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.
