  Two optimization algorithms are proposed for solving a stochastic programming
problem for which the objective function is given in the form of the
expectation of convex functions and the constraint set is defined by the
intersection of fixed point sets of nonexpansive mappings in a real Hilbert
space. This setting of fixed point constraints enables consideration of the
case in which the projection onto each of the constraint sets cannot be
computed efficiently. Both algorithms use a convex function and a nonexpansive
mapping determined by a certain probabilistic process at each iteration. One
algorithm blends a stochastic gradient method with the Halpern fixed point
algorithm. The other is based on a stochastic proximal point algorithm and the
Halpern fixed point algorithm; it can be applied to nonsmooth convex
optimization. Convergence analysis showed that, under certain assumptions, any
weak sequential cluster point of the sequence generated by either algorithm
almost surely belongs to the solution set of the problem. Convergence rate
analysis illustrated their efficiency, and the numerical results of convex
optimization over fixed point sets demonstrated their effectiveness.
