ABSTRACT 

Many problems in operations research, machine learning, statistics, and engineering share a common structure: optimization over infinitedimensional function spaces subject to stochastic dynamic constraints. The infinite-dimensional and stochastic nature of these problems poses fundamental computational challenges, as standard finite-dimensional optimization methods do not directly apply. This dissertation develops solution methodologies that exploit the analytic and probabilistic structure of these dynamics, combining tools from functional analysis, Monte Carlo simulation, and nonlinear programming to produce implementable algorithms with provable approximation g uarantees. We develop these ideas through two main studies: optimizing the drift of a regulated diffusion process, and computing determi nistic transport maps in the Monge optimal transport problem.

In the first study, we consider stochastic optimization problems driven by regulated stochastic processes. In applications su ch as queueing network control, inventory management, and scheduling, system dynamics are often modeled by regulated Brownian motion, whose drift function can be controlled. We formulate a more general problem of optimizing the drift function by minimizing a cost functional over an infinite-dimensional function space subject to hard constraints, such as the non-negativity of the paths. We develop a Sample Average Approximation (SAA) approach and derive unbiased pathwise directional derivatives of the cost functional. We prove equiconvergence of the SAA approximation and characterize an explicit error decomposition. This analysis enables asymptotically optimal budget allocation across the computational parameters.

In the second study, we turn to a fundamentally different problem that shares the same infinite-dimensional optimization structure: the Monge optimal transport problem, finding a deterministic transport map that pushes a source probability measure onto a target measure while minimizing a given transportation cost. Recent computational optimal transport methods largely target the Kantorovich r elaxation, which optimizes over probabilistic couplings and enjoys convex structure. In contrast, our formulation works directly with th e Monge problem, posed over infinite-dimensional Banach spaces of transport maps, and accommodates general, possibly nonconvex costs. We develop a Sequential Quadratic Programming (SQP) framework that solves a sequence of local quadratic subproblems using explic it analytical derivations of the first- and second-order variations of the objective and constraint functionals. A merit functional with sufficient constraint penalization guarantees global convergence toward critical points. To the best of our knowledge, this is the first globally convergent SQP operator recursion over infinite-dimensional spaces. We also propose an implementable discretization pipeline, and numerical experiments substantiate the practicality of the framework.

Finally, we present a synthesis connecting these two studies with modern generative modeling through a single dynamic coupling framework: optimizing a path measure on continuous trajectories subject to endpoint marginal constraints and governing dynami cs. This template unifies Monge optimal transport, continuous normalizing flows, score-based diffusion models, flow matching, stochastic interpolants, and Schrödinger bridges as specializations of common design choices. The drift optimization formalism shares this pathmeasure optimization structure, as it optimizes over the law of a controlled diffusion in the same sense that modern generati ve models do. The optimization and approximation techniques developed here provide a stochastic optimization perspective on these generative models by directly exploiting the geometric and probabilistic structure of the underlying dynamics. Collectively, this dissertation contributes to the theory and practice of infinite-dimensional stochastic optimization by providing broadly applicable methodologies that capitaliz e on the stochastic structures underlying diverse problem domains.

BIOGRAPHY

Zihe Zhou is a Ph.D. Candidate in the Edwardson School of Industrial Engineering at Purdue University. She received her B.S. in Statistics from the University of Science and Technology of China and her M.S. in Industrial Engineering from Purdue University. Her research interests include stochastic optimization, stochastic optimal control, optimal transport, and simulation-based optimization. Her dissertation work develops computational and theoretical frameworks for solving optimization problems constrained or driven by stochastic system dynamics.