Abstract
This paper considers the connections between the local extrema of a function f:D→R and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C 1 or C 2-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C 1 or C 2-curves. Surprisingly, what is true for C 1-curves fails to be true in part for C 2-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.
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Dogaru, O., Ţevy, I. & Udrişte, C. Extrema Constrained by a Family of Curves and Local Extrema. Journal of Optimization Theory and Applications 97, 605–621 (1998). https://doi.org/10.1023/A:1022642126176
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DOI: https://doi.org/10.1023/A:1022642126176