1. 1 Linear Programming:Duality theory

2. Duality Theory The theory of duality is a very elegant and important concept within the field of operations research. This theory was first developed in relation to linear programming, but it has many applications, and perhaps even a more natural and intuitive interpretation, in several related areas such as nonlinear programming, networks and game theory.

3. The notion of duality within linear programming asserts that every linear program has associated with it a related linear program called its dual. The original problem in relation to its dual is termed the primal. it is the relationship between the primal and its dual, both on a mathematical and economic level, that is truly the essence of duality theory.

4. Examples There is a small company which has recently become engaged in the production of office furniture. The company manufactures tables, desks and chairs. The production of a table requires 8 kgs of wood and 5 kgs of metal and is sold for $80; a desk uses 6 kgs of wood and 4 kgs of metal and is sold for $60; and a chair requires 4 kgs of both metal and wood and is sold for $50. We would like to determine the revenue maximizing strategy for this company, given that their resources are limited to 100 kgs of wood and 60 kgs of metal.

5. Problem P1 

6. Problem1 Now consider that there is a much bigger company which has been the lone producer of this type of furniture for many years. They don't appreciate the competition from this new company; so they have decided to tender an offer to buy all of their competitor's resources and therefore put them out of business.

7. Problem D1 

8. A Diet Problem An individual has a choice of two types of food to eat, meat and potatoes, each offering varying degrees of nutritional benefit. He has been warned by his doctor that he must receive at least 400 units of protein, 200 units of carbohydrates and 100 units of fat from his daily diet. Given that a kg of steak costs $10 and provides 80 units of protein, 20 units of carbohydrates and 30 units of fat, and that a kg of potatoes costs $2 and provides 40 units of protein, 50 units of carbohydrates and 20 units of fat, he would like to find the minimum cost diet which satisfies his nutritional requirements

9. Problem P2 

10. Examples Now consider a chemical company which hopes to attract this individual away from his present diet by offering him synthetic nutrients in the form of pills. This company would like determine prices per unit for their synthetic nutrients which will bring them the highest possible revenue while still providing an acceptable dietary alternative to the individual.

11. Problem D2 

12. FINDING THE DUAL OF A STANDARD LINEAR PROGRAM In this section we formalize the intuitive feelings we have with regard to the relationship between the primal and dual versions of the two illustrative examples The important thing to observe is that the relationship - for the standard form - is given as a definition.

13. Standard form of the Primal Problem 

14. Standard form of the Dual Problem 

15. L.P. 15 Linear Programming Duality max becomes min RHS coefficients swap places with objective function coefficients sense changes x variables go away y variables appear

16. L.P.16 Duality Example

17. FINDING THE DUAL OF NONSTANDARD LINEAR PROGRAMS The approach here is similar when we dealt with non-standard formulations in the context of the simplex method. There is one exception: we do not add artificial variables. We handle “=“ constraints by writing them as “<=“ constraints.

18. Possibility This is possible here because we do not require here that the RHS is non-negative.

19. Standard form!

20. Example 

21. Conversion Multiply through the greater-than-or- equal-to inequality constraint by -1 Use the approach described above to convert the equality constraint to a pair of inequality constraints. Replace the variable unrestricted in sign,, by the difference of two nonnegative variables.

22. 

23. Dual

24. Streamlining the conversion... An equality constraint in the primal generates a dual variable that is unrestricted in sign. An unrestricted in sign variable in the primal generates an equality constraint in the dual.

25. Example

26. + + correction

27. Primal-Dual relationship Primal Problem opt=max Constraint i : <= form = form Variable j: x j >= 0 x j urs opt=min Dual Problem Variable i : y i >= 0 y i urs Constraint j: >= form = form

28. Example 

29. equivalent non-standard form

30. Dual from the recipe 

31. L.P.Jose Rolim31 Weak Linear Programming Duality Any feasible solution to primal LP has value no greater than that of any feasible solution to the dual LP.

32. L.P.32 Weak Linear Programming Duality (continued)

33. L.P.33 Finding a Dual Solution Finding a dual solution whose value is equal to that of an optimal primal solution…

34. L.P.34 Optimality.

35.  As you remember  Given a directed graph, two vertices and a capacity for each edge  We want to find a flow function so that the flow value is maximal Maximum Flow

36.  But, we are subject to some rules:   What goes in must come out:  Capacity restrictions: Maximum Flow

37.  We’ll show by a simple example: Max Flow with Linear Programming 3 2 1 3 1

38.  Formally: Max Flow with Linear Programming 3 2 1 3 1 But we don’t permit equalities

39.  So we’ll add another edge, and change the problem’s representation a little. Max Flow with Linear Programming 3 2 1 3 1 ∞

42.  As you remember from Algorithms I  Given a directed graph, two vertices and a weight for each edge  We want to find a minimal-weight subset of edges such that if we’ll remove them, we won’t be able to travel from to. Minimum Cut

43.  In other words:  We’ll choose, where:,, such that the cut value, is minimal. Minimum Cut

44.  Going back to the example: Min Cut with Linear Programming 3 2 1 3 1

45. 3 2 1 3 1  What about:  Is that enough?  We haven’t ensured paths from to are cut. This look like And we know that

46. Min Cut with Linear Programming 3 2 1 3 1  Beside a variable for every edge, we’ll want a variable for every vertex, such that:

47. Min Cut with Linear Programming 3 2 1 3 1  Let’s take for instance:  If is in the cut that means that,  If is not in the cut, that means that either or or  So it is the same to constraint:

48.  Formally: Max Flow with Linear Programming 3 2 1 3 1 In order to ensure

52.  We now see that the problems are dual  So also according to the Strong Duality Theorem, Max Flow = Min Cut Max Flow – Min Cut Theorem