# Book:Claude Berge/Programming, Games and Transportation Networks

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## Claude Berge and A. Ghouila-Houri:

## Claude Berge and A. Ghouila-Houri: *Programming, Games and Transportation Networks*

Published $\text {1965}$, **Methuen and Co. Ltd.** (translated by Maxine Merrington and C. Ramanujacharyulu)

Translated from:

### Subject Matter

### Contents

**Preface**

**Translator's Note**(Maxine Merrington)

- Part I: General Theory of Convex Programming (
*by A. Ghouila-Houri*)

- Part I: General Theory of Convex Programming (

**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**

- Part II: Problems of Transportation and of Potential (
*by C. Berge*)

- Part II: Problems of Transportation and of Potential (

**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

- 1965: Claude Berge and A. Ghouila-Houri:
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