Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

Bayesian Networks (BNs) represent conditional probability relations among a set of random variables (nodes) in the form of a directed acyclic graph (DAG), and have found diverse applications in knowledge discovery. We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The central problem can be modeled as a mixed-integer program with an objective function composed of a convex quadratic loss function and a regularization penalty subject to linear constraints. The optimal solution to this mathematical program is known to have desirable statistical properties under certain conditions. However, the state-of-the-art optimization solvers are not able to obtain provably optimal solutions to the existing mathematical formulations for medium-size problems within reasonable computational times. To address this difficulty, we tackle the problem from both computational and statistical perspectives. On the one hand, we propose a concrete early stopping criterion to terminate the branch-and-bound process in order to obtain a near-optimal solution to the mixed-integer program, and establish the consistency of this approximate solution. On the other hand, we improve the existing formulations by replacing the linear "big-$M$" constraints that represent the relationship between the continuous and binary indicator variables with second-order conic constraints. Our numerical results demonstrate the effectiveness of the proposed approaches

Capacity Fade Analysis and Model Based Optimization of Lithium-ion Batteries

Electrochemical power sources have had significant improvements in design, economy, and operating range and are expected to play a vital role in the future in a wide range of applications. The lithium-ion battery is an ideal candidate for a wide variety of applications due to its high energy/power density and operating voltage. Some limitations of existing lithium-ion battery technology include underutilization, stress-induced material damage, capacity fade, and the potential for thermal runaway. This dissertation contributes to the efforts in the modeling, simulation and optimization of lithium-ion batteries and their use in the design of better batteries for the future. While physics-based models have been widely developed and studied for these systems, the rigorous models have not been employed for parameter estimation or dynamic optimization of operating conditions. The first chapter discusses a systems engineering based approach to illustrate different critical issues possible ways to overcome them using modeling, simulation and optimization of lithium-ion batteries. The chapters 2-5, explain some of these ways to facilitate: i) capacity fade analysis of Li-ion batteries using different approaches for modeling capacity fade in lithium-ion batteries,: ii) model based optimal design in Li-ion batteries and: iii) optimum operating conditions: current profile) for lithium-ion batteries based on dynamic optimization techniques. The major outcomes of this thesis will be,: i) comparison of different types of modeling efforts that will help predict and understand capacity fade in lithium-ion batteries that will help design better batteries for the future,: ii) a methodology for the optimal design of next-generation porous electrodes for lithium-ion batteries, with spatially graded porosity distributions with improved energy efficiency and battery lifetime and: iii) optimized operating conditions of batteries for high energy and utilization efficiency, safer operation without thermal runaway and longer life

A hyperbolastic type-I diffusion process: Parameter estimation by means of the firefly algorithm

A stochastic diffusion process, whose mean function is a hyperbolastic curve of type I, is presented. The main characteristics of the process are studied and the problem of maximum likelihood estimation for the parameters of the process is considered. To this end, the firefly metaheuristic optimization algorithm is applied after bounding the parametric space by a stagewise procedure. Some examples based on simulated sample paths and real data illustrate this development

Data-driven satisficing measure and ranking

We propose an computational framework for real-time risk assessment and prioritizing for random outcomes without prior information on probability distributions. The basic model is built based on satisficing measure (SM) which yields a single index for risk comparison. Since SM is a dual representation for a family of risk measures, we consider problems constrained by general convex risk measures and specifically by Conditional value-at-risk. Starting from offline optimization, we apply sample average approximation technique and argue the convergence rate and validation of optimal solutions. In online stochastic optimization case, we develop primal-dual stochastic approximation algorithms respectively for general risk constrained problems, and derive their regret bounds. For both offline and online cases, we illustrate the relationship between risk ranking accuracy with sample size (or iterations).Comment: 26 Pages, 6 Figure