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

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...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2010-10, Vol.147 (1), p.93-112
Main Authors: Bonnel, Henri, Yalçın Kaya, C.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Applied sciences
Calculus of variations and optimal control
Calculus of Variations and Optimal Control
Optimization
Construction
Control theory
Decision making
Decision theory. Utility theory
Engineering
Exact sciences and technology
Hilbert space
Mathematical analysis
Mathematical functions
Mathematical models
Mathematics
Mathematics and Statistics
Numerical analysis
Operational research and scientific management
Operational research. Management science
Operations Research/Decision Theory
Optimal control
Optimization
Pareto optimum
Sciences and techniques of general use
Studies
Theory of Computation
Vectors (mathematics)
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Summary: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.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-010-9709-y