# Foundations of Mathematical Programming (6 Parts)

Synopsis
1: Introduces the subject of mathematical programming and establishes national conventions, which appear as a general statement and in specialised application to linear minimum problem. The geometry of this problem is developed and the necessary conditions are given both graphical and analytic expression. The intuitive geometric discussion leads to the formulation of separating hyperplane theorems for convex sets. One such theorem is proved graphically and is seen to imply the necessary conditions for the linear minimum problem.
2: The theory of duality for linear programming. The necessary conditions for the linear minimum problem are used to construct a new problem, a maximum problem. The pair of problems, minimum and maximum, enjoy a set of relations that constitute duality. All of the basic theorems are built up as a 3 x 3 table of logical possibilities. By specialisation of the form of the pair of dual programs, a number of basic theorems from linear algebra result.
3: The derivation of the Kuhn-Tucker necessary conditions for non-linear programs. The initial segment lists parallels and differences between linear and non-linear programs. These are illustrated graphically and are used to motivate the formal statement of a non-linear program and the appropriate necessary conditions for an optimum. Two simple examples, discussed both graphically and analytically, introduce a restriction on the statement, called the constraint qualification. With this preparation, the proof follows the pattern of the first tape exactly. In a final segment, the simple ideas that led to the discovery of this theorem are revealed.
4: A simple example, using physical objects and numerical data, introduces the formal statement of the von Neumann expanding economy model. The basic problem posed for this model is recognised as a non-linear program and is reformulated by simple manipulations to fit the format of the previous tape. The necessary conditions for an optimum are derived on screen and are given economic interpretation. Returning to the numerical example, a geometric realisation is developed that is illustrated by a physical model and by animation. The geometric model suggests a different argument using a separating hyperplane. A final segment cites von Neumann’s original proof.
5: The Brouwer fixed-point theorem is stated, explained and proved by entirely elementary techniques. The statemente is illustrated graphically. The proof is developed with a strong emphasis on motivation which isolates a single combinational result, Sperner’s Lemma, as the core of the proof. This result has a simple graphical realisation that is used to illustrated its content and to provide a rigorous demonstration. A final proof is given along constructive lines that give an indication of current research in this area. Brouwer’s theorem is compared with Kakutani’s theorem, which provides a direct connection with part 4 and with applications to programming and economics.
6: Uses the physical apparatus of a game to illustrate the basic theorem of optimal control theory. The object of the game is abstracted as the control of an object on a line, using forces between two limits, so as to reach a prescribed point with zero velocity in minimum time. This problem, a prototype example in control theory, is completely analysed on the tape. The analysis consists of the development of necessary conditions and is done with graphical, algebraic and analytical arguments. After illustrating the solution by animation, cites the connection between the Pontryagin maximum principle and the subjects of the previous tapes.
Language
English
Country
Great Britain
Medium
Video; Videocassette. U-matic, VHS. b&w. 27. 23, 26, 38, 35, 35 min.
Year of production
1972
Availability
Sale
Notes
Archival interest.
Uses
Postgraduate students in economics; mathematics students and students of operational research dealing with the mathematics of economic models.
Subjects
Mathematics
Keywords
duality theory; fixed point theorems; hyperplane theorems; mathematical programming; Von Neumann expanding theory

Writer
H Kuhn

### Sections

Title
Separation of convex sets by hyperplanes, The
Synopsis
1: Introduces the subject of mathematical programming and establishes national conventions, which appear as a general statement and in specialised application to linear minimum problem. The geometry of this problem is developed and the necessary condition

Title
Duality theory
Synopsis
2: The theory of duality for linear programming. The necessary conditions for the linear minimum problem are used to construct a new problem, a maximum problem. The pair of problems, minimum and maximum, enjoy a set of relations that constitute duality. A

Title
Non-linear programs
Synopsis
3: The derivation of the Kuhn-Tucker necessary conditions for non-linear programs. The initial segment lists parallels and differences between linear and non-linear programs. These are illustrated graphically and are used to motivate the formal statement

Title
Von Neumann expanding economy, The
Synopsis
4: A simple example, using physical objects and numerical data, introduces the formal statement of the von Neumann expanding economy model. The basic problem posed for this model is recognised as a non-linear program and is reformulated by simple manipula

Title
Fixed point theorems
Synopsis
5: The Brouwer fixed-point theorem is stated, explained and proved by entirely elementary techniques. The statemente is illustrated graphically. The proof is developed with a strong emphasis on motivation which isolates a single combinational result, Sper

Title
Pontryagin’s maximum
Synopsis
6: Uses the physical apparatus of a game to illustrate the basic theorem of optimal control theory. The object of the game is abstracted as the control of an object on a line, using forces between two limits, so as to reach a prescribed point with zero ve

Name

Notes
Closed down.

### Distributor

Name

#### Learning on Screen - the British Universities and Colleges Film and Video Council

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