Convergence proof for a projected gradient optimization algorithm

In a previous paper [1] we discussed a stochastic projected gradient algorithm (in the context of an adaptive signal detection problem) which can be used to find a constrained optimum point for a concave or convex objective function subject to linear or nonlinear constraints which form a connected region, even when we do not have the objective function available, but only have a noisy estimate of the objective function available. When the constraint consists of only one linear equation, we said that one can prove convergence to the constrained optimum value and bound the rate of convergence of the algorithm to the constrained optimum value. A proof was given in [1] under noise free conditions. The purpose of this short paper is to extend the proof to the noisy case.