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Finite Dimensions Necessary and Sufficient Conditions: Global extrema in finite dimensions

Theorem (First order necessary conditions): Let \((x^{i;*})\) be an interior point in \(\mathcal{D}\) at which \(f: \mathcal{D}\subseteq \mathbb{R}^n \to \mathbb{R}\) has partial derivatives \(D_{,i}f\) (that are continuous in a neighborhood of \((x^{i;*})\)). A necessary condition for \((x^{i;*})\) to be a minimizer or maximizer is that \(D_{,i}f[(x^{i;*})]=0\).

Theorem (Extreme value theorem): Let \(f\) be a continuous function on a closed and bounded set \(\overline{\mathcal{D}}\subset \mathbb{R}^n\). Then \(f\) has both a maximum and minimium point in \(\overline{\mathcal{D}}\).

The above theorem is also called Weistrass theorem.

First order necessary and sufficiency conditions for a global minimizer.

Theorem Suppose that the function \(f\) is defined on a convex set \(\mathcal{D}\subset \mathbb{R}^n\) and let \((x^{i;*})\in \text{int}(\mathcal{D})\). Assume that \(f\) is \(C^1\) on a (open) ball around \((x^{i;*})\). If \(f\) is convex in \(\mathcal{D}\), then \((x^{i;*})\) is a (global) minimizer of \(f\) in \(\mathcal{D}\) if and only if \((x^{i;*})\) is a stationary point of \(f\).

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