Download or read book Maximal Lattice Free Polyhedra in Mixed Integer Cutting Plane Theory written by Christian Wagner and published by Cuvillier Verlag. This book was released on 2011-11-15 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis deals with the generation, evaluation, and analysis of cutting planes for mixed-integer linear programs (MILP's). Such optimization problems involve finitely many variables, some of which are required to be integer. The aim is to maximize or minimize a linear objective function over a set of finitely many linear equations and inequalities. Many industrial problems can be formulated as MILP's. The presence of both, discrete and continuous variables, makes it difficult to solve MILP's algorithmically. The currently available algorithms fail to solve many real-life problems in acceptable time or can only provide heuristic solutions. As a consequence, there is an ongoing interest in novel solution techniques. A standard approach to solve MILP's is to apply cutting plane methods. Here, the underlying MILP is used to construct a sequence of linear programs whose formulations are improved by successively adding linear constraints – so-called cutting planes – until one of the linear programs has an optimal solution which satisfies the integrality conditions on the integer constrained variables. For many combinatorial problems, it is possible to immediately deduce several families of cutting planes by exploiting the inherent combinatorial structure of the problem. However, for general MILP's, no structural properties can be used. The generation of cutting planes must rather be based on the objective function and the given, unstructured set of linear equations and inequalities. On the one hand, this makes the derivation of strong cutting planes for general MILP's more difficult than the derivation of cutting planes for structured problems. On the other hand, for this very reason, the analysis of cutting plane generation for general MILP's becomes mathematically interesting. This thesis presents an approach to generate cutting planes for a general MILP. The cutting planes are obtained from lattice-free polyhedra, that is polyhedra without interior integer point. The point of departure is an optimal solution of the linear programming relaxation of the underlying MILP. By considering multiple rows of an associated simplex tableau, a further relaxation is derived. The first part of this thesis is dedicated to the analysis of this relaxation and it is shown how cutting planes for the general MILP can be deduced from the considered relaxation. It turns out that the generated cutting planes have a geometric interpretation in the space of the discrete variables. In particular, it is shown that the strongest cutting planes which can be derived from the considered relaxation correspond to maximal lattice-free polyhedra. As a result, problems on cutting planes are transferable into problems on maximal lattice-free polyhedra. The second part of this thesis addresses the evaluation of the generated cutting planes. It is shown that the cutting planes which are important, are at the same time the cutting planes which are difficult to derive in the sense that they correspond to highly complex maximal lattice-free polyhedra. In addition, it is shown that under certain assumptions on the underlying system of linear equations and inequalities, the important cutting planes can be approximated with cutting planes which correspond to less complex maximal lattice-free polyhedra. A probabilistic model is used to complement the analysis. Moreover, a geometric interpretation of the results is given. The third part of this thesis focuses on the analysis of lattice-free polyhedra. In particular, the class of lattice-free integral polyhedra is investigated, a class which is important within a cutting plane framework. Two different notions of maximality are introduced. It is distinguished into the class of lattice-free integral polyhedra which are not properly contained in another lattice-free integral polyhedron, and the class of lattice-free integral polyhedra which are not properly contained in another lattice-free convex set. Both classes are analyzed, especially with respect to the properties of their representatives and the relation between the two classes. It is shown that both classes are of large cardinality and that they contain very large elements. For the second as well as the third part of this thesis, statements about two-dimensional lattice-free convex sets are needed. For that reason, the fourth part of this thesis is devoted to the derivation of these results.
Download or read book Proceedings Of The International Congress Of Mathematicians 2010 Icm 2010 In 4 Volumes Vol I Plenary Lectures And Ceremonies Vols Ii iv Invited Lectures written by Rajendra Bhatia and published by World Scientific. This book was released on 2011-06-06 with total page 4137 pages. Available in PDF, EPUB and Kindle. Book excerpt: ICM 2010 proceedings comprises a four-volume set containing articles based on plenary lectures and invited section lectures, the Abel and Noether lectures, as well as contributions based on lectures delivered by the recipients of the Fields Medal, the Nevanlinna, and Chern Prizes. The first volume will also contain the speeches at the opening and closing ceremonies and other highlights of the Congress.
