Block Lipschitz constant from coordinate smoothness ($L_{\mathcal{B}}\le {\sum }_{i\in \mathcal{B}}{L}_i$)

For a continuously differentiable convex function on ${\mathbb{R}}^n$ with coordinate Lipschitz continuity constants $L_i$, its block coordinate gradient is Lipschitz continuous along each corresponding coordinate subspace, with constant at most $S_{\mathcal{B}}={\sum }_{i\in \mathcal{B}}{L}_i$.

To prove this without assuming a Hessian matrix, put ${\delta }_{\mathcal{B}}={\nabla }_{\mathcal{B}}f(x)-{\nabla }_{\mathcal{B}}f(y)$. Sum the coordinate derivative bound by Bregman divergence over the block, then repeat with $x,y$ interchanged and average:If $x-y$ is supported in the block, the dot product uses only ${\delta }_{\mathcal{B}}$. The Cauchy-Schwarz inequality and cancellation of its Euclidean norm give the result, including the zero case separately. Taking the full block supplies a global Lipschitz gradient constant ${\sum }_i{L}_i$.

Target: convex-optimization.bigb, under Coordinate descent.