Toward the Direct Analytical Determination of the Pareto Optima of a Differentiable Mapping, I: Domains in Finite-Dimensional Spaces

Dallas, A.G.
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Department of Mathematical Sciences
The problem of locating the Pareto-optimal points of a differentiable mapping $F: {\mathcal M}^N \to {I\kern-.30em R}^n$ is studied, with the domain ${\cal M}^N$ a differentiable N-dimensional submanifold-without-boundary in a euclidean space ${I\kern-.30em R}^{N_{0}}$ and $N_0 \ge N \ge n$. The case in which the domain is the closure of a bounded, regular, open subset of ${I\kern-.30em R}^N$ is also discussed. The search is initiated from these observations: for a manifold-domain, (1) the image of any Pareto optimum lies in the boundary of the range of F; (2) a point of the boundary of the range of F that also lies in the range must be the image of a singular point of F, i.e., must appear amongst the singular values of the map. Further conditions are then needed to distinguish which of the singular values should be discarded because they belong to the interior of the range; local tests of this sort are given for the bicriterial case (n = 2). A search procedure based on the present developments can systematically determine all of the Pareto optima for sufficiently simple F. The conditions established here may be regarded as analogues of the classical ones for the determination of the global extrema of a real-valued differentiable function. The results proven are illustrated with single examples, including plots of the ranges, singular points, and singular values.