ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5144 on D5606: Subconvex real function
Subresult of R5142: Subconvex real function iff first-order Taylor approximates underestimates function globally
Vanishing gradient identifies a minimizer for differentiable subconvex real function
Formulation 0
Let $U \subseteq \mathbb{R}^D$ be a D5007: Standard open euclidean real set such that
(i) \begin{equation} U \neq \emptyset \end{equation}
(ii) $U$ is a D5623: Convex euclidean real set
(iii) $f : U \to \mathbb{R}$ is a D5606: Subconvex real function
(iv) $f : U \to \mathbb{R}$ is a D5614: Differentiable real function
(v) \begin{equation} x \in U \end{equation}
(vi) \begin{equation} \nabla f(x) = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \end{equation}
Then \begin{equation} \forall \, y \in U : f(x) \leq f(y) \end{equation}
Proofs
Proof 0
Let $U \subseteq \mathbb{R}^D$ be a D5007: Standard open euclidean real set such that
(i) \begin{equation} U \neq \emptyset \end{equation}
(ii) $U$ is a D5623: Convex euclidean real set
(iii) $f : U \to \mathbb{R}$ is a D5606: Subconvex real function
(iv) $f : U \to \mathbb{R}$ is a D5614: Differentiable real function
(v) \begin{equation} x \in U \end{equation}
(vi) \begin{equation} \nabla f(x) = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \end{equation}