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

Abstract

We consider a misspecified optimization problem that requires minimizing of a convex function $f(x;θ^*)$ in x over a constraint set represented by $h(x;θ^*)\leq 0$, 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 $θ_k$, 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 $θ^*$.