# Convergence of KKT equations for discrete parameter estimation problems

Consider a discrete constrained optimization problem:

$$\mathbf{q}_*^h= \arg \min {\cal J}^h(\mathbf{x}^h[\mathbf{q}^h],\mathbf{q}^h)$$ subject to the (weak-form) constraint $$F^h[\mathbf{x}^h(\mathbf{q}^h),\mathbf{q}^h](\mathbf{w}^h) = 0 \; \forall \, \mathbf{w}^h$$ with $\mathbf{x}^h \in \Omega^h$ (the discrete domain) and $\mathbf{w}^h$ the discrete test functions. We can assume $F^h$ is a consistent and stable discretization of a corresponding continuous problem on the discrete domain $\Omega^h$. The first order necessary conditions that define our local minimum $\mathbf{x}^*$, with $\lambda_*^h$ the discrete adjoint variable (or Lagrange multiplier) are:

$$F^h[\mathbf{x}^h,\mathbf{q}_*^h](\mathbf{w}^h) = 0 \;\; \forall \, \mathbf{w}^h \\ F_{\mathbf{x}}^h[\mathbf{x}^h,\mathbf{q}_*^h](\mathbf{w}^h,\lambda_*^h) = {\cal J}_\mathbf{x}[\mathbf{x}^h,\mathbf{q}_*^h] \;\; \forall \, \mathbf{w}^h \\ F_{\mathbf{q}}^h[\mathbf{x}^h,\mathbf{q}_*^h](\mathbf{w}^h,\lambda_*^h) = {\cal J}_\mathbf{q}[\mathbf{x}^h,\mathbf{q}_*^h] \;\; \forall \, \mathbf{w}^h$$

How sensitive is the discrete optimum $\mathbf{q}^h_*$ to consistent perturbations to the primal problem? By consistent I mean a discrete perturbation that vanishes when evaluated at the exact (continuous) KKT triplet $(\mathbf{u},\lambda,\mathbf{q})$:

$$F^h[\mathbf{x}^h(\mathbf{q}^h),\mathbf{q}^h](\mathbf{w}^h) + \Delta^h [\mathbf{x}^h,\mathbf{q}^h](\mathbf{w}^h) = 0 \; \forall \, \mathbf{w}^h$$

What bounds do we need to prove on $\Delta^h$ (or its derivative) to guarantee the perturbed KKT system still converges to $(\mathbf x_*^h,\mathbf q_*^h)$? I'm aware the answer may well be problem dependent, so I'd welcome some references to existing literature for discrete model problems (linear diffusion, etc.).