Numerical noise is a prevalent concern in many practical optimization problems. Convergence of gradient based optimization algorithms in the presence of numerical noise is not always assured. One way to improve optimization algorithm performance in the presence of numerical noise is to adjust the method of gradient computation. This study investigates the use of Continuous Sensitivity Equation (CSE) gradient approximations in the context of numerical noise and optimization. Three problems are considered: a problem with a system of ODE constraints, a single parameter flow problem constrained by the Navier-Stokes equations, and a multiple parameter flow problem constrained by the Navier-Stokes equations. All three problems use adaptive methods in the simulation of the constraint and are numerically noisy. Gradients for each problem are computed with both CSE and finite difference methods. The gradients are analyzed and compared. The two flow problems are optimized with a trust region optimization algorithm using both sets of gradient calculations. Optimization results are also compared, and the CSE gradient approximation yields impressive results for these examples. / Master of Science

We build a variety of customer booking choice models for a major airline that operates in a very competitive origin-destination market. Some of the models are aimed at incorporating unobserved heterogeneous customer preferences for different departure times. The estimation results show that including these factors into choice models dramatically affects price sensitivity estimates, and therefore matters. We present a stochastic trust region algorithm for estimating ML-type models that involve high-dimensional integrals. The algorithm embeds two sampling processes: (i) a data sampling process and (ii) a Monte Carlo sampling process, and the algorithm dynamically controls sample sizes based on the magnitude of the errors incurred due to the two sampling processes. The first-order convergence is proved based on generalized uniform law of large numbers theories for both the average log-likelihood function and its gradient. The efficiency of the algorithm is tested with real data and compared with existing algorithms. We also study how a specific behavioral phenomenon, called the decoy effect, affects the decisions of sellers in product assortment competition in a duopoly. We propose a discrete choice model to capture decoy effects, and we provide a complete characterization of the Nash equilibria and their dependence on choice model parameters. For the cases in which there are multiple equilibria, we consider dynamical systems models of the sellers responding to their competitors using Cournot adjustment or fictitious play to study the evolution of the assortment competition and the stability of the equilibria. We provide a simple geometric characterization of the dynamics of fictitious play for 2×2 games that is more complete than previous characterizations.