Data Analytical Approach for Large-scale Optimization

大规模优化问题在从制造到运输到医疗保健系统的广泛实际应用中很常见。通常使用启发式方法来解决此类问题，以提供良好的（尽管不是最优的）解决方案。该奖项支持的研究的基本前提是，通过结合启发式和统计学的数据分析方法，可以在解决大规模优化问题方面取得重大进展。如果成功，这项研究不仅会以更低的计算成本带来改进的解决方案，而且所得到的解决方案的质量将得到统计理论的支持，而统计理论通常无法用于启发式。通过将该项目的成果纳入现有的研究生课程，以及通过将作为该奖项的一部分开发的数据分析优化的新研究生课程，预计将产生重大的教育影响。在这个项目中，将研究两种特定的数据分析方法。第一种方法将使用最近开发的峰值-超过阈值推理方法估计搜索区域的极限极值分布。第二种方法将利用尾商相关系数（TQCC）来搜索全局最小值。在统计推断中，TQCC的主要用途是在非常高的向下阈值下测量尾巴区域的尾巴依赖性（尾巴联合运动）。在优化问题中，目标函数的值在尾部区域（山谷）的随机样本点处评估，表现为尾部协同运动。在具有多个局部最优目标函数的单变量优化问题的初步实验中，使用TQCC来搜索最有希望的区域有望导致高效率的搜索。在高维优化问题中，由于在任何局部谷中尾巴在所有方向上的协同运动的性质，期望有更好的性能。

Large-scale optimization problems are common in a broad range of practical applications from manufacturing to transportation to healthcare systems. Such problems are routinely attacked using heuristics to provide good, although suboptimal, solutions. The basic premise of the research supported by this award is that significant advances in solving large-scale optimization problems can be achieved by combining heuristics and data analytical approaches from statistics. If successful, not only will this research lead to improved solutions at a lower computational cost, but the quality of the resulting solutions will be supported by statistical theory, which is not generally available for heuristics. Significant educational impact is also expected, through incorporating the results of this project into existing graduate courses and through a new graduate course in data analytical optimization that will be developed as part of the award.In this project, two particular data analytical approaches will be investigated. The first approach will estimate the limiting extreme value distributions in the search region using the recently developed Peaks-Over-Threshold inference method. The second approach will utilize the tail quotient correlation coefficient (TQCC) in the search for global minimizers. In statistical inference, the main usage of TQCC is to measure the tail dependence (tail co-movement) in tail regions under very high downward threshold values. In optimization problems, values of the objective function evaluated at random sample points in tail regions (valleys) behave like tail co-movement. Using TQCC in searching the most promising regions is expected to lead to highly efficient search, a phenomenon observed in preliminary experimental trials on univariate optimization problems with objective functions with multiple local optima. Even better performance is expected in high-dimensional optimization problems due to the nature of tail co-movements in all directions in any local valley.