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In this dissertation we study several non-convex and stochastic optimization problems. The common theme is the use of mixed-integer programming (MIP) techniques including valid inequalities and reformulation to solve these problems.
In this dissertation we study several nonconvex and combinatorial optimization problems with applications in production planning, machine learning, advertising, statistics, and computer vision. The common theme is the use of algorithmic and modelling techniques from mixed-integer programming (MIP) which include formulation strengthening, decomposition, and linear programming (LP) rounding. We first consider MIP formulations for piecewise linear functions (PLFs) that are evaluated when an indicator variable is turned on. We describe modifications to standard MIP formulations for PLFs with desirable theoretical properties and superior computational performance in this context. Next, we consider a production planning problem where the production process creates a mixture of desirable products and undesirable byproducts. In this production process, at any point in time the fraction of the mixture that is an undesirable byproduct increases monotonically as a function of the cumulative mixture production up to that time. The mathematical formulation of this continuous-time problem is nonconvex. We present a discrete time mixed-integer nonlinear programming (MINLP) formulation that exploits the increasing nature of the byproduct ratio function. We demonstrate that this new formulation is more accurate than a previously proposed MINLP formulation. We describe three different mixed-integer linear programming (MIP) approximation and relaxation models of this nonconvex MINLP, and derive modifications that strengthen the LP-relaxations of these models. We provide computational experiments that demonstrate that the proposed formulation is more accurate than the previous formulation, and that the strengthened MIP approximation and relaxation models can be used to obtain near-optimal solutions for large instances of this nonconvex MINLP. We then study production planning problems in the presence of realistic business rules like taxes, tariffs, and royalties. We propose two different solution techniques. The first is a MIP formulation while the second is a search algorithm based on a novel continuous domain formulation. We then discuss decomposition methods to compute bounds on the optimal solution. Our computational experiments demonstrate the impact of our formulations, solution techniques, and algorithms on a sample application problem. Finally, we study three classes of combinatorial optimization problems: set packing, set covering, and multiway-cut. Near-optimal solutions of these combinatorial problems can be computed by rounding the solution of an LP. We show that one can recover solutions of comparable quality by rounding an approximate LP solution instead of an exact one. These approximate LP solutions can be computed efficiently by solving a quadratic-penalty formulation of the LP using a parallel stochastic coordinate descent method. We derive worst-case runtime and solution quality guarantees of this scheme using novel perturbation and convergence analyses. Our experiments demonstrate that on these combinatorial problems our rounding scheme is up to an order of magnitude faster than Cplex (a commercial LP solver) while producing solutions of similar quality.
This thesis concerns the application of mixed-integer programming techniques to solve special classes of network flow problems and stochastic integer programs. We draw tools from complexity and polyhedral theory to analyze these problems and propose improved solution methods. In the first part, we consider semi-continuous network flow problems, that is, a class of network flow problems where some of the variables are required to take values above a prespecified minimum threshold whenever they are not zero. These problems find applications in management and supply chain models where orders in small quantities are undesirable. We introduce the semi-continuous inflow set with variable upper bounds as a relaxation of general semi-continuous network flow problems. Two particular cases of this set are considered, for which we present complete descriptions of the convex hull in terms of linear inequalities and extended formulations. We also consider a class of semi-continuous transportation problems where inflow systems arise as substructures, for which we investigate complexity questions. Finally, we study the computational efficacy of the developed polyhedral results in solving randomly generated instances of semi-continuous transportation problems. In the second part, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of optimizing a linear function on the subset of vertices of P that are not contained in X. This problem is closely related to finding the k-best basic solutions to a linear problem and finds applications in stochastic integer programming. We observe that the complexity of the problem depends on how P and X are specified. For instance, P can be explicitly given by its linear description, or implicitly by an oracle. Similarly, X can be explicitly given as a list of vectors, or implicitly as a face of P. While removing vertices turns to be hard in general, it is tractable for tractable 0-1 polytopes, and compact extended formulations can be obtained. Some extensions to integral polytopes are also presented. The third part is devoted to the integer L-shaped method for two-stage stochastic integer programs. A widely used model assumes that decisions are made in a two-step fashion, where first-stage decisions are followed by second-stage recourse actions after the uncertain parameters are observed, and we seek to minimize the expected overall cost. In the case of finitely many possible outcomes or scenarios, the integer L-shaped method proposes a decomposition scheme akin to Benders' decomposition for linear problems, but where a series of mixed-integer subproblems have to be solved at each iteration. To improve the performance of the method, we devise a simple modification that alternates between linear and mixed-integer subproblems, yielding significant time savings in instances from the literature. We also present a general framework to generate optimality cuts via a cut-generating problem. Using an extended formulation of the forbidden-vertices problem, we recast our cut-generating problem as a linear problem and embed it within the integer L-shaped method. Our numerical experiments suggest that this approach can prove beneficial when the first-stage set is relatively complicated.
