Robust Design of Single-Commodity Networks

The results in the present work were obtained in a collaboration with Eduardo Álvarez- Miranda, Valentina Cacchiani, Tim Dorneth, Michael Jünger, Frauke Liers, Andrea Lodi and Tiziano Parriani. The subject of this thesis is a robust network design problem, i.e., a problem of the type “dimension a network such that it has sufficient capacity in all likely scenarios.” In our case, we model the network with an undirected graph in which each scenario defines a supply or demand for each node. We say that a flow in the network is feasible for a scenario if it can balance out its supplies and demands. A scenario polytope B defines which scenarios are relevant. The task is now to find integer capacities that minimize the total installation costs while allowing for a feasible flow in each scenario. This problem is called Single-Commodity Robust Network Design Problem (sRND) and was introduced by Buchheim, Liers and Sanità (INOC 2011). The problem contains the Steiner Tree Problem (given an undirected graph and a terminal set, find a minimum cost subtree that connects all terminals) and therefore is N P-hard. The problem is also a natural extension of minimum cost flows. The network design literature treats the case that the scenario polytope B is given as the finite set of its extreme points (finite case) and that it is given as the feasible region of finitely many linear inequalities (polyhedral case). Both descriptions are equivalent, however, an efficient transformation is not possible in general. Buchheim, Liers and Sanità (INOC 2011) propose a Branch-and-Cut algorithm for the finite case. In this case, there exists a canonical problem formulation as a mixed integer linear program (MIP). It contains a set of flow variables for every scenario. Buchheim, Liers and Sanità enhance the formulation with general cutting planes that are called target cuts. The first part of the dissertation considers the problem variant where every scenario has exactly two terminal nodes. If the underlying network is a complete, unweighted graph, then this problem is the Network Synthesis Problem as defined by Chien (IBM Journal of R&D 1960). There exist polynomial time algorithms by Gomory and Hu (SIAM J. of Appl. Math 1961) and by Kabadi, Yan, Du and Nair (SIAM J. on Discr. Math.) for this special case. However, these algorithms are based on the fact that complete graphs are Hamiltonian. The result of this part is a similar algorithm for hypercube graphs that assumes a special distribution of the supplies and demands. These graphs are also Hamiltonian. The second part of the thesis discusses the structure of the polyhedron of feasible sRND solutions. Here, the first result is a new MIP-based capacity formulation for the sRND problem. The size of this formulation is independent of the number of extreme points of B and therefore, it is also suited for the polyhedral case. The formulation uses so-called cut-set inequalities that are known in similar form from other network design problems. By adapting a proof by Mattia (Computational Optimization and Applications 2013), we show that cut-set inequalities induce facets of the sRND polyhedron. To obtain a better linear programming relaxation of the capacity formulation, we interpret certain general mixed integer cuts as 3-partition inequalities and show that these inequalities induce facets as well. The capacity formulation has exponential size and we therefore need a separation algorithm for cut-set inequalities. In the finite case, we reduce the cut-set separation problem to a minimum cut problem that can be solved in polynomial time. In the polyhedral case, however, the separation problem is N P-hard, even if we assume that the scenario polytope is basically a cube. Such a scenario polytope is called Hose polytope. Nonetheless, we can solve the separation problem in practice: We show a MIP based separation procedure for the Hose scenario polytope. Additionally, the thesis presents two separation methods for 3-partition inequalities. These methods are independent of the encoding of the scenario polytope. Additionally, we present several rounding heuristics. The result is a Branch-and-Cut algorithm for the capacity formulation. We analyze the algorithm in the last part of the thesis. There, we show experimentally that the algorithm works in practice, both in the finite and in the polyhedral case. As a reference point, we use a CPLEX implementation of the flow based formulation and the computational results by Buchheim, Liers and Sanità. Our experiments show that the new Branch-and-Cut algorithm is an improvement over the existing approach. Here, the algorithm excels on problem instances with many scenarios. In particular, we can show that the MIP separation of the cut-set inequalities is practical

