 Solution

If the Strong Lagrangian property holds with multiplier $\lambda$, its infimum identity implies, for every $x\in X$,For any feasible right-hand side $u$, take the infimum over $h(x)=u$ to obtainThe inequality also holds at infeasible $u$, where $\varphi (u)=+\infty$, and it is equality at $u=b$. Thus $r=\varphi (b)+{\lambda }^T(u-b)$ is the required non-vertical supporting hyperplane of a value function. This proves the converse and completes the equivalence. Neither implication requires that $X$ be a convex set.