This book covers not only foundational materials but also the most recent progresses made during the past few years on the area of machine learning algorithms. In spite of the intensive research and development in this area, there does not exist a systematic treatment to introduce the fundamental concepts and recent progresses on machine learning algorithms, especially on those based on stochastic optimization methods, randomized algorithms, nonconvex optimization, distributed and online learning, and projection free methods. This book will benefit the broad audience in the area of machine learning, artificial intelligence and mathematical programming community by presenting these recent developments in a tutorial style, starting from the basic building blocks to the most carefully designed and complicated algorithms for machine learning.
Linear programming has attracted the interest of mathematicians since World War II when the first computers were constructed. Early attempts to apply linear programming methods practical problems failed, in part because of the inexactness of the data used to create the models. This book presents a comprehensive treatment of linear optimization with inexact data, summarizing existing results and presenting new ones within a unifying framework.
On March 15, 2002 we held a workshop on network interdiction and the more general problem of stochastic mixed integer programming at the University of California, Davis. Jesús De Loera and I co-chaired the event, which included presentations of on-going research and discussion. At the workshop, we decided to produce a volume of timely work on the topics. This volume is the result. Each chapter represents state-of-the-art research and all of them were refereed by leading investigators in the respective fields. Problems - sociated with protecting and attacking computer, transportation, and social networks gain importance as the world becomes more dep- dent on interconnected systems. Optimization models that address the stochastic nature of these problems are an important part of the research agenda. This work relies on recent efforts to provide methods for - dressing stochastic mixed integer programs. The book is organized with interdiction papers first and the stochastic programming papers in the second part. A nice overview of the papers is provided in the Foreward written by Roger Wets.
Risk-averse stochastic optimization problems widely exist in practice, but are generally challenging computationally. In this dissertation, we conduct both theoretical and computational research on these problems. First, we study chance-constrained two-stage stochastic optimization problems where second-stage feasible recourse decisions incur additional cost. We also propose a new model, where recovery decisions are made for the infeasible scenarios, and develop strong decomposition algorithms. Our computational results show the effectiveness of the proposed method. Second, we study the static probabilistic lot-sizing problem (SPLS), as an application of a two-stage chance-constrained problem in supply chains. We propose a new formulation that exploits the simple recourse structure, and give two classes of strong valid inequalities, which are shown to be computationally effective. Third, we study two-sided chance-constrained programs with a finite probability space. We reformulate this class of problems as a mixed-integer program. We study the polyhedral structure of the reformulation and propose a class of facet-defining inequalities. We propose a polynomial dynamic programming algorithm for the separation problem. Preliminary computational results are encouraging. Finally, we study risk-averse models for multicriteria stochastic optimization problems. We propose a new model that optimizes the worst-case multivariate conditional value-at-risk (CVaR), and develop a finitely convergent delayed cut generation algorithm.
Operations Research is a field whose major contribution has been to propose a rigorous fonnulation of often ill-defmed problems pertaining to the organization or the design of large scale systems, such as resource allocation problems, scheduling and the like. While this effort did help a lot in understanding the nature of these problems, the mathematical models have proved only partially satisfactory due to the difficulty in gathering precise data, and in formulating objective functions that reflect the multi-faceted notion of optimal solution according to human experts. In this respect linear programming is a typical example of impressive achievement of Operations Research, that in its detenninistic fonn is not always adapted to real world decision-making : everything must be expressed in tenns of linear constraints ; yet the coefficients that appear in these constraints may not be so well-defined, either because their value depends upon other parameters (not accounted for in the model) or because they cannot be precisely assessed, and only qualitative estimates of these coefficients are available. Similarly the best solution to a linear programming problem may be more a matter of compromise between various criteria rather than just minimizing or maximizing a linear objective function. Lastly the constraints, expressed by equalities or inequalities between linear expressions, are often softer in reality that what their mathematical expression might let us believe, and infeasibility as detected by the linear programming techniques can often been coped with by making trade-offs with the real world.
Branch and bound experiments in 0-1 programming; A subadditive approach to the group problem of integer programming; Two computationaly difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems; Lagrangean relaxation for integer programming; A heuristic algorithm for mixed-integer programming problems; On the group problem for mixed integer programming; Experiments in the formulation of integer programming problems.
Motivation Stochastic Linear Programming with recourse represents one of the more widely applicable models for incorporating uncertainty within in which the SLP optimization models. There are several arenas model is appropriate, and such models have found applications in air line yield management, capacity planning, electric power generation planning, financial planning, logistics, telecommunications network planning, and many more. In some of these applications, modelers represent uncertainty in terms of only a few seenarios and formulate a large scale linear program which is then solved using LP software. However, there are many applications, such as the telecommunications planning problem discussed in this book, where a handful of seenarios do not capture variability well enough to provide a reasonable model of the actual decision-making problem. Problems of this type easily exceed the capabilities of LP software by several orders of magnitude. Their solution requires the use of algorithmic methods that exploit the structure of the SLP model in a manner that will accommodate large scale applications.