Coordinate-block smoothness descent model

Given: The function, block and Lipschitz bound in the Statement.

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Laws of finite sums and finite products: identify block label $j\in \{1,\dots ,d\}$ with standard coordinate $j-1$ and order each block increasingly; then ${\mathrm{\nabla }}_{i}f(x)$ is the corresponding restriction of $\mathrm{\nabla }f(x)$ and $⟨\mathrm{\nabla }f(x),U_{i}h⟩=⟨{\mathrm{\nabla }}_{i}f(x),h⟩$ by the finite-coordinate definition of the Euclidean inner product and finite-sum splitting.

Continuously differentiable maps, local inverses, and local diffeomorphisms: continuously differentiable means totally differentiable at every point, with continuous derivative entries.

The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$: along a differentiable curve, the derivative of the composition is the derivative of the outer function applied to the curve derivative.

For a differentiable scalar field, $D_{v}f(a)=⟨\mathrm{\nabla }f(a),v⟩$ and the unit direction of steepest ascent is the normalized gradient: for a totally differentiable scalar function, the directional derivative in direction $v$ is $⟨\mathrm{\nabla }f,v⟩$.

Total differentiability gives a local $O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ increment bound and therefore continuity: a totally differentiable function is continuous at each point.

The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$: if a real function is continuous on a closed interval and differentiable in its interior, its endpoint difference is its derivative at an interior point times the interval length.

Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2$.

The principle of mathematical induction: an identity that holds at zero and is preserved at successors holds for every natural number.

Finite sums and finite products, by recursion and Laws of finite sums and finite products: finite sums satisfy the zero base case, successor recursion, additivity, telescoping, scaling, and the constant-sum law.

For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$: for every $\eta >0$ there is an integer $N\ge 1$ with $1\mathrm{/}N<\eta$.

Every natural-number-indexed list of nonempty sets has a choice function on its family of values: a finite indexed list of nonempty witness sets has a choice function in ZF.