Springer Series in Operations Research and Financial Engineering. Series Editors: Thomas V. Mikosch Sidney I. Resnick Stephen M.

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1 Springer Series in Operations Research and Financial Engineering Series Editors: Thomas V. Mikosch Sidney I. Resnick Stephen M. Robinson

2 Springer Series in Operations Research and Financial Engineering Altiok: Performance Analysis of Manufacturing Systems Birge and Louveaux: Introduction to Stochastic Programming Bonnans and Shapiro: Perturbation Analysis of Optimization Problems Bramel, Chen, and Simchi-Levi: The Logic of Logistics: Theory, Algorithms, and Applications for Logistics and Supply Chain Management (second edition) Dantzig and Thapa: Linear Programming 1: Introduction Dantzig and Thapa: Linear Programming 2: Theory and Extensions de Haan and Ferreira: Extreme Value Theory: An Introduction Drezner (Editor): Facility Location: A Survey of Applications and Methods Facchinei and Pang: Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I Facchinei and Pang: Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume II Fishman: Discrete-Event Simulation: Modeling, Programming, and Analysis Fishman: Monte Carlo: Concepts, Algorithms, and Applications Haas: Stochastic Petri Nets: Modeling, Stability, Simulation Klamroth: Single-Facility Location Problems with Barriers Muckstadt: Analysis and Algorithms for Service Parts Supply Chains Nocedal and Wright: Numerical Optimization Olson: Decision Aids for Selection Problems Pinedo: Planning and Scheduling in Manufacturing and Services Pochet and Wolsey: Production Planning by Mixed Integer Programming Resnick: Extreme Values, Regular Variation, and Point Processes Resnick: Heavy Tail Phenomena: Probabilistic and Statistical Modeling Simchi-Levi: The Logic of Logistics: Theory, Algorithms, and Applications for Logistics and Supply Chain Management Whitt: Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues Yao (Editor): Stochastic Modeling and Analysis of Manufacturing Systems Yao and Zheng: Dynamic Control of Quality in Production-Inventory Systems: Coordination and Optimization Yeung and Petrosyan: Cooperative Stochastic Differential Games

3 Jean Bernard Lasserre Linear and Integer Programming vs Linear Integration and Counting A Duality Viewpoint 123

4 Jean Bernard Lasserre LAAS-CNRS and Institute of Mathematics University of Toulouse LAAS, 7 avenue du Colonel Roche Toulouse Cédex 4 France lasserre@laas.fr Series Editors: Thomas V. Mikosch University of Copenhagen Laboratory of Actuarial Mathematics DK-1017 Copenhagen Denmark mikosh@act.ku.dk Sidney I. Resnick Cornell University School of Operations Research and Industrial Engineering Ithaca, NY U.S.A. sirl@cornell.edu Stephen M. Robinson University of Wisconsin-Madison Department of Industrial Engineering Madision, WI U.S.A. smrobins@facstaff.wise.edu ISSN ISBN e-isbn DOI / Springer Dordrecht Heidelberg London New York Library of Congress Control Number: Mathematics Subject Classification (2000): 60J05, 37A30, 28A33, 47A35 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To my daughter Julia and to Carole

6 Preface Integer programming (IP) is a fascinating topic. Indeed, while linear programming (LP), its continuous analogue, is well understood and extremely efficient LP software packages exist, solving an integer program can remain a formidable challenge, even for some small size problems. For instance, the following small (5-variable) IP problem (called the unbounded knapsack problem) min{213x x x x x 5 } s.t x x x x x 5 = , x 1,x 2,x 3,x 4,x 5 N, taken from a list of difficult knapsack problems in Aardal and Lenstra [2], is not solved even by hours of computing, using for instance the last version of the efficient software package CPLEX. However, this is not a book on integer programming, as very good ones on this topic already exist. For standard references on the theory and practice of integer programming, the interested reader is referred to, e.g., Nemhauser and Wolsey [113], Schrijver [121], Wolsey [136], and the more recent Bertsimas and Weismantel [21]. On the other hand, this book could provide a complement to the above books as it develops a rather unusual viewpoint. Indeed, one first goal of this book is to analyze and develop some striking analogies between four problems, all related to an integer program P d (the subscript d being for discrete); namely its associated linear programming problem P, its associated linear integration problem I, and its associated linear summation (or linear counting) problem I d.infact,whilep and P d are the respective (max,+)-algebra analogues of I and I d, P d, and I d are the respective discrete analogues of P and I. In addition, the same simple relationship links the value of P with that of I on the one hand, and the value of P d with that of I d on the other hand. If LP duality is of course well understood (as a special case of Legendre Fenchel duality in convex analysis), IP duality is much less developed although there is a well-known superadditive dual associated with P d. On the other hand, the linear integration and linear counting problems I and I d have well-defined respective dual problems I and I d, although they are not qualified as such in the literature. Indeed, I (resp., I d ) is obtained from the inverse Laplace transform (resp., inverse Z-transform) applied to the Laplace transform (resp., Z-transform) of the value function, exactly in the same way the LP dual P is obtained from the Legendre Fenchel transform applied to the Legendre Fenchel transform of the value function. Moreover, recent results by people like

