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.).



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