On the rate analysis of inexact augmented Lagrangian schemes for convex optimization problems with misspecified constraints

We consider a misspecified optimization problem that requires minimizing of a convex function f(x;θ^*) in x over a constraint set represented by h(x;θ^*)≤ 0, where θ^* is an unknown (or misspecified) vector of parameters. Suppose θ^* can be learnt by a distinct process that generates a sequence of estimators θk, each of which is an increasingly accurate approximation of θ^*. We develop a first-order augmented Lagrangian scheme for computing an optimal solution x^* while simultaneously learning θ^*.