Download or read book Facets of Combinatorial Optimization written by Michael Jünger and published by Springer Science & Business Media. This book was released on 2013-07-03 with total page 510 pages. Available in PDF, EPUB and Kindle. Book excerpt: Martin Grötschel is one of the most influential mathematicians of our time. He has received numerous honors and holds a number of key positions in the international mathematical community. He celebrated his 65th birthday on September 10, 2013. Martin Grötschel’s doctoral descendant tree 1983–2012, i.e., the first 30 years, features 39 children, 74 grandchildren, 24 great-grandchildren and 2 great-great-grandchildren, a total of 139 doctoral descendants. This book starts with a personal tribute to Martin Grötschel by the editors (Part I), a contribution by his very special “predecessor” Manfred Padberg on “Facets and Rank of Integer Polyhedra” (Part II), and the doctoral descendant tree 1983–2012 (Part III). The core of this book (Part IV) contains 16 contributions, each of which is coauthored by at least one doctoral descendant. The sequence of the articles starts with contributions to the theory of mathematical optimization, including polyhedral combinatorics, extended formulations, mixed-integer convex optimization, super classes of perfect graphs, efficient algorithms for subtree-telecenters, junctions in acyclic graphs and preemptive restricted strip covering, as well as efficient approximation of non-preemptive restricted strip covering. Combinations of new theoretical insights with algorithms and experiments deal with network design problems, combinatorial optimization problems with submodular objective functions and more general mixed-integer nonlinear optimization problems. Applications include VLSI layout design, systems biology, wireless network design, mean-risk optimization and gas network optimization. Computational studies include a semidefinite branch and cut approach for the max k-cut problem, mixed-integer nonlinear optimal control, and mixed-integer linear optimization for scheduling and routing of fly-in safari planes. The two closing articles are devoted to computational advances in general mixed integer linear optimization, the first by scientists working in industry, the second by scientists working in academia. These articles reflect the “scientific facets” of Martin Grötschel who has set standards in theory, computation and applications.
Download or read book New Developments on Polyhedral Methods for Mixed integer Programming written by Ming Zhao and published by . This book was released on 2008 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: Branch-and-cut is today the premier method for solving mixed-integer programming (MIP) problems. The tremendous success of branch-and-cut software is due, in great part, to the developments in the theory of inequality description for MIP polyhedra that took place in the last twenty years. This dissertation is about polyhedral theory for MIP and its use within branch-and-cut to solve difficult instances of MIP efficiently. We focus on two directions. The first is cutting planes for general, unstructured MIP. The second takes advantage of combinational structures that are pervasive in applications. Specifically, we study the polyhedron defined by a knapsack constraint and by special ordered sets of type 2 (SOS2), and by a knapsack and cardinality constraints. A family of inequalities of great importance for solving general MIP is mixed integer rounding (MIR). Here we investigate extensions of MIR inequalities, in the hope of learning more about the structure of general MIP polyhedra, the intractability of MIP, and how MIR may be generalized to give inequalities that are more efficient than the state-of-the-art. We accomplish this by studying extensions of the mixing-MIR set. Specifically, we consider the mixing-MIR set with divisible capacities, the mixing-MIR set with two nondivisible capacities, and the continuous mixing set. We establish the computational complexity of optimization over these sets, the structure of their convex hulls, and we point out how to derive cutting planes valid for them efficiently. A nonlinear function can be approximated to an arbitrary degree of accuracy by a piecewise linear function. This is the main application of piecewise linear optimization (PLO), although it arise on its own in a number of important applications, e.g. economics of scale. In the case of separable continuous PLO, it is best to model the piecewise linear function through SOS2. We give several families of inequalities valid for a set defined by a knapsack and SOS2 constraints. We give separation heuristics for the families of inequalities. Finally, we use them in a branch-and-cut algorithm to solve difficult instances of PLO. As our computational results clearly show, the use of such inequalities can improve considerably our ability to solve such problems, to proven optimality or even heuristically, by branch-and-cut. One of the most difficult, and at the same time important, constraint in operations research is that at most a fixed number of variables in the model is nonzero, such cardinality constraint arises, for example in portfolio selection, data mining, and metabolic engineering. We study the polyhedron defined by a knapsack and a cardinality constraints. Our model and our results generalize significantly the state-of-the art. In particular, our knapsack set proves to be considerably richer than the ones studied so far in the literature. Finally, we present conclusions and discuss directions for further research, some of which we are currently investigating.
Download or read book Integer Points in Polyhedra written by Alexander Barvinok and published by European Mathematical Society. This book was released on 2008 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a self-contained exposition of several core aspects of the theory of rational polyhedra with a view towards algorithmic applications to efficient counting of integer points, a problem arising in many areas of pure and applied mathematics. The approach is based on the consistent development and application of the apparatus of generating functions and the algebra of polyhedra. Topics range from classical, such as the Euler characteristic, continued fractions, Ehrhart polynomial, Minkowski Convex Body Theorem, and the Lenstra-Lenstra-Lovasz lattice reduction algorithm, to recent advances such as the Berline-Vergne local formula. The text is intended for graduate students and researchers. Prerequisites are a modest background in linear algebra and analysis as well as some general mathematical maturity. Numerous figures, exercises of varying degree of difficulty as well as references to the literature and publicly available software make the text suitable for a graduate course.
