Abstract
Let U be an open set in ℝn; this means that to every point u ∈ U there corresponds a ball N(u) = {y∈ℝn: ‖y-u‖ < γ(u)}, for some γ(u) > 0, such that N(u) ⊂ U. This concept is illustrated in Fig. 3.1. Let f: U → ℝ be a (Fréchet) differentiable function. (Note that the definition of f’(u) at u ∈ U requires that the domain of f contains some ball N(u). An open set U, from the definition, does not contain the points of its boundary.)
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Further Reading
Bazaraa, M.S. and Shetty, C.M. (1976), Foundations of Optimization, Springer-Verlag, Berlin. (Vol. 122 of Lecture Notes in Economics and Mathematical Systems). [For more general versions of the Kuhn-Tucker theorem.]
Ben-Israel, A., Ben-Tal, A. and Zlobec, S. (1976), Optimality conditions in convex programming, IX International Symposium on Mathematical Programming (Budapest, 1976), A. Prekopa (ed.), North-Holland, Amsterdam (1979), pp. 177–92. [For the last example in Section 3.5.]
Craven, B.D. (1978), Mathematical Programming and Control Theory, Chapman and Hall, London. [For more general constrained minimization theory, no longer restricted to finite dimensions.]
Craven, B.D. (1979), On constrained maxima and minima, Austral Math. Soc. Gazette, 6 (2), 46–50. [For the discrimination of constrained stationary points.]
Hancock, H. (1917), Theory of Maxima and Minima (reprinted in 1960 by Dover, New York). [A classic account.]
Kuhn, H.W. and Tucker, A.W. (1951), Nonlinear programming, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, J. Neyman (ed.), University of California Press, Berkeley, pp. 481–92. [For the original version of the Kuhn-Tucker theorem.]
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© 1981 B.D. Craven
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Craven, B.D. (1981). Maxima and minima. In: Functions of several variables. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9347-7_3
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DOI: https://doi.org/10.1007/978-94-010-9347-7_3
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