Characteristic of Optimal Points for Unconstraind Optimization

Characteristic of Optimal Points for Unconstraind Optimization

Unconstrained Optimization for smooth function $f(x)$

Taylor expansion plays an important role in unconstrained optimization

first order necessary condition:

if $x ^ *$ is local minimizer and continuously differentiable in an open neighborhood -> $\nabla f(x ^ *)$ = 0

second order necessary condition:

if $x ^ *$ is local minimizer and $\nabla ^ 2 f(x)$ exists and is continuous in an open neighborhood => $\nabla f(x ^ *)$ = 0 and $\nabla ^ 2 f(x ^ *)$ is positive semidefinite.

sufficient condition

if $\nabla ^ 2 f(x)$ is continuous in an open neighborhood of $x$, $\nabla f(x) = 0$ and $\nabla ^ 2 f(x)$ is positive definite => x is a strict local minimizer.

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