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.
翻译:我们设计了最优化系统(SBOS)问题,并为这些问题提供了解决办法。在一个SBOS问题中,有一定数目的系统是竞争者。在每个系统中,一个连续的决定变量会影响系统的预期业绩。一个SBOS问题根据它们自己最佳选择的、最佳选择选择最佳的系统的预期业绩,比较了不同的系统。一个SBOS问题根据它们自己最佳选择的、选择最佳选择最佳的系统(SBOS)问题,没有预先了解系统的预期性能,也没有优化每个系统内部的决定。我们设计了容易执行的算法,根据适应性选择一个系统和选择决定来评价系统噪音性能,先后消除低级系统,最终建议一个系统在花费用户指定预算后成为最佳系统。提议的算法结合了随机分析梯度梯度梯度下降法和顺序消除方法,同时利用每个系统内部的结构,并进行系统之间的比较。关于拟议的算法,我们证明随着预算增长到无限,误选的可能性会达到指数的趋同为零。我们举三个数字例子,代表SBOS问题的三个实际案例。我们提议的算法在抽样的概率方面显示了一致和较强的业绩。