Book:Claude Berge/Programming, Games and Transportation Networks

Subject Matter

 * Topology
 * Game Theory
 * Graph Theory

Contents

 * Preface


 * Translator's Note


 * (by )


 * 1. Preliminary ideas: sets, vector spaces
 * 1.1 Sets
 * 1.2 Vector spaces
 * 1.3 Dimension of a vector space
 * 1.4 Linear manifolds. Cones
 * 1.5 Convex sets
 * 1.6 Convex functions


 * 2. Preliminary ideas: topological properties of the space $\R^n$
 * 2.1 Scalar product. Norm
 * 2.2 Open and closed sets
 * 2.3 Limits. Continuous functions
 * 2.4 Compact sets
 * 2.5 Numerical semicontinuous functions


 * 3. Properties of complex sets and functions in the space $\R^n$
 * 3.1 Separation theorems
 * 3.2 Theorem on supporting planes. Polytopes
 * 3.3 Intersections of convex sets
 * 3.4 Minimax theorem. Farkas-Minkowski theorem
 * 3.5 Sion's theorem


 * 4. Programming and associated problems
 * 4.1 Differentiable functions
 * 4.2 Kuhn and Tucker multipliers


 * 5. Convex programmes with linear constraints
 * 5.1 Associated problem
 * 5.2 Duality in linear programming
 * 5.3 The simplex method
 * 5.4 Non-linear programming
 * 5.5 Quadratic programming
 * 5.6 Conjugate functions
 * 5.7 Duality in certain non-linear programmes


 * 6. Games of strategy
 * 6.1 Introduction
 * 6.2 Strictly determined games
 * 6.3 Eluding games
 * 6.4 Solution of a game by linear programming
 * 6.5 Solution of a game by successive approximations


 * Bibliography to Part I


 * (by )


 * 7. Cycles and coboundaries of a graph
 * 7.1 General remarks on graphs
 * 7.2 Cycles and coboundaries, lemma on coloured arcs
 * 7.3 Strongly connected graphs and graphs without circuits
 * 7.4 Trees and cotrees
 * 7.5 Planar graphs


 * 8. General study of flows and potential differences
 * 8.1 Flow and potential difference
 * 8.2 Matrix analysis of flows and potential differences
 * 8.3 The transportation problem
 * 8.4 New formulation of the transportation problem
 * 8.5 The fundamental duality theorem


 * 9. Flow algorithms
 * 9.1 The path problem
 * 9.2 The problem of the shortest path
 * 9.3 The problem of the shortest spanning tree
 * 9.4 The generalized problem of the shortest path
 * 9.5 The problem of the maximum potential difference
 * 9.6 The problem of the maximum flow
 * 9.7 The generalized problem of maximum flow
 * 9.8 The trans-shipment problem
 * 9.9 The restricted problem of potential
 * 9.10 The general transportation problem


 * 10. Problems related to the transportation problem
 * 10.1 General study of a symmetric transportation network
 * 10.2 The transportation network with multipliers
 * 10.3 Special problems of maximum flow
 * 10.4 Special problems of flow and potential difference


 * Exercises


 * Bibliography to Part II


 * Index



Source work progress
* : $1$. Preliminary ideas; sets, vector spaces: $1.2$. Vector Spaces