An augmented Lagrangian stochastic approximation method for convex stochastic conic optimization

In this paper, we consider the convex stochastic conic optimization defined by expectations. An augmented Lagrangian stochastic approximation (ALSAcp) method is proposed to solve this convex stochastic conic optimization problem and the regrets of this method are analyzed. Under mild conditions, we show that this method exhibits ${\rm~O}(T^{-1/2})$ regret for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where $T$ denotes the number of iterations. Moreover, we show that, with at least $1-1/T$ probability, the method has no more than ${\rm~O}(\log~(T)/\sqrt{T})$ for both objective descent regret and constraint violation regret.