Generating general-purpose cutting planes for mixed-integer programs

Franz WesselmannPaderborn, Univ., Diss., 201

When Less Is More: Consequence-Finding in a Weak Theory of Arithmetic

This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication symbol, which retains many of the desirable algorithmic properties of linear arithmetic. In particular, we show that the conjunctive fragment of the theory can be effectively manipulated (analogously to the usual operations on convex polyhedra, the conjunctive fragment of linear arithmetic). As a result, we can solve the following consequence-finding problem: given a ground formula F, find the strongest conjunctive formula that is entailed by F. As an application of consequence-finding, we give a loop invariant generation algorithm that is monotone with respect to the theory and (in a sense) complete. Experiments show that the invariants generated from the consequences are effective for proving safety properties of programs that require non-linear reasoning

A Novel Approach to Tightening Semidefinite Relaxations for Certain Combinatorial Problems

RÉSUMÉ : Ce mémoire présente une nouvelle famille de coupes nommées contraintes polytopiques kprojection (kPPCs) qui peuvent être utilisées pour résoudre certains problèmes quadratiques binaires. Notamment les problèmes qui satisfont une propriété de projection pour les solutions réalisables sur un sous-graphe induit ont la même structure que les solutions faisables sur le graphe entier. Parmi ces problèmes se trouvent le problème max-cut et le problème d’ensemble stable (stable set problem). Les coupes sont généralement des inégalités, cependant les kPPCs s’en distinguent par le fait qu’elles sont formées d’un ensemble d’inégalités. De plus, elle peuvent être définies pour un seul sous-graphe induit ou pour un ensemble de sous-graphes induits, et sont utilisées pour resserrer les relaxations en programmation semi-définie. Trois aspects des kPPCs sont examinés dans ce mémoire : une hiérarchie qui converge vers une formulation exacte, une formulation pour trouver la contrainte kPPC la plus violée, et l’amélioration de la borne supérieure (pour un problème de maximisation) d’une implémentation pratique de kPPCs pour le problème max-cut. La relaxation SDP avec kPPCs forme une hiérarchie. Le kème niveau de la hiérarchie est la relaxation SDP avec kPPCs pour tous les sous-graphes induits de taille k. Lorsque k augmente, l’intensité de la relaxation augmente également puisque CUTk ⊆ CUTk+1 où CUTk est le polytope de coupe de taille k. Au nème niveau, la formulation n’est plus une relaxation et rejoint exactement le problème d’origine CUTn. Il existe n/k sous-graphes induits uniques pour un graphe à n noeuds. Par conséquent, il n’est possible d’énumérer explicitement les niveaux de la hiérarchie que pour de petits exemples. Cependant, la force de la hiérarchie des kPPCs est que la matrice semi-définie positive, qui est variable dans la relaxation SDP, n’augmente pas en taille lorsque le niveau augmente, contrairement aux hiérarchies de Lasserre. Pour un sous-graphe induit donné I, un modèle d’optimisation (nommé distance-au-polytope) est présenté pour déterminer si la solution optimale de la relaxation SDP viole les kPPCs pour I et, dans l’affirmative, pour quantifier la violation. Le modèle distance-au-polytope a une fonction objectif quadratique, des contraintes linéaires et se résout rapidement. La solution optimale est la distance euclidienne entre le mineur principal de la solution optimale de la relaxation (X*I) et le polytope de coupe (CUT|I|). Si la distance est égale à zéro, alors l’inclusion de kPPCs pour I dans la relaxation SDP ne resserrera pas la borne. Si la distance est strictement supérieure à zéro, alors les kPPCs pour I ne sont pas satisfaites par la solution courante. Par conséquent, leur inclusion dans la relaxation SDP changera la solution courante X* (bien qu’une amélioration de la borne ne soit pas garantie). Ce mémoire présente un modèle d’optimisation binaire-mixte dans un cône de second ordre (SOC) qui, pour un k donné, trouve la kPPC la plus éloignée du polytope de coupe. Le problème interne est le modèle distance-au-polytope. Le problème externe comporte des variables binaires qui prennent en compte tous les sous-graphes induits de taille k. Les problèmes à deux niveaux sont intrinsèquement difficiles à résoudre. Une reformulation est donc présentée qui change le problème à deux niveaux en un problème SOC équivalent à un seul niveau. La reformulation utilise des techniques telles que les conditions KKT, les contraintes disjointes et le saut de dualité. De plus, nous montrons comment renforcer le modèle à un seul niveau en incluant des contraintes de bris de symétrie et en incluant des variables binaires additionnelles qui réduisent la taille de l’arbre d’énumération. MOSEK est utilisé pour résoudre le problème et les résultats sont présentés jusqu’à la taille 20. À chaque itération d’une méthode de plan sécant, une relaxation est résolue et, si un critère d’arrêt n’est pas atteint, une procédure de séparation cherche les coupes violées ou valides à ajouter à la relaxation. Ce mémoire présente un algorithme de plan sécant utilisant les kPPCs pour le problème max-cut. Notre méthode de plan sécant comporte 3 étapes. La première résout la relaxation SDP simple pour fournir une solution optimale initiale. La seconde résout itérativement la relaxation SDP simple à laquelle s’ajoute des inégalités triangulaires. À chaque itération, l’ensemble des inégalités triangulaires est composé, d’une part, de certaines inégalités triangulaires qui sont violées par la solution précédente et, d’autre part, des inégalités triangulaires actives de l’itération précédente. Les inégalités non actives ne sont pas saturées et ne sont par conséquent pas conservées. La troisième étape débute quand l’étape 2 n’apporte plus d’amélioration significative : des kPPCs sont ajoutées au modèle (relaxation SDP simple avec inégalités triangulaires fournies par la dernière itération de l’étape 2). Pour trouver les kPPCs violées, la procédure de séparation résout le problème distance-aupolytope pour les indices générés à partir des inégalités triangulaires violées. Cette méthode donne de meilleurs résultats que la sélection aléatoire des sous-graphes induits pour en tester la violation. En particulier, nous montrons que davantage de kPPCs violées sont trouvées et que la violation est plus grande. Finalement, nous présentons des résultats numériques (pour n = 500 − 1000) montrant que, lorsque l’amélioration de la borne à partir d’inégalités triangulaires est faible, les kPPCs sont encore capables de resserrer la relaxation.