Abstract
Let ℝn be the n-dimensional Euclidean space with the usual inner product 〈·, ·〉, 〈x, y〉 = x ⊤ y, and norm \(\left\| x \right\| = \sqrt {\left\langle {x,x} \right\rangle } \). Let M ⊂ ℝn, M ≠ ∅ and f : M → ℝ a function. An \(\bar x \in M\) M having the property: \(f\left( {\bar x} \right)f(x)\) holds for all x ∈ M, is called a global minimum for f . If \(f\left( {\bar x} \right) \leqslant f(x)\) holds for all x ∈ M ∩ O, O being a ℝn-neighborhood of \(\bar x\), then \(\bar x\) is called a local minimum for f. Obviously, a global minimum is also a local minimum. Local (global) maxima of f are defined to be local (global) minima for the function -f.
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© 2001 Springer Science+Business Media Dordrecht
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Jongen, H.T., Jonker, P., Twilt, F. (2001). Introduction. In: Nonlinear Optimization in Finite Dimensions. Nonconvex Optimization and Its Applications, vol 47. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0017-9_1
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DOI: https://doi.org/10.1007/978-1-4615-0017-9_1
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