On the Analysis of Inexact Augmented Lagrangian Schemes for Misspecified Conic Convex Programs

In this article, we consider the misspecified optimization problem of minimizing a convex function $f(x;\theta ^*)$ in notation="LaTeX">$x$ over conic constraint set represented by notation="LaTeX">$h(x;\theta ^*) \in \mathcal {K}$ , where notation="LaTeX">$\theta ^*$ is an unknown (or misspecified) vector parameters, notation="LaTeX">$\mathcal closed cone, and notation="LaTeX">$h$ affine . Suppose that unavailable but may be learnt separate process generates sequence estimators _k$ each which increasingly accurate approximation We develop first-order inexact augmented Lagrangian (AL) scheme for computing optimal solution notation="LaTeX">$x^*$ corresponding to while simultaneously learning particular, derive rate statements such schemes when penalty parameter either constant or increasing bounds on overall complexity terms proximal gradient steps AL subproblems are inexactly solved via accelerated scheme. Numerical results portfolio with covariance matrix suggest these perform well practice, naive sequential poorly comparison.