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We consider optimizing a stochastic system, given only a simulation model that is parameterized by continuous decision variables. The model is assumed to produce unbiased point estimates of the system performance measure(s), which must be expected values. The performance measures may appear in the objective function and/or in the constraints. We develop a family of retrospective-optimization (RO) algorithms based on a sequence of sample-path approximations to the original problem with increasing sample sizes. Each approximation problem is obtained by substituting point estimators for each performance measure and using common random numbers over all values of the decision variables. We assume that these approximation problems can be deterministically solved within a specified error in the decision variables, and that this error is decreasing to zero. The computational efficiency of RO arises from being able to solve the next approximation problem efficiently based on knowledge gained from the earlier, easier approximation problems.