We consider optimization problems with an objective function that is estimable using a Monte Carlo oracle, constraint functions that are known deterministically through a constraint-satisfaction oracle, and integer decision variables. Seeking an appropriately defined local minimum, we propose an iterative adaptive sampling algorithm that, during each iteration, performs a statistical local optimality test followed by a line search when the test detects a stochastic descent direction. We prove a number of results. First, the true function values at the iterates generated by the algorithm form an almost-supermartingale process, and the iterates are absorbed with probability one into the set of local minima in finite time. Second, such absorption happens exponentially fast in iteration number and in oracle calls. This result is analogous to non-standard rate guarantees in stochastic continuous optimization contexts that involve sharp minima. Third, the oracle complexity of the proposed algorithm increases linearly in the dimensionality of the local neighborhood. As a solver, primarily due to combining line searches that use common random numbers with statistical tests for local optimality, the proposed algorithm is effective on a variety of problems. We illustrate such performance using three problem suites, on problems ranging from 25 to 200 dimensions.

我们考虑使用蒙特卡洛预言机可估计的目标函数、通过约束满足预言机确定性已知的约束函数和整数决策变量的优化问题。为了寻求适当定义的局部最小值，我们提出了一种迭代自适应采样算法，该算法在每次迭代期间执行统计局部最优性测试，然后在测试检测到随机下降方向时进行线搜索。我们证明了一些结果。首先，算法生成的迭代中的真实函数值形成了一个几乎超鞅的过程，并且迭代在有限时间内以概率为 1 吸收到局部最小值的集合中。其次，这种吸收在迭代次数和预言机调用中以指数方式快速发生。这个结果类似于随机连续优化上下文中涉及尖锐最小值的非标准速率保证。第三，所提出算法的预言机复杂度在局部邻域的维数上线性增加。作为求解器，主要是由于将使用常见随机数的线搜索与局部最优性的统计测试相结合，所提出的算法对各种问题都有效。我们使用三个问题套件来说明这种性能，问题范围从 25 到 200 维。主要是由于将使用常见随机数的线搜索与局部最优性的统计测试相结合，所提出的算法对各种问题都很有效。我们使用三个问题套件来说明这种性能，问题范围从 25 到 200 维。主要是由于将使用常见随机数的线搜索与局部最优性的统计测试相结合，所提出的算法对各种问题都很有效。我们使用三个问题套件来说明这种性能，问题范围从 25 到 200 维。