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

We consider the misspecified optimization problem of minimizing a convex function $f(x;{\theta }^{\ast })$ in $x$ over a conic constraint set represented by $h(x;{\theta }^{\ast })\in \mathcal{K}$, where ${\theta }^{\ast }$ is an unknown (or misspecified) vector of parameters, $\mathcal{K}$ is a closed convex cone and $h$ is affine in $x$. Suppose ${\theta }^{\ast }$ 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 ${\theta }^{\ast }$. We develop a first-order inexact augmented Lagrangian (AL) scheme for computing an optimal solution $x^{\ast }$ corresponding to ${\theta }^{\ast }$ while simultaneously learning ${\theta }^{\ast }$. 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.