Linear and Integer Programming Vs Linear Integration and Counting: A Duality Viewpoint

ISBN-13: 9781441918536 / Angielski / Miękka / 2010 / 168 str.

Jean-Bernard Lasserre

Opis

Integer programming (IP) is a fascinating topic. Indeed, while linear programming (LP), its c- tinuous analogue, is well understood and extremely ef?cient 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?1928x?11111x?2345x +9123x} 1 2 3 4 5 s.t. 12223x +12224x +36674x +61119x +85569x = 89643482, 1 2 3 4 5 x, x, x, x, x?N, 1 2 3 4 5 taken from a list of dif?cult knapsack problems in Aardal and Lenstra 2], is not solved even by hours of computing, using for instance the last version of the ef?cient software package CPLEX. However, thisisnotabookonintegerprogramming, asverygoodonesonthistopicalreadyexist. 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.

Recenzje

From the reviews:

"Lasserre has produced a fascinating slim ... monograph (much of the work his own) looking at the parallels between linear (respectively integer) programming on the one hand and integration (respectively integer counting) problems on the other hand. ... An appendix on various transforms a hundred references and a brief index complete the work which is a welcome addition to an important set of topics." (J. Borwein, Mathematical Reviews, Issue 2010 f)

"This book is devoted to analysing four important problems: integer programming problem, linear programming problem, linear integration problem, and linear counting problem. ... a very specialized book on the integer programming problem and its dual variants. ... can be very helpful for researchers working in developing algorithms for the integer programming problem which is a formidable challenging problem. This is a clear and well-written book ... ." (E. Almehdawe, Journal of the Operational Research Society, Vol. 61 (12), 2010)

Spis treści

I Linear Integration and Linear Programming.- The Linear Integration Problem I.- Comparing the Continuous Problems P and I.- II Linear Counting and Integer Programming.- The Linear Counting Problem I.- Relating the Discrete Problems P and I with P.- III Duality.- Duality and Gomory Relaxations.- Barvinok#x2019;s Counting Algorithm and Gomory Relaxations.- A Discrete Farkas Lemma.- The Integer Hull of a Convex Rational Polytope.- Duality and Superadditive Functions.

Blurb

In this book the author analyzes and compares four closely related problems, namely linear programming, integer programming, linear integration, linear summation (or counting). The focus is on duality and the approach is rather novel as it puts integer programming in perspective with three associated problems, and permits one to define discrete analogues of well-known continuous duality concepts, and the rationale behind them. Also, the approach highlights the difference between the discrete and continuous cases. Central in the analysis are the continuous and discrete Brion and Vergne's formulae for linear integration and counting. This approach provides some new insights on duality concepts for integer programs, and also permits to retrieve and shed new light on some well-known results. For instance, Gomory relaxations and the abstract superadditive dual of integer programs are re-interpreted in this algebraic approach.

This book will serve graduate students and researchers in applied mathematics, optimization, operations research and computer science. Due to the substantial practical importance of some presented problems, researchers in other areas will also find this book useful.