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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.

Integer programming. --- Linear programming. --- Linear systems --Mathematical models. --- Mathematical optimization. --- Neoclassical school of economics. --- Operations research --Mathematical models. --- Rothschild, Kurt W. --(Kurt Wilhelm), --1914-. --- Steindl, Josef, --1912-1993. --- Linear programming --- Integer programming --- Mathematical optimization --- Linear systems --- Operations research --- Civil & Environmental Engineering --- Operations Research --- Engineering & Applied Sciences --- Business & Economics --- Economic Theory --- Mathematical models --- Mathematical models. --- Operational analysis --- Operational research --- Systems, Linear --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- 631.11 <213> --- 631.445.5 --- 631.5 --- 631.5 Agricultural operations --- Agricultural operations --- 631.445.5 Arid soils. Desert soils --- Arid soils. Desert soils --- 631.11 <213> Farms, farm properties and farming systems--Warme, tropische en subtropische zones --- Farms, farm properties and farming systems--Warme, tropische en subtropische zones --- Mathematics. --- Computer science --- Convex geometry. --- Discrete geometry. --- Operations research. --- Management science. --- Operations Research, Management Science. --- Optimization. --- Discrete Mathematics in Computer Science. --- Convex and Discrete Geometry. --- Arid regions agriculture --- Dry farming --- Dryland farming --- Farming, Dry --- Rainfed agriculture --- Tillage --- Dryland agriculture --- Agriculture --- Crop zones --- Desert reclamation --- Pedology --- Mathematical analysis --- Maxima and minima --- Simulation methods --- System analysis --- Quantitative business analysis --- Management --- Problem solving --- Statistical decision --- Industrial engineering --- Management science --- Research --- System theory --- Geometry --- Combinatorial geometry --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Math --- Science --- Mathematics --- Programming (Mathematics) --- Production scheduling --- Differential equations, Linear --- Basic Sciences. Agriculture --- Agricultural Systems. --- Computational complexity. --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Complexity, Computational --- Machine theory --- Computer science—Mathematics. --- Convex geometry . --- Rothschild, Kurt W. --- Steindl, Josef, --- Rothschild, Kurt Wilhelm --- Rothschild, K. W.