Today: Linear Programming (con t.)

Transcription

1 Today: Linear Programming (con t.) COSC 581, Algorithms April 10, 2014 Many of these slides are adapted from several online sources

2 Reading Assignments Today s class: Chapter 29.4 Reading assignment for next class: Chapter 9.3 (Selection in Linear Time) Chapter 34 (NP Completeness)

3 Optimality of SIMPLEX Duality is a way to prove that a solution is optimal Max-Flow, Min-Cut is an example of duality Duality: given a maximization problem, we define a related minimization problem s.t. the two problems have the same optimal objective value

4 Duality in LP Given an LP, we ll show how to formulate a dual LP in which the objective is to minimize, and whose optimal value is identical to that of the original LP (now called primal LP)

5 Primal Dual LPs: Primal: maximize c T x subject to: Ax b x 0 (standard form) n y x T A b Dual: minimize y T b subject to: y T A c T y 0 c T d (standard form)

6 Forming dual Change maximization to minimization Exchange roles of coefficients on RHSs and the objective function Replace each with Each of the m constraints in primal has associated variable y i in the dual Each of the n constraints in the dual as associated variable x i in the primal

7 Example : Primal-Dual PRIMAL: max 16 x 1-23 x x x 4 subject to: 3 x x 2-9 x x x x x 3-14 x 4 = x x x x x 1 0, x 2 0, x 4 0 DUAL: min 239 y y y 3 subject to: 3 y 1-9 y y y y y y y y 3 = 43 4 y 1-14 y y 3 82 y 1 0, y 3 0

8 Think about bounding optimal solution Is optimal solution 30? Yes, consider (2,1,3)

9 Think about bounding optimal solution Is optimal solution 5? Yes, because x3 1. Is optimal solution 6? Yes, because 5x1 + x2 6. Is optimal solution 16? Yes, because 6x1 + x2 +2x3 16.

10 Strategy for bounding solution? What is the strategy we re using to prove lower bounds? Take a linear combination of constraints!

11 Strategy for bounding solution? 5 Don t reverse inequalities. What s the objective?? Optimal solution = 26 To maximize the lower bound.

12 Note: Use of primal as minimization Just to show you something a bit different from the text, the following discussion assumes the primal is a minimization problem, and thus the dual is a maximization problem Doesn t change the meaning (compared to text)

13 Primal-Dual Programs Primal Program Dual Program Dual solutions Primal solutions

14 Primal Dual Weak Duality Theorem If x and y are feasible primal and dual solutions, then any solution to the primal has a value no less than any feasible solution to dual. Proof

15 Primal Dual Programs Primal Program Dual Program Dual solutions Primal solutions Von Neumann [1947] Primal optimal = Dual optimal Dual solutions Primal solutions

16 Strong Duality Prove that if primal solution = dual solution, then the solution is optimal for both PROVE:

17 Farka s Lemma Exactly one of the following is solvable: AA 0 c T x > 0 and: A T y = c y 0 where: x and c are n-vectors y is an m-vector A is m n matrix

18 Fundamental Theorem on Linear Inequalities

19 Proof of Fundamental Theorem

20 Strong Duality PROVE: In other words, the optimal value for the primal is the optimal value for the dual.

21 Example Objective: max

22 Example Objective: max

23 Geometric Intuition

24 Geometric Intuition Intuition: There exist nonnegative y 1, y 2 so that The vector c can be generated by a 1, a 2. Y = (y 1, y 2 ) is the dual optimal solution!

25 Strong Duality Intuition: There exist y 1, y 2 so that Y = (y 1, y 2 ) is the dual optimal solution! Primal optimal value

26 Here s another analogy: 2 Player Game Column player Row player Strategy: A probability distribution Row player tries to maximize the payoff, column player tries to minimize

27 2 Player Game Strategy: A probability distribution Column player Row player A(i,j) Is it fair?? You have to decide your strategy first.

28 Von Neumann Minimax Theorem Strategy set Which player decides first doesn t matter!

29 Key Observation If the row player fixes his strategy, then we can assume that y chooses a pure strategy Vertex solution is of the form (0,0,,1, 0), i.e. a pure strategy

30 Key Observation similarly

31 Primal-Dual Programs duality

32 Reading Assignments Reading assignment for next class: Chapter 9.3 (Selection in Linear Time) Chapter 34 (NP Completeness)