Local Minimum

tags:definition |variational

Local Minimum

Context

Statements

Definition. (Citation Needed) Let $\mathbf{V}$ be a vector space of functions equipped with a norm $||\cdot ||$, let $\mathbf{A}$ be a subset of $\mathbf{V}$, and let $J$ be a real-valued Functional defined on $\mathbf{A}$. A function $y^{\ast }\in \mathbf{A}$ is a local minimum of $J$ over $\mathbf{A}$ if there exists $\epsilon >0$ such that the inequality

$$J(y^{\ast })\le J(y)$$

holds, for every $y\in \mathbf{A}$ satisfying $||y-y^{\ast }||<\epsilon$.

References

Cite Liberzon’s Book