Download or read book INFORMS Annual Meeting written by Institute for Operations Research and the Management Sciences. National Meeting and published by . This book was released on 2009 with total page 644 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Theory and Computation for Cutting plane Methods in Discrete Optimization written by Haoran Zhu (Ph.D.) and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation is about cutting-planes for discrete optimization problems. We study cutting-planes to understand their structure and to provide strong formulations for two specific problem classes: linear programs with complementarity constraints and multidimensional knapsack problems.We begin by investigating the polyhedrality of the cutting-plane closure with respect to some particular families of cuts. For one specific type of cutting-planes, the cutting-plane closure refers to the intersection of all of these cutting-planes. By showing that the set of all integer packing sets is well-quasi-ordered by inclusion, we directly answer and generalize the first half of an open question proposed by Bodur et al. [27], establishing that the k-aggregation closure of packing sets are polyhedral. To answer a different open question in [27], we give a characterization of when a cutting-plane closure is polyhedron. As applications of such general characterization, later we specifically prove a more general version for the second half of the open question in Bodur et al. [27], showing that the k-aggregation closure of a covering set is polyhedral, and we show that the Chva̹tal-Gomory closure of a Motzkin-decomposable set with a rational polyhedral cone is a rational polyhedron. The latter result generalizes and unifies all the currently known results in the literature, for the case of a rational polyhedron and a compact convex set. The second problem we study is a linear program with complementarity constraints (LPCC). The complementarity constraints we study here can have overlapping variables. This is a more general setting than the widely-studied KKT-type of complementarity constraints, in which each variable appears in at most one complementarity constraint. We provide a novel extended relaxation of the problem that is based on the graph theoretic concept of a vertex cover. Our work generalizes the extended reformulation-linearization technique in the literature [123] to problems with more general complementarity constraints. We then discuss how to obtain strong cutting-planes for our extended formulation. Many of our proposed cutting-planes arise from the boolean quadric polytope associated with a complete bipartite graph. For three types of practical problems, computational results demonstrate that the combination of our proposed linear relaxation and new cutting-planes is significantly better than other relaxations in terms of optimality gap closed. As a special case of LPCC, and as a bridging work to the knapsack problem studied in the last chapters, we perform a polyhedral study for the complementarity knapsack problem. The feasible set of this problem is given by a single knapsack constraint and complementarity conditions [51, 65]. For this problem, we propose three new families of cutting-planes that are obtained from a combinatorial concept known as a pack. Sufficient conditions for these inequalities to be facet-defining, based on the new concept of a maximal switching pack, are also provided. These inequalities are fundamentally different from those in literature, in the sense that they cannot be obtained from lifting. Moreover, we answer positively a conjecture by de Farias et al. [65] about the separation complexity of the inequalities introduced in their work. The third problem we study is the 0-1 multi-dimensional knapsack problem. We propose a new method to generate cutting-planes from multiple covers of knapsack constraints. The valid inequalities we derive for the knapsack problem have a number of interesting properties. First, they generalize the well-known (1,k)-configuration inequalities. Second, they are not aggregation cuts. Third, they cannot be generated as rank-1 Chva̹tal-Gomory cut from the inequality system consisting of the knapsack constraints and all their minimal cover inequalities. We also provide conditions under which the inequalities are facets for the convex hull of the feasible set as well as conditions for those inequalities to fully characterize the convex hull. Later we give an integer program to solve the associated separation problem and provide numerical experiments to showcase the strength of these new inequalities. In the final chapter, we close three open problems in the separation complexity of valid inequalities for the knapsack polytope. Specifically, we establish that the separation problems for extended cover inequalities, (1,k)-configuration inequalities, and weight inequalities are all NP-complete. We also give a number of important special cases where the separation problems can be solved in polynomial time.
Download or read book Graphs and Polyhedra written by A. M. H. Gerards and published by . This book was released on 1990 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mixed discrete Programming Over a Convex Polyhedron written by Mohammad Reza Shaahinfar and published by . This book was released on 1981 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mixed Integer Nonlinear Programming written by Jon Lee and published by Springer Science & Business Media. This book was released on 2011-12-02 with total page 687 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many engineering, operations, and scientific applications include a mixture of discrete and continuous decision variables and nonlinear relationships involving the decision variables that have a pronounced effect on the set of feasible and optimal solutions. Mixed-integer nonlinear programming (MINLP) problems combine the numerical difficulties of handling nonlinear functions with the challenge of optimizing in the context of nonconvex functions and discrete variables. MINLP is one of the most flexible modeling paradigms available for optimization; but because its scope is so broad, in the most general cases it is hopelessly intractable. Nonetheless, an expanding body of researchers and practitioners — including chemical engineers, operations researchers, industrial engineers, mechanical engineers, economists, statisticians, computer scientists, operations managers, and mathematical programmers — are interested in solving large-scale MINLP instances.
