Local minimum

From DDWiki

In an optimization problem of the form

maximize	$f(\mathbf{x})$
with respect to	$\mathbf{x}$
subject to	$\mathbf{g(x) \leq 0}$
	$\mathbf{h(x)=0}$
	$\mathbf{x}\in\mathcal{X}$

a local minimum is conceptually the objective function value of a feasible point that is not inferior to any of its immediate feasible neighbors. If the problem is continuous ( $\mathcal{X}=\Re^n$ ), a local minimizer is a point $\mathbf{x}^*$ such that $\exists \epsilon > 0 : f(\mathbf{x + \partial x}) \geq f(\mathbf{x}^*);$ $\forall \mathbf{\partial x}: ||\mathbf{\partial x}|| \leq \epsilon$ . The objective function value $f(\mathbf{x}^*)$ of the local minimizer is called the local minimum.