PŁ-like gradient-mapping relation for trust-region optimisation

Let $E$ be the objective function, let $\boldsymbol{s}^{(k)}$ be the iterate at iteration $k$, let $E^*$ denote the optimal objective value, and let $\boldsymbol{g}_{\beta_k}^{(k)}$ be the gradient mapping vector. Suppose Assumption holds, with convex feasible set $\mathscr{C}$ and parameter $\mu$. PŁ-like relation. There exists a constant $\mu_p$ satisfying $0<\mu_p\leq\mu$ such that, for every $\boldsymbol{s}\in\mathscr{C}$ and every $\beta\in\mathbb{R}^+$,

$$\lVert \boldsymbol{g}_{\beta_k}^{(k)}\rVert_2^2\geq 2\mu_p\bigl(E(\boldsymbol{s}^{(k)})-E^*\bigr).$$

This relation is proposed as a projected-gradient analogue of the Polyak–Łojasiewicz inequality, connecting the gradient-mapping norm to the suboptimality gap in the trust-region iteration. Its status is not established in the supplied text.