Approximation schemes for network, clustering and queueing models

In this dissertation, we consider important optimization problems that arise in three different domains, namely network models, clustering problems and queueing models. To be more specific, we focus on devising efficient traffic routing models, deriving exact convex reformulation to the well-known K-means clustering problem and studying the classical Naor’s observable queues under uncertain parameters. In the following chapters, we discuss these problems in detail, design efficient and tractable solution methodologies, and assess the quality of proposed solutions. In the first part of the dissertation, we analyze a limited-adaptability traffic routing model for the Austin road network. Routing a person through a traffic network presents a tension between selecting a fixed route that is easy to navigate and selecting an aggressively adaptive route that minimizes the expected travel time. We develop non-aggressive adaptive routes in the middle-ground seeking the best of both these extremes. Specifically, these routes still adapt to changing traffic condition, however we limit the total number of allowable adjustments. This improves the user experience, by providing a continuum of options between saving travel time and minimizing navigation. We design strategies to model single and multiple route adjustments, and investigate enumerative techniques to solve these models. We also develop tractable algorithms with easily computable lower and upper bounds to handle real-size traffic data. We finally present the numerical results highlighting the benefit of different levels of adaptability in terms of reducing the expected travel time. In the second part of the dissertation, we study the well-known classical K-means clustering problem. We show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formulation gives rise to solutions that can be leveraged to identify the clusters. We devise a new approximation algorithm for K-means clustering that utilizes the improved formulation and empirically illustrate its superiority over the state-of-the-art solution schemes. Finally, we study an extension of Naor’s analysis [74] on the joining or balking problem in observable M/M/1 queues, relaxing the principal assumption of deterministic arrival and service rates. While all the Markovian assumptions still hold, we assume the arrival and service rates are uncertain and study this problem under stochastic and distributionally robust settings. In the former setting, the exact rates are unknown but we assume the distribution of rates are known to all the decision makers. We derive the optimal joining threshold strategies from the perspective of an individual customer, a social optimizer and a revenue maximizer, such that expected profit rate is maximized. In the distributionally robust setting, we go a step further to assume the true distributions are unknown and the decision makers have access to only a finite set of training samples. Similar to the stochastic setting, we derive optimal thresholds such that the worst-case expected profit rates are maximized. Finally, we compare our observations, both theoretically and numerically, with Naor’s classical results.Operations Research and Industrial Engineerin

Computing approximate PSD factorizations

We give an algorithm for computing approximate PSD factorizations of nonnegative matrices. The running time of the algorithm is polynomial in the dimensions of the input matrix, but exponential in the PSD rank and the approximation error. The main ingredient is an exact factorization algorithm when the rows and columns of the factors are constrained to lie in a general polyhedron. This strictly generalizes nonnegative matrix factorizations which can be captured by letting this polyhedron to be the nonnegative orthant.Comment: 10 page

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page

(Global) Optimization: Historical notes and recent developments

Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning