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

TL;DR

This paper develops a first-order augmented Lagrangian method for convex optimization problems with misspecified constraints, integrating parameter learning to improve solution accuracy.

Contribution

It introduces a novel scheme that jointly optimizes and learns parameters in convex problems with misspecified constraints.

Findings

Converges to optimal solutions despite misspecification

Effectively learns parameters during optimization process

Provides theoretical guarantees for convergence and accuracy

Abstract

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