Penalty relaxation feasibility bound for convex constrained optimization

Let $f$ and $g$ define a convex constrained optimization problem of the form

$$\min_x f(x)\quad\text{subject to}\quad g(x)\leq 0.$$

For a penalty parameter $\Lambda$, consider the penalty relaxation

$$\min_x f(x)+\Lambda [g(x)]_+,$$

where $[g(x)]_+$ denotes the positive part of $g(x)$, and let $\Delta(\delta)$ denote the quantity appearing in the source's Slater-type bound. Penalty relaxation feasibility conjecture. The penalty relaxation can guarantee $\delta$-feasibility whenever $\delta$ solves

$$\delta\geq\frac{\Delta(\delta)}{\Lambda}.$$

This is presented as following from a Slater-type argument, but the source does not provide enough surrounding definitions to determine the precise meaning of $\Delta(\delta)$ or whether the claim has been established as a theorem.