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Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

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Abstract

The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints induce discontinuity points in the time evolution. A numerical method based on an operator splitting scheme and a fixed point algorithm is advocated. The ordinary differential equations are approximated by the Crank-Nicolson scheme, while a primal-dual interior-point method with warm-starts is used to solve the minimization problem. The computation of the discontinuity points is based on geometric arguments, extrapolation polynomials and sensitivity analysis. Second order convergence of the method is proved when an inequality constraint is activated. Numerical results for atmospheric particles confirm the theoretical investigations.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX, 77204-3008, USA

    A. Caboussat

  2. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    C. Landry

  3. Institute of Analysis and Scientific Computing, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    J. Rappaz

Authors
  1. A. Caboussat
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  2. C. Landry
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  3. J. Rappaz
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Corresponding author

Correspondence to A. Caboussat.

Additional information

Communicated by R. Glowinski.

This work was partially supported by US Environmental Protection Grant X-83234201. The first author gratefully acknowledges the support of the Institute of Analysis and Scientific Computing (EPFL); part of this work has been achieved during his sabbatical leave in 2009–2010 at EPFL. The authors thank Prof. Ernst Hairer (University of Geneva) and Prof. Jiwen He (University of Houston) for fruitful discussions.

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Caboussat, A., Landry, C. & Rappaz, J. Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection. J Optim Theory Appl 147, 141–156 (2010). https://doi.org/10.1007/s10957-010-9714-1

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  • DOI: https://doi.org/10.1007/s10957-010-9714-1

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