Download or read book Theory of Linear and Integer Programming written by Alexander Schrijver and published by John Wiley & Sons. This book was released on 1998-06-11 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: Als Ergänzung zu den mehr praxisorientierten Büchern, die auf dem Gebiet der linearen und Integerprogrammierung bereits erschienen sind, beschreibt dieses Werk die zugrunde liegende Theorie und gibt einen Überblick über wichtige Algorithmen. Der Autor diskutiert auch Anwendungen auf die kombinatorische Optimierung; neben einer ausführlichen Bibliographie finden sich umfangreiche historische Anmerkungen.
Download or read book Disjunctive Programming written by Egon Balas and published by Springer. This book was released on 2018-11-27 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: Disjunctive Programming is a technique and a discipline initiated by the author in the early 1970's, which has become a central tool for solving nonconvex optimization problems like pure or mixed integer programs, through convexification (cutting plane) procedures combined with enumeration. It has played a major role in the revolution in the state of the art of Integer Programming that took place roughly during the period 1990-2010. The main benefit that the reader may acquire from reading this book is a deeper understanding of the theoretical underpinnings and of the applications potential of disjunctive programming, which range from more efficient problem formulation to enhanced modeling capability and improved solution methods for integer and combinatorial optimization. Egon Balas is University Professor and Lord Professor of Operations Research at Carnegie Mellon University's Tepper School of Business.
Download or read book Integer Programming written by Michele Conforti and published by Springer. This book was released on 2014-11-15 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an elegant and rigorous presentation of integer programming, exposing the subject’s mathematical depth and broad applicability. Special attention is given to the theory behind the algorithms used in state-of-the-art solvers. An abundance of concrete examples and exercises of both theoretical and real-world interest explore the wide range of applications and ramifications of the theory. Each chapter is accompanied by an expertly informed guide to the literature and special topics, rounding out the reader’s understanding and serving as a gateway to deeper study. Key topics include: formulations polyhedral theory cutting planes decomposition enumeration semidefinite relaxations Written by renowned experts in integer programming and combinatorial optimization, Integer Programming is destined to become an essential text in the field.
Download or read book Integer Points in Polyhedra Geometry Number Theory Representation Theory Algebra Optimization Statistics written by Matthias Beck and published by American Mathematical Soc.. This book was released on 2008 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume is a cross section of recent advances connected to lattice-point questions. Topics range from commutative algebra to optimization, from discrete geometry to statistics, from mirror symmetry to geometry of numbers. The book is suitable for resarchers and graduate students interested in combinatorial aspects of the above fields.
Download or read book The Gomory Chv tal Closure written by Juliane Dunkel and published by . This book was released on 2011 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we examine theoretical aspects of the Gomory-Chvátal closure of polyhedra. A Gomory-Chvátal cutting plane for a polyhedron P is derived from any rational inequality that is valid for P by shifting the boundary of the associated half-space towards the polyhedron until it intersects an integer point. The Gomory-ChvAital closure of P is the intersection of all half-spaces defined by its Gomory-Chvátal cuts. While it is was known that the separation problem for the Gomory-Chvátal closure of a rational polyhedron is NP-hard, we show that this remains true for the family of Gomory-Chvátal cuts for which all coefficients are either 0 or 1. Several combinatorially derived cutting planes belong to this class. Furthermore, as the hyperplanes associated with these cuts have very dense and symmetric lattices of integer points, these cutting planes are in some- sense the "simplest" cuts in the set of all Gomory-Chvátal cuts. In the second part of this thesis, we answer a question raised by Schrijver (1980) and show that the Gomory-Chvátal closure of any non-rational polytope is a polytope. Schrijver (1980) had established the polyhedrality of the Gomory-Chvdtal closure for rational polyhedra. In essence, his proof relies on the fact that the set of integer points in a rational polyhedral cone is generated by a finite subset of these points. This is not true for non-rational polyhedral cones. Hence, we develop a completely different proof technique to show that the Gomory-Chvátal closure of a non-rational polytope can be described by a finite set of Gomory-Chvátal cuts. Our proof is geometrically motivated and applies classic results from polyhedral theory and the geometry of numbers. Last, we introduce a natural modification of Gomory-Chvaital cutting planes for the important class of 0/1 integer programming problems. If the hyperplane associated with a Gomory-Chvátal cut for a polytope P C [0, 1]' does not contain any 0/1 point, shifting the hyperplane further towards P until it intersects a 0/1 point guarantees that the resulting half-space contains all feasible solutions. We formalize this observation and introduce the class of M-cuts that arises by strengthening the family of Gomory- Chvátal cuts in this way. We study the polyhedral properties of the resulting closure, its complexity, and the associated cutting plane procedure.
Download or read book Solving Pseudo convex Mixed Integer Problems by Cutting Plane Techniques written by Tapio Westerlund and published by . This book was released on 2001 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
