Theorem 8, p. 14 — g(Ax) with g strongly convex and smooth is in qS(X) with L_f = L_g‖A‖², κ_f = σ_g/θ²(A, C)

Let $X=\{x\in\mathbb R^n: Cx\le d\}$ be a polyhedral set, $g:\mathbb R^m\to\mathbb R$ differentiable and $\sigma_g$-strongly convex ($\sigma_g>0$) with $L_g$-Lipschitz continuous gradient, and $A\in\mathbb R^{m\times n}$ a nonzero matrix. Assume problem (38), $\min_{x\in X} g(Ax)$, has an optimal point $x^*$, and let $\theta(A,C)>0$ be a Hoffman constant for the polyhedral optimal set $\{x: Ax=Ax^*,\ Cx\le d\}$. Then $f(x)=g(Ax)$ belongs to the class $q\mathcal S_{L_f,\kappa_f}(X)$ with

that is, $f$ is convex on $X$, $\|\nabla f(x)-\nabla f(y)\|\le L_f\|x-y\|$ for $x,y\in X$, and for every $x\in X$ and $\bar x=[x]_{X^*}$

The theorem shows that strong convexity of $g$ survives composition with a rank-deficient linear map and restriction to a polyhedron in the weaker quasi-strong form, with a constant governed by Hoffman's constant. It is the main example of a non-strongly convex class on which the paper's first order methods converge linearly.

Formalization Note $\|A\|$ is the operator (spectral) norm. Strong convexity is Mathlib's StrongConvexOn on all of $\mathbb R^m$, which is (5). The page's "$L_g$-Lipschitz continuous gradient on $X$" is read on $\mathbb R^m$, the domain of $g$. The Hoffman constant is a hypothesis (its existence is Hoffman's theorem, cited by the page); it is stated for the polyhedron written with $Ax^*$, which the proof identifies with $X^*$. The page's "simple" set and "closed" function add no mathematical hypothesis here.