3.6 Optimality Conditions and Lagrange Multipliers

The gradient must respect feasible directions¶

For an unconstrained differentiable problem, a local optimum usually satisfies $\nabla J(z^*)=0$. With constraints, the objective may still decrease in an infeasible direction. Optimality instead requires that no feasible first-order direction improves the objective.

Define the Lagrangian

where $\lambda$ and $\mu$ are Lagrange multipliers. Under standard regularity assumptions, the Karush–Kuhn–Tucker conditions include

Complementarity means an inactive inequality has zero multiplier; only an active limit can exert first-order influence on the optimum.

Multipliers as sensitivity information¶

Suppose a constraint is written $q(z)\le q_{\max}$. Its multiplier estimates how the optimal objective changes when the limit is relaxed slightly. A large displacement multiplier says that additional suspension travel is valuable. A small actuator-force multiplier says that increasing actuator rating offers little immediate benefit at the current optimum.

Multiplier magnitude depends on scaling and sign convention. Interpret it only after checking units, formulation, numerical convergence, and whether constraint qualifications hold.

Why CCD needs this interpretation¶

CCD often improves performance by reallocating scarce resources between the plant and controller. Multipliers identify those scarce resources. They also help explain why a design changed, prioritize model refinement around active constraints, and reveal whether an expensive component is actually limiting performance.

Activity 3.6: interpret shadow prices¶