Stochastic and Simulation Optimization Over Function Spaces: Theory, Methods, and Algorithms
Abstract
We propose to investigate the theoretical, methodological, and algorithmic underpinnings of a large class of stochastic optimization/control problems (i) having decision variables lying in function spaces; (ii) driven by an underlying stochastic model amenable to Monte Carlo path generation; and (iii) needing the control or decision function to be open loop, that is, the solution functioncan be time-dependent but not adapted to the filtration generated by the underlying stochastic process. This challenging class of problems subsumes various disparate but critically important optimization settings under uncertainty. The defining feature is the need to identify solutions that are time-dependent Rd-valued or measure-valued functions with specified structural features. Many natural and man-made systems have probabilistic time dynamics that can be well-modeled by non closed-form, non-stationary stochastic models that are amenable to Monte Carlo path generation. Examples include unmanned vehicles in a stochastic ambient environment, airflow in a large building, molecular dynamics within a material, weather and hurricane models, vehicular/packet flow in networks, and heat flow in physical objects. One may then wonder, can these systems be nudged to evolve in a favorable way, without violating the regular operational constraints of the system? We formally investigate this question by posing a stochastic optimization problem overfunction space subject to structural constraints, and seeking solver-based algorithmic frameworks that can reliably produce solutions without the need for human intervention.The principal challenges in constructing solver-based solutions are three-fold.(i) Computability: The underlying model and optimization space are infinite-dimensional, rendering standard algorithmic recursions hypothetical since they will involve objects that cannot implemented on a finite computer. Computable algorithms will thus necessarily rely on finite-dimensional approximations of infinite-dimensional objects; any such approximation,and the ensuing analysis, needs to be performed in a mathematically principled manner. (ii) Adaptivity: Given our goal of solver-based solutions, the algorithms we construct should operate without the need for user-tuning of parameters. Such adaptivity is key to ready adoption within a wide variety of application domains. (iii) Statistical Inference: Since the underlying problem we consider is stochastic optimization, there is a need for inferential statistics on the quality of solutions reported by the algorithms to facilitate termination and assess risk. Statistical inference is challenging even in Euclidean optimization settings due to issues relating to bias and autocorrelation of iterates. Such challenges are vastly compounded when operating in infinite-dimensional spaces.The proposed investigation accordingly emphasizes developing computable methods leading to solvers that exploit theoretical advances resulting from the investigation. The proposed solvers rely heavily on convergence rate and complexity calculations for adaptive algorithmic decisions such as trading-off statistical (model) learning and optimization, and theories on infinite-dimensional statistical inference for solver termination with probabilistic guarantees on solution quality.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 06, 2021
- Source ID
- N000142112287
Entities
People
- Raghu Pasupathy
Organizations
- Office of Naval Research
- Purdue University
- United States Navy