An augmented Lagrangian-type stochastic approximation method for convex stochastic semidefinite programming defined by expectations

Abstract

An augmented Lagrangian-type stochastic approximation method (ALSAssdp) is proposed to solve the convex stochastic semidefinite optimization problem defined by expectations and regrets of this method are analyzed. Under mild conditions, we show that this method exhibits $O\left(T^{-1/2}\right)$ regret for both objective reduction and constraint violation. Moreover, we show that, with at least $1-1/T$ probability, the method has no more than $O(\log (T)/\sqrt{T})$ for both objective regret and constraint violation regret.