In this thesis I focus on cutting planes for large Mixed Integer Programming (MIP) problems. More specifically, I focus on two independent cutting planes studies. The first of these deals with cutting planes for the Traveling Salesman Problem (TSP), and the second with cutting planes for general MIPs. In the first study I introduce a new class of cutting planes which I call the Generalized Domino Parity (GDP) inequalities. My main achievements with regard to these are: (1) I show that these are valid for the TSP and for the graphical TSP. (2) I show that they generalize most well-known TSP inequalities (including combs, domino-parity constraints, clique-trees, bipartitions, paths and stars). (3) I show that a sub-class of these (which contains all clique-tree inequalities w/ a fixed number of handles) can be separated in polynomial time, on planar graphs. My second study can be subdivided in two parts. In the first of these I study the Mixed Integer Knapsack Problem (MIKP) and develop a branch-and-bound based algorithm for solving it. The novelty of the approach is that it exploits the notion of "dominance" in order to effectively prune solutions in the branch-and-bound tree. In the second part, I develop a Mixed Integer Rounding (MIR) cut separation heuristic, and embed the MIKP solver in a column generation algorithm in order to assess the performance of said heuristic. The goal of this study is to understand why no other class of inequalities derived from single-row systems has been able to outperform the MIR. Computational results are presented.

We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv´atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included.

In this Ph.D. dissertation research, we lay the mathematical foundations of various fundamental concepts in convex mixed-integer programs (MIPs), that is, optimization problems where all the decision variables belong to a given convex set and, in addition, a subset of them are required to be integer. In particular, we study properties of their feasible region and properties of cutting planes. The main contribution of this work is the extension of several fundamental results from the theory of linear MIPs to the case of convex MIPs. In the first part, we study properties of general closed convex sets that determine the closedness and polyhedrality of their integer hulls. We first present necessary and sufficient conditions for the integer hull of a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the integer hulls of general convex sets to be polyhedra. This result generalizes the well-known result due to Meyer in the case of linear MIPs. Under a simple technical assumption, we show that these sufficient conditions are also necessary for the integer hull of general convex sets to be polyhedra. In the second part, we apply the previous results to mixed-integer second order conic programs (MISOCPs), a special case of nonlinear convex MIPs. We show that there exists a polynomial time algorithm to check the closedness of the mixed integer hulls of simple MISOCPs. Moreover, in the special case of pure integer problems, we present sufficient conditions for verifying the closedness of the integer hull of intersection of simple MISOCPs that can also be checked in polynomial time. In the third part, we present an extension of the duality theory for linear MIPs to the case of conic MIPs. In particular, we construct a subadditive dual to conic MIPs. Under a simple condition on the primal problem, we are able to prove strong duality. In the fourth part, we study properties of maximal S-free convex sets, where S is a subset of integers contained in an arbitrary convex set. An S-free convex set is a convex set not containing any points of S in its interior. In this part, we show that maximal S-free convex sets are polyhedra and discuss some properties of these sets. In the fifth part, we study some generalizations of the split closure in the case of linear MIPs. Split cuts form a well-known class of valid inequalities for linear MIPs. Cook et al. (1990) showed that the split closure of a rational polyhedron - that is, the set of points in the polyhedron satisfying all split cuts - is again a polyhedron. In this thesis, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We also show how this result can be used to prove that some generalizations of split cuts, namely cross cuts, also yield closures that are rational polyhedra.

