We formulate selecting the best optimizing system (SBOS) problems and provide solutions for those problems. In an SBOS problem, a finite number of systems are contenders. Inside each system, a continuous decision variable affects the system's expected performance. An SBOS problem compares different systems based on their expected performances under their own optimally chosen decision to select the best, without advance knowledge of expected performances of the systems nor the optimizing decision inside each system. We design easy-to-implement algorithms that adaptively chooses a system and a choice of decision to evaluate the noisy system performance, sequentially eliminates inferior systems, and eventually recommends a system as the best after spending a user-specified budget. The proposed algorithms integrate the stochastic gradient descent method and the sequential elimination method to simultaneously exploit the structure inside each system and make comparisons across systems. For the proposed algorithms, we prove exponential rates of convergence to zero for the probability of false selection, as the budget grows to infinity. We conduct three numerical examples that represent three practical cases of SBOS problems. Our proposed algorithms demonstrate consistent and stronger performances in terms of the probability of false selection over benchmark algorithms under a range of problem settings and sampling budgets.