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

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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.

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Authors and Affiliations

  1. Université de la Nouvelle-Calédonie, ERIM, BP. R4, 98851, Nouméa Cédex, New Caledonia

    Henri Bonnel

  2. School of Mathematics and Statistics, University of South Australia, Mawson Lakes, SA, 5095, Australia

    C. Yalçın Kaya

Authors
  1. Henri Bonnel
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  2. C. Yalçın Kaya
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Correspondence to C. Yalçın Kaya.

Additional information

Communicated by H.P. Benson.

The authors would like to thank the Associate Editor and a Co-editor for their comments and suggestions which improved the exposition of the paper.

C. Yalçın Kaya is grateful to Helmut Maurer for fruitful discussions which helped to detail the use of solution differentiability results and sensitivity derivatives.

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Bonnel, H., Yalçın Kaya, C. Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems. J Optim Theory Appl 147, 93–112 (2010). https://doi.org/10.1007/s10957-010-9709-y

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