Single-row mixed-integer programming (MIP) problems have been studied thoroughly under many different perspectives over the years. While not many practical applications can be modeled as a single-row MIP, their importance lies in the fact that they are simple, natural and very useful relaxations of generic MIPs. This thesis will focus on such MIPs and present theoretical and computational advances in their study. Chapter 1 presents an introduction to single-row MIPs, a historical overview of results and a motivation of why we will be studying them. It will also contain a brief review of the topics studied in this thesis as well as our contribution to them. In Chapter 2, we introduce a generalization of a very important structured single-row MIP: Gomory's master cyclic group polyhedron (MCGP). We will show a structural result for the generalization, characterizing all facet-defining inequalities for it. This structural result allows us to develop relationships with MCGP, extend it to the mixed-integer case and show how it can be used to generate new valid inequalities for MIPs. Chapter 3 presents research on an algorithmic view on how to maximally lift continuous and integer variables. Connections to tilting and fractional programming will also be presented. Even though lifting is not particular to single-row MIPs, we envision that the general use of the techniques presented should be on easily solvable MIP relaxations such as single-row MIPs. In fact, Chapter 4 uses the lifting algorithm presented. Chapter 4 presents an extension to the work of Goycoolea (2006) which attempts to evaluate the effectiveness of Mixed Integer Rounding (MIR) and Gomory mixed-integer (GMI) inequalities. By extending his work, natural benchmarks arise, against which any class of cuts derived from single-row MIPs can be compared. Finally, Chapter 5 is dedicated to dealing with an important computational problem when developing any computer code for solving MIPs, namely the problem of numerical accuracy. This problem arises due to the intrinsic arithmetic errors in computer floating-point arithmetic. We propose a simple approach to deal with this issue in the context of generating MIR/GMI inequalities.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd W. Easton / Integer programming is an important discipline in operation research that positively impacts society. Unfortunately, no algorithm currently exists to solve IP's in polynomial time. Researchers are constantly developing new techniques, such as cutting planes, to help solve IPs faster. For example, DeLissa discovered the existence of equality cuts limited to zero and one coefficients for the multiple knapsack equality problem (MKEP). An equality cut is an improper cut because every feasible point satisfies the equality. However, such a cut always reduces the dimension of the linear relaxation space by at least one. This thesis introduces lifted equality cuts, which can have coefficients greater than or equal to two. Two main theorems provide the conditions for the existence of lifted equalities. These theorems provide the foundation for The Algorithm of Lifted Equality Cuts (ALEC), which finds lifted equality cuts in quadratic time. The computational study verifies the benefit of lifted equality cuts in random MKEP instances. ALEC generated millions of lifted equality cuts and reduced the solution time by an average of 15%. To the best of the author's knowledge, ALEC is the first algorithm that has found over 30.7 million cuts on a single problem, while reducing the solving time by 18%.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programs (IPs) are mathematical models that can provide an optimal solution to a variety of different problems. They have been used to reduce costs and optimize organizations. Additionally, IPs are NP-complete resulting in many IPs that cannot be solved. Cutting planes or valid inequalities have been used to decrease the time required to solve IPs. Lifting is a technique that strengthens existing valid inequalities. Lifting inequalities can result in facet defining inequalities, which are the theoretically strongest valid inequalities. Because of these properties, lifting procedures are used in software to reduce the time required to solve an IP. The thesis introduces a new algorithm for exact synchronized simultaneous uplifting over an arbitrary initial inequality for knapsack problems. Synchronized Simultaneous Lifting (SSL) is a pseudopolynomial time algorithm requiring O(nb+n[superscript]3) effort to solve. It exactly uplifts two sets simultaneously into an initial arbitrary valid inequality and creates multiple inequalities of a particular form. This previously undiscovered class of inequalities generated by SSL can be facet defining. A small computational study shows that SSL is quick to execute, requiring on average less than a quarter of a second. Additionally, applying SSL inequalities to a knapsack problem enabled commercial software to solve problems that it could not solve without them.

ropositional proof complexity is the study of the resources that are needed to prove formulas in propositional logic. In this thesis we are concerned with the size and space of proofs, and in particular with the latter. Different approaches to reasoning are captured by corresponding proof systems. The simplest and most well studied proof system is resolution, and we try to get our understanding of other proof systems closer to that of resolution. In resolution we can prove a space lower bound just by showing that any proof must have a large clause. We prove a similar relation between resolution width and polynomial calculus space that lets us derive space lower bounds, and we use it to separate degree and space. For cutting planes we show length-space trade-offs. This is, there are formulas that have a proof in small space and a proof in small length, but there is no proof that can optimize both measures at the same time. We introduce a new measure of space, cumulative space, that accounts for the space used throughout a proof rather than only its maximum. This is exploratory work, but we can also prove new results for the usual space measure. We define a new proof system that aims to capture the power of current SAT solvers, and we show a landscape of length-space trade-offs comparable to those in resolution. To prove these results we build and use tools from other areas of computational complexity. One area is pebble games, very simple computational models that are useful for modelling space. In addition to results with applications to proof complexity, we show that pebble game cost is PSPACE-hard to approximate. Another area is communication complexity, the study of the amount of communication that is needed to solve a problem when its description is shared by multiple parties. We prove a simulation theorem that relates the query complexity of a function with the communication complexity of a composed function. / <p>QC 20170509</p>

