3), pages 575-613, July.
    3. Joseph Scott & Paul Barton, 2013. "Improved relaxations for the parametric solutions of ODEs using differential inequalities," Journal of Global Optimization, Springer, vol. 57(1), pages 143-176, September.
    4. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    5. Joseph K. Scott & Paul I. Barton, 2013. "Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 617-649, March.
    6. Chrysoula D. Kappatou & Dominik Bongartz & Jaromił Najman & Susanne Sass & Alexander Mitsos, 2022. "Global dynamic optimization with Hammerstein–Wiener models embedded," Journal of Global Optimization, Springer, vol. 84(2), pages 321-347, October.

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    1. Ellison, D., 1981. "Efficient automatic integration of ordinary differential equations with discontinuities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 23(1), pages 12-20.

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