= Strong Lagrangian property
{c}
{title2=$\phi(b)=\inf_{x\in X}L_b(x,\lambda)$}
= Strong Lagrangian principle
{c}
{synonym}
For a finite constrained optimum value $\phi(b)=\inf\{f(x):h(x)=b,x\in X\}$ 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: $\inf_XL_b=\phi(b)$. 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 $f(x)\geq\phi(h(x))\geq\phi(b)+\lambda^T(h(x)-b)$, hence $\inf_XL_b\geq\phi(b)$; <weak duality> gives the opposite inequality. Conversely the infimum identity bounds every $L_b(x,\lambda)$ below by $\phi(b)$; taking the infimum over $h(x)=u$ gives the supporting inequality. No convexity assumption is needed for this equivalence.
Back to article page