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer Programming is used to solve numerous optimization problems. This class of mathematical models aims to maximize or minimize a cost function restricted to some constraints and the solution must be integer. One class of widely studied Integer Program (IP) is the Multiple Knapsack Problem (MKP). Unfortunately, both IPs and MKPs are NP-hard, potentially requiring an exponential time to solve these problems. Utilization of cutting planes is one common method to improve the solution time of IPs. A cutting plane is a valid inequality that cuts off a portion of the linear relaxation space. This thesis presents a new class of cutting planes referred to as merged knapsack cover inequalities (MKCI). These valid inequalities combine information from a cover inequality with a knapsack constraint to generate stronger inequalities. Merged knapsack cover inequalities are generated by the Merging Knapsack Cover Algorithm (MKCA), which runs in linear time. These inequalities may be improved by the Exact Improvement Through Dynamic Programming Algorithm (EITDPA) in order to make them stronger inequalities. Theoretical results have demonstrated that this new class of cutting planes may cut off some space of the linear relaxation region. A computational study was performed to determine whether implementation of merged knapsack cover inequalities is computationally effective. Results demonstrated that MKCIs decrease solution time an average of 8% and decrease the number of ticks in CPLEX, a commercial IP solver, approximately 4% when implemented in appropriate instances.

Le problème de tournées de véhicules (Vehicle Routing Problem - VRP) est un problème d'optimisation combinatoire utilisé généralement pour modéliser et résoudre des différents problèmes rencontrés dans les systèmes logistiques et de transport. Dans cette thèse, nous nous sommes intéressés à l'étude et la résolution d'une classe de problèmes du VRP appelée les problèmes de courses d'orientation (Team Orienteering Problem - TOP). Dans cette catégorie de problèmes, il est a priori impossible de visiter tous les clients en raison de ressources limitées. On associe plutôt un profit à chaque client qui représente sa valeur. Ce profit est collecté lorsque le client est visité par l'un des véhicules disponibles. L'objectif est donc de sélectionner un sous ensemble de clients à servir tout en maximisant le profit total collecté. Dans un premier temps, nous avons introduit une nouvelle généralisation pour le TOP que nous avons appelé le Clustered TOP ou CluTOP. Dans cette variante, les clients sont regroupés en sous-ensembles appelés clusters auxquels nous associons des profits. Pour résoudre cette variante, nous avons proposé un schéma exact basé sur l'approche des plans sécants avec des inégalités valides supplémentaires et des pré-traitements. Nous avons également conçu une méthode heuristique basée sur l'approche order first-cluster second. Cette heuristique hybride combine une heuristique de type Adaptive Large Neighborhood Search qui explore l'espace des solutions et une procédure de découpage qui explore l'espace de recherche des tours géants. De plus, la procédure de découpage est renforcée par une recherche locale afin de mieux explorer l'espace de recherche. Le deuxième problème traité dans ce travail s'appelle le Synchronized Team Orienteering Problem with Time Windows (STOPTW). Cette variante avait été initialement proposée afin de modéliser des scénarios liés à la protection des infrastructures stratégiques menacées par l'avancée des feux de forêts. En plus des contraintes de fenêtres de temps et des visites synchronisées, cette variante considère le cas d'une flotte de véhicules hétérogène. Pour résoudre ce problème, nous avons proposé une méthode heuristique basée sur l'approche GRASP×ILS qui est parvenue à dominer la seule approche existante dans la littérature. La dernière variante du TOP abordée dans cette thèse s'appelle le Set Orienteering Problem (SOP). Les clients dans cette variante sont regroupés en sous-ensembles appelés clusters. Un profit est associé à chaque groupe qui n'est obtenu que si au moins un client est desservi par le véhicule disponible. Nous avons proposé une méthode de coupes avec deux procédures de séparation pour séparer les contraintes d'élimination des sous-tours. Nous avons également proposé un algorithme Mémétique avec une procédure de découpage optimale calculée à l'aide de la programmation dynamique. / The Vehicle Routing Problem (VRP) is a family of Combinatorial Optimization Problems generally used to solve different issues related to transportation systems and logistics. In