----------ABSTRACT : This thesis introduces a new family of cuts called k-projection polytope constraints (kPPCs)that can be used to solve certain binary quadratic problems. Specifically those problems that satisfy a projection property in which feasible solutions on an induced subgraph have the same structure as feasible solutions on the full graph, such as the max-cut problem and the stable set problem. Typically cuts (also called valid inequalities) are inequalities, however kPPCs differ as they are a set of equalities. Furthermore they can be defined for a single induced subgraph or a set of induced subgraphs and are used to tighten semidefinite programming (SDP) relaxations. Three aspects of kPPCs are examined in this thesis: a hierarchy that converges to an exact formulation, a formulation to find the most violated kPPC and a practical implementation of a cutting plane algorithm using kPPCs that improves the upper bound (of a maximization problem) for the max-cut problem. The SDP relaxation with kPPCs forms a hierarchy. The kth level of the hierarchy is the SDP relaxation with kPPCs for all induced subgraphs of size k. As k increases, the strength of the relaxation also increases since CUTk ⊆ CUTk+1 where CUTk is the cut polytope of size k. At the nth level the formulation is no longer a relaxation and defines the original problem, CUTn, exactly. There are n/K unique induced subgraphs for a graph with n vertices. Therefore explicitly producing the levels of the hierarchy is only possible for small examples. However the strength of the hierarchy of kPPCs is that the positive semidefinite matrix variable in the SDP relaxation does not grow in size as the level is increased. This is in contrast to other hierarchies including the Lasserre hierarchy. For a given induced subgraph I, an optimization model (denoted distance-to-polytope) is presented to determine if the optimal solution to an SDP relaxation violates the kPPC for I and, if so, to quantify the violation. The distance-to-polytope model has a quadratic objective function, linear constraints and solves quickly. The optimal solution is the euclidean distance between the principal minor of the optimal solution to the relaxation (X*I ) and the cut polytope (CUT|I|). If the distance equals zero then including the kPPC for I in the SDP relaxation will not tighten the bound. If the distance is strictly greater than zero then the kPPC for I is not satisfied by the current solution. Therefore including it in the SDP relaxation will change the current solution X* (although a strict improvement in the bound is not guaranteed). The maximally violated valid inequality problem (MVVIP) determines the valid inequality from a family of cuts that is most violated. This thesis examines this problem for kPPCs. Specifically we present a mixed-binary second order cone optimization model that, for a given k, finds the kPPC that is furthest from the cut polytope. The inner problem is the distance-to-polytope model. The outer problem includes binary variables that consider all induced subgraphs of size k. Bilevel problems are inherently hard to solve. A reformulation is presented that changes the bilevel model into an equivalent single level second order cone problem. The reformulation uses techniques such as KKT conditions, disjunctive constraints and the duality gap. Moreover we show how to strengthen the single level model by including symmetry breaking constraints and including additional binary variables that reduce the size of the enumeration tree. MOSEK is used to solve the problem and results are presented up to size 20. At each iteration of a cutting plane method a relaxation is solved and if a stopping criteria is not met a separation procedure looks for violated and valid cuts to add to the relaxation. This thesis presents a cutting plane algorithm using kPPCs for the max-cut problem. There are 3 stages in our cutting plane method. The first solves the basic SDP relaxation to give an initial optimal solution. The second stage iteratively solves the basic SDP relaxation plus some triangle inequalities. At each iteration the set of triangle inequalities is composed of some triangle inequalities that are violated by the previous solution and the triangle inequalities from the previous iteration that are active. The non-active inequalities are not binding and therefore are not kept. When there are no more violated triangle inequalities (or the improvement has stalled) we begin the third stage in which kPPCs are added to the model (basic SDP relaxation plus triangle inequalities from the last iteration of stage 2). The separation procedure to find violated kPPCs solves the distance-to-polytope problem for indices generated from violated triangle inequalities. Compared to randomly selecting induced subgraphs to test for violation, generating them from the indices used in triangle inequalities gives better results. Specifically we show that more violated kPPCs are found and that the amount of violation is larger. Finally we examine dense graphs of size 500 to 1000 and present computational results showing that kPPCs are able to improve the bound even after triangle inequalities can no longer tighten the relaxation

