A desire with iterative optimization techniques is that the algorithm reach the global optimum rather than get stranded at a local optimum value. One method used to try to assure global convergence is the injection of extra noise terms into the recursion, which may allow the algorithm to escape local optimum points. The amplitude of the injected noise is decreased over time (a process called "annealing"), so that the algorithm can finally converge when it reaches the global optimum point. In this context, we examine a certain "gradient free" stochastic approximation algorithm called "SPSA," that has performed well in complex optimization problems. We discuss conditions under which SPSA will converge globally using injected noise. In a separate section, we show that, under different conditions, "basic" SPSA (i.e., without injected noise) can achieve a standard type of convergence to a global optimum. The discussion is supported by a numerical study.