7 viii Preface Barvinok, Brion, and Vergne, and Pukhlikov and Khovanskii have provided nice and elegant exact formulas for I and I d. One purpose of this book is to show that a careful analysis of these formulas permit us to shed some interesting light on the links and analogies between the (continuous) integration and (discrete) counting programs I and I d. In addition, in view of connections and analogies among P,P d,i, and I d on the one hand, and duality results already available for P,I, and I d on the other, another goal is to provide new insights and results on duality for integer programming, and to reinterpret some already existing results in light of these new results and analogies. This book is an attempt to reach this goal, and throughout all chapters our investigation is guided by these analogies, which are not just formal but rest on a rigorous mathematical analysis. We hope to convince the reader that they are also useful to better understand in particular the difference between the discrete and continuous cases and reasons why the former is significantly more difficult. We also hope that some of the material presented here could be introduced in graduate courses in optimization and operations research, as this new viewpoint makes a link between problems that after all are not so different when looked at through a distinct lens. Indeed, very often the discrete and continuous cases are treated separately (as are integration and optimization) and taught in different courses. The associated research communities are also distinct. On a more practical side, some duality results presented in the last chapters of the book may also provide new ideas for generating efficient cuts for integer programs, a crucial issue for improving the powerful software packages already available, like CPLEX and XPRESS-MP codes. Finally, let us mention that IP is also studied in the algebraic geometry research community, and standard algebraic tools like Gröbner basis, Gröbner fan, and toric ideals permit us to better understand IP as an arithmetic refinement of LP and to re-interpret some known (algebraic) concepts, like Gomory relaxations, for example. The plan of the book is as follows: We first introduce the four related problems P,P d,i, and I d in Chapter 1. Next, in Part I we analyze problem I and compare with P in Chapters 2 and 3. In Part II, we do the same with I d and P d in Chapters 4 and 5. Part III is then devoted to various duality results, including (i) the link between the old concept of Gomory relaxations and exact formula obtained by Brion, and Brion and Vergne, in Chapters 6 and 7, (ii) a discrete Farkas lemma and a characterization of the integer hull in Chapters 8 and 9, and (iii) the link with the superadditive dual in Chapter 10. Some elementary background on Legendre Fenchel, Laplace, and Z-transforms, as well as Cauchy residue theorem in complex analysis, can be found in the Appendix. Acknowledgements This book benefited from several stimulating discussions with (among others) Alexander Barvinok, Matthias Beck, Jesus de Loera, Jon Lee, Serkan Hosten, Bernd Sturmfels, Rekha Thomas, and Michelle Vergne during several visits and/or workshops at the Mathematical Sciences Research Institute (MSRI, Berkeley), the Fields Institute for Research in Mathematical Sciences (Toronto), and the Institute for Mathematics and Its Applications (IMA, Minneapolis). I gratefully acknowledge financial support from the AMS and the above institutes, which made these visits possible.

8 Preface ix Some results presented in this book were obtained in collaboration with Eduardo Santillan Zeron from Cinvestav-IPN (México D.F.) with financial support from a research cooperation program between CNRS (France) and CONACYT (Mexico), which is also gratefully acknowledged. Finally, I also want to thank the CNRS institution for providing a very pleasant working environment at the LAAS-CNRS laboratory in Toulouse. Toulouse 24 July 2008 Jean B. Lasserre

9 Contents Preface... vii 1 Introduction The four problems P,P d,i,i d Summary of content Part I Linear Integration and Linear Programming 2 The Linear Integration Problem I Introduction Primal methods A dual approach A residue algorithm for problem I Notes Comparing the Continuous Problems P and I Introduction Comparing P,P,I, and I Notes Part II Linear Counting and Integer Programming 4 The Linear Counting Problem I d Introduction A primal approach: Barvinok s counting algorithm A dual approach Inversion of the Z-transform by residues An algebraic method Asimpleexplicitformula Notes... 69

10 xii Contents 5 Relating the Discrete Problems P d and I d with P Introduction Comparing the dual problems I and I d A dual comparison of P and P d Proofs Notes Part III Duality 6 Duality and Gomory Relaxations Introduction Gomoryrelaxations Brion and Vergne s formula and Gomory relaxations The Knapsack Problem A dual of P d Proofs Notes Barvinok s Counting Algorithm and Gomory Relaxations Introduction Solving P d via Barvinok s counting algorithm ThelinkwithGomoryrelaxations Notes A Discrete Farkas Lemma Introduction AdiscreteFarkaslemma A discrete theorem of the alternative The knapsack equation Notes The Integer Hull of a Convex Rational Polytope Introduction Theintegerhull Notes Duality and Superadditive Functions Introduction Preliminaries Duality and superadditivity Notes

11 Contents xiii Appendix Legendre Fenchel, Laplace, Cramer, and Z Transforms A.1 The Legendre Fenchel transform A.2 Laplacetransform A.3 The Z-transform A.4 Notes References Glossary Index