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;θ^*) in x over a conic constraint set represented by h(x;θ^*) ∈ K, where θ^* is an unknown (or misspecified) vector of parameters, K is a closed convex cone and h is affine in x. Suppose θ^* is unavailable but may be learnt by a separate process that generates a sequence of estimators θk, each of which is an increasingly accurate approximation of θ^*. We develop a first-order inexact augmented Lagrangian (AL) scheme for computing an optimal solution x^* corresponding to θ^* while simultaneously learning θ^*. 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.