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We consider the misspecified optimization problem of minimizing a convexfunction $f(x;\theta^*)$ in $x$ over a conic constraint set represented by$h(x;\theta^*) \in \mathcal{K}$, where $\theta^*$ is an unknown (ormisspecified) 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 aseparate process that generates a sequence of estimators $\theta_k$, each ofwhich is an increasingly accurate approximation of $\theta^*$. We develop afirst-order inexact augmented Lagrangian (AL) scheme for computing an optimalsolution $x^*$ corresponding to $\theta^*$ while simultaneously learning$\theta^*$. In particular, we derive rate statements for such schemes when thepenalty parameter sequence is either constant or increasing, and derive boundson the overall complexity in terms of proximal-gradient steps when ALsubproblems are inexactly solved via an accelerated proximal-gradient scheme.Numerical results for a portfolio optimization problem with a misspecifiedcovariance matrix suggest that these schemes perform well in practice whilenaive sequential schemes may perform poorly in comparison.

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