High-Dimensional Optimization Problems in Decision-Making and Discrete Geometry

Abstract

This dissertation is organized in two separate parts focusing on two optimization problems; a framework for scheduling of modern telescopes, and optimization problems with Fourier-analytic structures.

In the first part, we show that traditional operational schemes cannot optimally utilize the new generation of fast astronomical instruments. Then we introduce an approximate Markovian Decision Process (MDP) to model the hybrid system of telescope-environment. Given the MDP model, we present an adaptive decision-making strategy to optimally operate a ground-based instrument. Our strategy is a framework that can be adopted and customized for a wide variety of astronomical missions. It can be automatically and efficiently trained with different sets of mission objectives and constraints. In addition to our theoretical work, we developed, based on the proposed decision-making framework, an open-source software that will be used to schedule the Large Synoptic Survey Telescope (LSST). LSST is the primary ground-based survey telescope of the next decade which is located in Chile. It will image half of the sky every few nights starting from 2021. We compare the performance of our scheduler with the previous LSST scheduler that is designed and engineered based on traditional methods.

In the second part, we discuss how optimization problems with Fourier-analytic structures appear in continuous relaxations of some fundamental combinatorial problems. Then we explain the problem of packing with convex bodies and Turan Extremal Problem. They can be expressed as Fourier-analytic optimization problems and appear in discrete geometry and number theory respectively. Then we introduce a framework and computational tool to bridge the gap between theoretical questions and computational intuitions. The problems that we address are notoriously difficult and have long been only a subject of theoretical approaches in pure mathematics. In this study we introduce a computational approach to provide approximations, insights and intuitions for the solution of these problems. Finally, we present a formulation of a more general set of Fourier-analytic optimization problems with applications in efficient utility allocation. We also present a proposal for the future studies that can be built upon the results of this dissertation.