Distances to lattice points in rational polyhedra

Let a ∈ Z n >0 , n ≥ 2 , gcd(a) := gcd(a1 , . . . , an ) = 1, b ∈ Z≥0 and denote by k · k∞ the ℓ∞-norm. Consider the knapsack polytope P(a, b) = { x ∈ R n ≥0 : a T x = b and assume that P(a, b) ∩ Z n 6= ; holds. The main result of this thesis states that for any vertex x ∗ of the knapsack polytope P(a, b) there exists a feasible integer point z ∗ ∈ P(a, b) such that, denoting by s the size of the support of z ∗ , i.e. the number of nonzero components in z ∗ and upon assuming s > 0 , the inequality kx ∗ − z ∗ k∞ 2 s−1 s < kak∞ holds. This inequality may be viewed as a transference result which allows strengthening the best known distance (proximity) bounds if integer points are not sparse and, vice versa, strengthening the best known sparsity bounds if feasible integer points are sufficiently far from a vertex of the knapsack polytope. In particular, this bound provides an exponential in s improvement on the previously best known distance bounds in the knapsack scenario. Further, when considering general integer linear programs, we show that a resembling inequality holds for vertices of Gomory’s corner polyhedra [49, 96]. In addition, we provide several refinements of the known distance and support bounds under additional assumption

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

Applications of simulation and optimization techniques in optimizing room and pillar mining systems

The goal of this research was to apply simulation and optimization techniques in solving mine design and production sequencing problems in room and pillar mines (R&P). The specific objectives were to: (1) apply Discrete Event Simulation (DES) to determine the optimal width of coal R&P panels under specific mining conditions; (2) investigate if the shuttle car fleet size used to mine a particular panel width is optimal in different segments of the panel; (3) test the hypothesis that binary integer linear programming (BILP) can be used to account for mining risk in R&P long range mine production sequencing; and (4) test the hypothesis that heuristic pre-processing can be used to increase the computational efficiency of branch and cut solutions to the BILP problem of R&P mine sequencing. A DES model of an existing R&P mine was built, that is capable of evaluating the effect of variable panel width on the unit cost and productivity of the mining system. For the system and operating conditions evaluated, the result showed that a 17-entry panel is optimal. The result also showed that, for the 17-entry panel studied, four shuttle cars per continuous miner is optimal for 80% of the defined mining segments with three shuttle cars optimal for the other 20%. The research successfully incorporated risk management into the R&P production sequencing problem, modeling the problem as BILP with block aggregation to minimize computational complexity. Three pre-processing algorithms based on generating problem-specific cutting planes were developed and used to investigate whether heuristic pre-processing can increase computational efficiency. Although, in some instances, the implemented pre-processing algorithms improved computational efficiency, the overall computational times were higher due to the high cost of generating the cutting planes --Abstract, page iii