OPTIMIZATION UNDER UNCERTAINTY USING EXPONENTIAL CONES

Abstract

This thesis introduces new models and methodology for optimization under uncertainty by exploiting the modeling power and associated computational advances of exponential cones. At a high level, we inject information of uncertain parameters into optimization models through moment generating functions, where exponentials arise naturally, and approximate the resulting problems by exponential cone programs (ECPs) to solve them efficiently. We start from an electric vehicle charging management problem under stochastic customer arrivals. We formulate the problem of scheduling vehicle charging to minimize the expected total cost as a large-scale stochastic optimization problem and solve it using ECP approximations. We demonstrate the advantages of our ECP approach both theoretically and numerically over existing approaches. Next, we generalize the ECP approximation technique to risk-averse dynamic decision making. We propose robust optimization models and their tractable approximations that cater for ambiguity-averse decision makers whose underlying risk preferences are consistent with constant absolute risk aversion (CARA). At last, we show the solution methodology in robust CARA optimization applies to more entropy-related models, which we refer to as entropic robust optimization. We develop polyhedral and second-order cone approximations to leverage the efforts in mixed-integer linear programming and second-order conic programming and demonstrate their effectiveness.