The Geometry of Single- and Multi-Objective Near-Optimization
Abstract
In this project, we develop theory and algorithms related to near-optimization, the problem of identifying regions of points in the feasible set of an optimization problem whose objective values are close to optimal. Our work specifically focuses on applying a geometric perspective to the problem, enabling efficient algorithms that leverage this added structure in the spaces of variables and objective values. Moreover, we apply the near-optimization perspective to multi-objective problems, revealing the space of trade-offs between multiple objectives for a given task. Our narrative describes a continuation of an ARO-funded research project launched in a 2019 proposal titled "Geometric Approaches to Near-Optimization." The current project, however, is completely new and inspired by our progress during the previous ARO-funded period. In particular, based on our most promising results, we now focus on several themes: The introduction of near-optimization to multi-objective problems by putting probability measures and geometries on the Pareto set and Pareto front, the comparison of stochastic (sampling-based) and deterministic approaches to near-optimization, and the incorporation of both discrete and continuous variables in our optimization problems. The majority of our work is divided among four tasks, which will take place in sequence through the period of performance: TASK I: Variational inference in multi-objective problem TASK II: Near-optimization of nested-problems TASK III: Near-optimization for geometrically-structured problems TASK IV: Near-optimization for spatiotemporal problems Each task is designed to uncover a central piece of the theory of near-optimization and to suggest practical tools that can be deployed for solving near-optimization problems in practice. We also describe a number of speculative secondary research directions designed to expand to distributionally-robust settings, mixed-integer programming, and optimization on manifolds. The research will be led by Prof. Justin Solomon and members of the MIT Geometric Data Processing Group. Solomon received the Army Young Investigator Award in 2017 and is a well-recognized member of the machine learning, optimization, and applied geometry communities.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 25, 2021
- Source ID
- W911NF2110293
Entities
People
- Justin Solomon
Organizations
- Army Contracting Command
- Massachusetts Institute of Technology
- United States Army