 Solution

Suppose the non-vertical supporting hyperplane of a value function has slope $\lambda$. Every $x\in X$ then satisfiesso $L_b(x,\lambda )\ge \varphi (b)$. Taking the infimum gives ${\inf }_X{L}_b\ge \varphi (b)$. Conversely, feasible points approaching the infimum of $f$ have $L_b=f$, so ${\inf }_X{L}_b\le \varphi (b)$ even if no primal minimizer exists. Thereforewhich is the Strong Lagrangian property.