On the rate analysis of inexact augmented Lagrangian schemes for convex optimization problems with misspecified constraints

We consider a misspecified optimization problem that requires minimizing of a convex function $f(x;\theta^*)$ in x over a constraint set represented by $h(x;\theta^*)\leq 0$, where $\theta^*$ is an unknown (or misspecified) vector of parameters. Suppose $\theta^*$ can be learnt by a distinct process that generates a sequence of estimators $\theta_k$, each of which is an increasingly accurate approximation of $\theta^*$. We develop a first-order augmented Lagrangian scheme for computing an optimal solution $x^*$ while simultaneously learning $\theta^*$.