Collaborative Proposal: Advanced Algorithms for Nonlinear Constrained Stochastic Optimization
Abstract
Mathematical optimization is a vital tool used in many diverse fields. More often than not, in real-world applications, optimization problems are nonlinear, nonconvex, stochastic and constrained. The objective of this project is to design, analyze, and implement advanced algorithms for solving constrained optimization problems. Specifically, we plan to develop sequential quadratic programming(SQP) and interior point (IP) methods to solve problems with deterministic general nonlinear and possibly nonconvex constraints anda stochastic objective function. These problems arise in a plethora of science and engineering applications, such as, statistics, optimal power flow, PDE constrained optimization and machine learning and deep learning, and many areas of interest to the Office of Naval Research (ONR) and the Department of Defense (DoD). While there has been a recent surge of interest in designing algorithms for this class of problems, most if not all of these methods do not incorporate advanced features that are of paramount importance in the deterministic setting, including robust inexact linear system solvers, fully adaptive and robust parameter estimation including merit parameters and step sizes, and the utilization of second-order information. The goal of the proposed project is to design, analyze and implement advanced SQP and IP algorithms for solving stochastic optimization problems with nonlinear constraints. We will build upon the existing literature for stochastic SQP methods and enhance existing algorithms with advanced features such as improvedsearch direction computation, adaptive step size rules, and utilization of second-order information. In addition, we will design the first nonlinearly constrained IP algorithm for stochastic optimization. This method will build upon the advancements created for stochastic SQP methods and integrate refined features, such as adaptive step sizes and step acceptance, adaptive barrier parameter updates, and the incorporation of second-order information. Moreover, we intend to design two different IP methods, each tailored to specific assumptions on the stochasticity of the objective function. Our efforts will be focused on these classes of algorithms, as they have a long history in deterministic optimization of superior practical performance on problems with general nonlinear constraints and have shown significant promise in recent work in the stochastic regime. Throughout the project, we will empirically validate the algorithms created on standard test problems as well as synthetic problems. In addition, each PI involved in the current proposal has established a local collaboration at their university where they will apply thedeveloped algorithms to real-world applicationsincluding aircraft design optimization and training physics informed neural networks to solve inverse problems. The proposed project provides the natural next step in the design of advanced sequential quadratic programming and interior point algorithms for solving challenging constrained stochastic optimization problems. The work outlined in this proposal aligns with several areas of interestof ONR#s Mathematics, Computer and Information Sciences (MCIS) Division due to the mathematical innovation and the applicability. One example is the Mathematical and Resource Optimization program, due to the focus on the design, analysis, and implementation of nonlinear optimization algorithms.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Nov 09, 2024
- Source ID
- N000142412670
Entities
People
- Michael Oneill
Organizations
- Office of Naval Research
- United States Navy
- University of North Carolina at Chapel Hill