Shadow price of an inequality constraint ($\partial V/\partial b_i=-{\lambda }_i$)

A shadow price measures the marginal improvement from relaxing an inequality optimization constraint. Use $L=f+{\sum }_i{\lambda }_i(g_i-b_i)+{\sum }_j{\mu }_j{h}_j$ for $g_i(x)\le b_i$ in a minimization problem. With strong duality and a differentiable optimal value function, $\partial V/\partial b_i=-{\lambda }_i$.^[ref]

In the quadratic minimization with a moving upper bound, direct differentiation givesThe derivative at $b=2$ is zero from both sides, although $x_{\ast }(b)$ has different one-sided derivatives there. The value of additional room has faded continuously to zero just as the active constraint stops restricting the optimizer.

At $b=1$, $x_{\ast }=1$, $V=1$ and ${\lambda }_{\ast }=2$. Raising the bound to $1.1$ lowers the value to $0.81$: the first-order prediction is a change of $-2(0.1)=-0.2$, while the exact change is $-0.19$. At $b=3$, the same relaxation leaves $x_{\ast }=2$ and $V=0$. At $b=2$, tightening by $0.1$ costs $0.01$, a second-order effect despite the zero Lagrange multiplier.

There is also a global statement in this example. Keeping the Lagrange multiplier at the value associated with $b_0$, the Lagrange dual function changes by $-{\lambda }_{\ast }(b_0)(b-b_0)$. Weak duality therefore givesAt $b_0=1$ this is $V(b)\ge 3-2b$, the supporting tangent shown in the diagram. This bound remains valid for large changes, whereas the derivative predicts only the first-order effect. If the value function is not differentiable, an optimal Lagrange multiplier can instead supply a supporting subgradient, rather than a unique marginal rate. This is the same idea used in sensitivity analysis in convex perturbation duality.