On the analysis of inexact augmented Lagrangian schemes for misspecified conic convex programs

We consider the misspecified optimization problem of minimizing a convex function $f(x;\theta^*)$ in $x$ over a conic constraint set represented by $h(x;\theta^*) \in \mathcal{K}$, where $\theta^*$ is an unknown (or misspecified) vector of parameters, $\mathcal{K}$ is a closed convex cone and $h$ is affine in $x$. Suppose $\theta^*$ is unavailable but may be learnt by a separate process that generates a sequence of estimators $\theta_k$, each of which is an increasingly accurate approximation of $\theta^*$. We develop a first-order inexact augmented Lagrangian (AL) scheme for computing an optimal solution $x^*$ corresponding to $\theta^*$ while simultaneously learning $\theta^*$. In particular, we derive rate statements for such schemes when the penalty parameter sequence is either constant or increasing, and derive bounds on the overall complexity in terms of proximal-gradient steps when AL subproblems are inexactly solved via an accelerated proximal-gradient scheme. Numerical results for a portfolio optimization problem with a misspecified covariance matrix suggest that these schemes perform well in practice while naive sequential schemes may perform poorly in comparison.