this thesis, we focused our attention on a variant of the VRP called the Team Orienteering Problem (TOP). In this family of problems, it is a priory impossible to visit all the customers due to travel time limitation on vehicles. Instead, a profit is associated with each customer to represent its value and it is collected once the customer is visited by one of the available vehicles. The objective function is then to maximize the total collected profit with respect to the maximum travel time. Firstly, we introduced a new generalization for the TOP that we called the Clustered TOP (CluTOP). In this variant, the customers are grouped into subsets called clusters to which we associate profits. To solve this variant, we proposed an exact scheme based on the cutting plane approach with additional valid inequalities and pre-processing techniques. We also designed a heuristic method based on the order first-cluster second approach for the CluTOP. This Hybrid Heuristic combines between an ANLS heuristic that explores the solutions space and a splitting procedure that explores the giant tours search space. In addition, the splitting procedure is enhanced by local search procedure in order to enhance its coverage of search space. The second problem treated in this work is called the Synchronized Team Orienteering Problem with Time Windows (STOPTW). This variant was initially proposed in order to model scenarios related to asset protection during escaped wildfires. It considers the case of a heterogeneous fleet of vehicles along with time windows and synchronized visits. To solve this problem, we proposed a heuristic method based on the GRASP×ILS approach that led to a very outstanding results compared to the literature. The last variant of the TOP tackled in this thesis called the Set Orienteering Problem (SOP). Customers in this variant are grouped into subsets called clusters. Each cluster is associated with a profit which is gained if at least one customer is served by the single available vehicle. We proposed a Branch-and-Cut with two separation procedures to separate subtours elimination constraints. We also proposed a Memetic Algorithm with an optimal splitting procedure based on dynamic programming.

The focus of this dissertation is solution strategies for stochastic mixed-integer programs with special structures. Motivation for the methods comes from the relatively sparse number of algorithms for solving stochastic mixed-integer programs. Two stage models with finite support are assumed throughout. The first contribution introduces the nodal decision framework under private information restrictions. Each node in the framework has control of an optimization model which may include stochastic parameters, and the nodes must coordinate toward a single objective in which a single optimal or close-to-optimal solution is desired. However, because of competitive issues, confidentiality requirements, incompatible database issues, or other complicating factors, no global view of the system is possible. An iterative methodology called the nodal decomposition-coordination algorithm (NDC) is formally developed in which each entity in the cooperation forms its own nodal deterministic or stochastic program. Lagrangian relaxation and subgradient optimization techniques are used to facilitate negotiation between the nodal decisions in the system without any one entity gaining access to the private information from other nodes. A computational study on NDC using supply chain inventory coordination problem instances demonstrates that the new methodology can obtain good solution values without violating private information restrictions. The results also show that the stochastic solutions outperform the corresponding expected value solutions. The next contribution presents a new algorithm called scenario Fenchel decomposition (SFD) for solving two-stage stochastic mixed 0-1 integer programs with special structure based on scenario decomposition of the problem and Fenchel cutting planes. The algorithm combines progressive hedging to restore nonanticipativity of the first-stage solution, and generates Fenchel cutting planes for the LP relaxations of the subproblems to recover integer solutions. A computational study SFD using instances with multiple knapsack constraint structure is given. Multiple knapsack constrained problems are chosen due to the advantages they provide when generating Fenchel cutting planes. The computational results are promising, and show that SFD is able to find optimal solutions for some problem instances in a short amount of time, and that overall, SFD outperforms the brute force method of solving the DEP.