  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 $\theta^*$.
