Optimization problems involving an infinite number of constraints are considered. This paper presents a general stochastic outer approximations method which incorporates mechanisms for active search of relevant constraints and for dropping of irrelevant constraints. The method extracts the characteristic features of several stochastic outer approximations algorithms suggested by Wardi [J. Optim. Theory Appl., 56 (1988), pp. 285-311; J. Optim. Theory Appl., 64 (1990), pp. 615-640] and furthermore develops the approach to get advantages of the Eaves-Zangwill scheme. Similarly to Gonzaga and Polak [SIAM J. Control Optim., 17 (1979), pp. 477-493] the method is based on the use of quasi-optimality functions satisfying some general unrestricted assumptions. These functions are usually employed in the stopping criteria of numerical techniques for solving simpler problems. It is shown that the method’s trajectories almost surely converge to the quasioptimal set. Following the proposed approach a stochastic algorithm for solving the approximation problem is constructed and studied. The proposed general method can be considered as a developed Eaves-Zangwill method applying the multistart technique at each iteration for the search of relevant constraints’ parameters.