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Constrained Optimization Involving Expensive Function Evaluations : A Sequential Approach

Author

Listed:
  • Brekelmans, R.C.M.

    (Tilburg University, Center For Economic Research)

  • Driessen, L.
  • Hamers, H.J.M.

    (Tilburg University, Center For Economic Research)

  • den Hertog, D.

    (Tilburg University, Center For Economic Research)

Abstract

This paper presents a new sequential method for constrained non-linear optimization problems.The principal characteristics of these problems are very time consuming function evaluations and the absence of derivative information. Such problems are common in design optimization, where time consuming function evaluations are carried out by simulation tools (e.g., FEM, CFD).Classical optimization methods, based on derivatives, are not applicable because often derivative information is not available and is too expensive to approximate through finite differencing.The algorithm first creates an experimental design. In the design points the underlying functions are evaluated.Local linear approximations of the real model are obtained with help of weighted regression techniques.The approximating model is then optimized within a trust region to find the best feasible objective improving point.This trust region moves along the most promising direction, which is determined on the basis of the evaluated objective values and constraint violations combined in a filter criterion.If the geometry of the points that determine the local approximations becomes bad, i.e. the points are located in such a way that they result in a bad approximation of the actual model, then we evaluate a geometry improving instead of an objective improving point.In each iteration a new local linear approximation is built, and either a new point is evaluated (objective or geometry improving) or the trust region is decreased.Convergence of the algorithm is guided by the size of this trust region.The focus of the approach is on getting good solutions with a limited number of function evaluations (not necessarily on reaching high accuracy).
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Suggested Citation

  • Brekelmans, R.C.M. & Driessen, L. & Hamers, H.J.M. & den Hertog, D., 2001. "Constrained Optimization Involving Expensive Function Evaluations : A Sequential Approach," Discussion Paper 2001-87, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:c94095bf-9069-47e6-b77e-5e08c1382439
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    1. Driessen, L. & Brekelmans, R.C.M. & Hamers, H.J.M. & den Hertog, D., 2001. "On D-Optimality Based Trust Regions for Black-Box Optimization Problems," Discussion Paper 2001-69, Tilburg University, Center for Economic Research.
    2. den Hertog, Dick & Stehouwer, Peter, 2002. "Optimizing color picture tubes by high-cost nonlinear programming," European Journal of Operational Research, Elsevier, vol. 140(2), pages 197-211, July.
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    1. Driessen, L. & Brekelmans, R.C.M. & Hamers, H.J.M. & den Hertog, D., 2001. "On D-Optimality Based Trust Regions for Black-Box Optimization Problems," Other publications TiSEM dcde83b3-cab6-48bd-b94d-7, Tilburg University, School of Economics and Management.
    2. Driessen, L. & Brekelmans, R.C.M. & Gerichhausen, M. & Hamers, H.J.M. & den Hertog, D., 2006. "Why Methods for Optimization Problems with Time-Consuming Function Evaluations and Integer Variables Should Use Global Approximation Models," Other publications TiSEM 45a73d28-9fed-4b4c-a909-1, Tilburg University, School of Economics and Management.
    3. ten Eikelder, S.C.M. & van Amerongen, J.H.M., 2023. "Resource allocation problems with expensive function evaluations," European Journal of Operational Research, Elsevier, vol. 306(3), pages 1170-1185.
    4. Regis, Rommel G. & Shoemaker, Christine A., 2007. "Parallel radial basis function methods for the global optimization of expensive functions," European Journal of Operational Research, Elsevier, vol. 182(2), pages 514-535, October.
    5. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    6. Miriyala, Srinivas Soumitri & Subramanian, Venkat & Mitra, Kishalay, 2018. "TRANSFORM-ANN for online optimization of complex industrial processes: Casting process as case study," European Journal of Operational Research, Elsevier, vol. 264(1), pages 294-309.

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    More about this item

    Keywords

    optimization; nonlinear programming;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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