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Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems

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  • Henri Bonnel

    (Université de la Nouvelle-Calédonie, ERIM)

  • C. Yalçın Kaya

    (University of South Australia)

Abstract

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach.

Suggested Citation

  • Henri Bonnel & C. Yalçın Kaya, 2010. "Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 93-112, October.
    • Handle: RePEc:spr:joptap:v:147:y:2010:i:1:d:10.1007_s10957-010-9709-y
    DOI: 10.1007/s10957-010-9709-y
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    References listed on IDEAS

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    1. C.J. Price & I.D. Coope & D. Byatt, 2002. "A Convergent Variant of the Nelder–Mead Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 5-19, April.
    2. R. Horst & N. V. Thoai & Y. Yamamoto & D. Zenke, 2007. "On Optimization over the Efficient Set in Linear Multicriteria Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 433-443, September.
    3. Horst, Reiner & Thoai, Nguyen V., 1999. "Maximizing a concave function over the efficient or weakly-efficient set," European Journal of Operational Research, Elsevier, vol. 117(2), pages 239-252, September.
    4. Regina Burachik & C. Kaya & Musa Mammadov, 2010. "An inexact modified subgradient algorithm for nonconvex optimization," Computational Optimization and Applications, Springer, vol. 45(1), pages 1-24, January.
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    Citations

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    Cited by:

    1. Henri Bonnel & Léonard Todjihoundé & Constantin Udrişte, 2015. "Semivectorial Bilevel Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 464-486, November.
    2. C. Yalçın Kaya, 2020. "Optimal Control of the Double Integrator with Minimum Total Variation," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 966-981, June.
    3. C. Kaya & Helmut Maurer, 2014. "A numerical method for nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 685-702, April.
    4. R. S. Burachik & C. Y. Kaya & M. M. Rizvi, 2014. "A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 428-446, August.
    5. Henri Bonnel & Christopher Schneider, 2019. "Post-Pareto Analysis and a New Algorithm for the Optimal Parameter Tuning of the Elastic Net," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 993-1027, December.
    6. Gokhan Kirlik & Serpil Sayın, 2015. "Computing the nadir point for multiobjective discrete optimization problems," Journal of Global Optimization, Springer, vol. 62(1), pages 79-99, May.

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