KKT Conditions

For ${\min }_{\sigma ,u}J(\sigma ,u)$ subject to $e(\sigma ,u)=0$, with the Lagrangian $\mathcal{L}=J+⟨\lambda ,e⟩$, a local minimiser $({\sigma }^{\ast },u^{\ast })$ satisfies, under a constraint qualification (here: the state equation is uniquely solvable for every $\sigma$), the first-order (KKT) conditions:

1. ${\partial }_{\lambda }\mathcal{L}=0$: the State Equation $e(\sigma ,u)=0$ (primal feasibility);
2. ${\partial }_u\mathcal{L}=0$: the Adjoint Equation;
3. ${\partial }_{\sigma }\mathcal{L}=0$: the gradient vanishes.

With the box constraints ${\sigma }_{\min }\le \sigma \le {\sigma }_{\max }$, condition 3 becomes a variational inequality:

$$⟨{\partial }_{\sigma }\mathcal{L},\tilde{\sigma }-{\sigma }^{\ast }⟩\ge 0\forall \tilde{\sigma }\in \Sigma .$$

Pointwise, the gradient is zero where the bound is inactive, non-negative at the lower bound and non-positive at the upper bound.

Relation to the adjoint method. An iterate is not a KKT point. The Adjoint State Method enforces conditions 1 and 2 exactly, by solving the state and adjoint equations, and evaluates the left-hand side of 3. The result equals the gradient of the reduced functional $\sigma \mapsto J(\sigma ,u(\sigma ))$, which the optimiser then drives to zero.

References

1. J. Nocedal, S. J. Wright (2006). Numerical Optimization, 2nd ed. Springer. doi:10.1007/978-0-387-40065-5
2. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1