 Strong Lagrangian property

For a finite constrained optimum value $\varphi (b)=\inf \{f(x):h(x)=b,x\in X\}$ and optimization Lagrangian $L_b(x,\lambda )=f(x)-{\lambda }^T(h(x)-b)$, the strong Lagrangian property means that some finite Lagrange multiplier attains the dual lower bound exactly: ${\inf }_X{L}_b=\varphi (b)$. It includes dual attainment, but need not include primal attainment. It is equivalent to a non-vertical supporting hyperplane of a value function: a supporting slope gives $f(x)\ge \varphi (h(x))\ge \varphi (b)+{\lambda }^T(h(x)-b)$, hence ${\inf }_X{L}_b\ge \varphi (b)$; weak duality gives the opposite inequality. Conversely the infimum identity bounds every $L_b(x,\lambda )$ below by $\varphi (b)$; taking the infimum over $h(x)=u$ gives the supporting inequality. No convexity assumption is needed for this equivalence.