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Linear Programming: Introduction and Duality

Linear Programming: Introduction and Duality. NTHU CS CSBB lab 劉至善. Outline. Linear programming The LP-duality theorem. Linear programming. Linear programming The problem of optimizing a linear function subject to linear inequality constraints. Minimize 7 x 1 + x 2 +5 x 3

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Linear Programming: Introduction and Duality

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  1. Linear Programming: Introduction and Duality NTHU CS CSBB lab 劉至善

  2. Outline • Linear programming • The LP-duality theorem

  3. Linear programming • Linear programming • The problem of optimizing a linear function subject to linear inequality constraints. Minimize 7x1+x2+5x3 Subject to x1-x2+3x3≧10 5x1+2x2-x3≧6 x1,x2,x3≧0

  4. Linear programming • Linear programming • Any solution satisfies all the constraints is a feasible solution. Is 7x1+x2+5x3≦30? Yes, (2,1,3) makes 7x1+x2+5x3=30. Minimize 7x1+x2+5x3= 7*2+1+5*3=30 Subject to x1-x2+3x3 = 2-1+3*3=10 ≧10 5x1+2x2-x3 =5*2+2*1-3=9≧6 x1,x2,x3≧0

  5. Linear programming • Linear programming • How to provide a “NO” certificate? Find an optimal lower bound. 7x1+x2+5x3 ≧ 6x1+x2+2x3 =(x1-x2+3x3)+(5x1+2x2-x3) ≧16.

  6. Linear programming Minimize 7x1+x2+5x3 Subject to x1-x2+3x3≧10 5x1+2x2-x3≧6 x1,x2,x3≧0 y1*(x1-x2+3x3 )+y2*(5x1+2x2-x3) ≧10y1+6y2 → (y1+5y2)*x1 +(-y1+2y2)*x2 +(3y1-y2)*x3 ≧10y1+6y2 Maximize 10y1+6y2 Subject to y1+5y2≦7 -y1+2y2≦1 3y1-y2≦5 y1,y2≧0 (y1+5y2)*x1≦7x1 (-y1+2y2)*x2≦1x2 (3y1-y2)*x3≦5x3

  7. Linear programming Primal program Dual program Minimize 7x1+x2+5x3 Subject to x1-x2+3x3≧10 5x1+2x2-x3≧6 x1,x2,x3≧0 Maximize 10y1+6y2 Subject to y1+5y2≦7 -y1+2y2≦1 3y1-y2≦5 y1,y2≧0 • What is the dual of the dual program?

  8. Linear programming Maximize 10y1+6y2 Subject to y1+5y2≦7 -y1+2y2≦1 3y1-y2≦5 y1,y2≧0 x1*(y1+5y2)+x2*(-y1+2y2)+x3*(3y1-y2)≦7x1+x2++5x3 →(x1-x2+3x3)*y1 +(5x1+2x2-x3)*y2≦7x1+x2+5x3 Minimize 7x1+x2+5x3 Subject to x1-x2+3x3≧10 5x1+2x2-x3≧6 x1,x2,x3≧0 (x1-x2+3x3)*y1≧10y1 (5x1+2x2-x3)*y2≧6y2

  9. Linear programming • Max flow

  10. Linear programming

  11. Linear programming • Every feasible solution to the dual program gives a lower bound on the optimum value of the primal. 0 26 ∞ dual solutions primal solutions 10y1+6y2 7x1+x2+5x3 (2,1) (7/4,0,11/4)

  12. The LP-duality theorem • (LP-duality theorem) The primal program has finite optimum iff its dual has finite optimum. Moreover, if x*=(x1*,…,xn*) and y*=(y1*,…,ym*) are optimal solutions for the primal and dual programs, respectively, then

  13. The LP-duality theorem

  14. The LP-duality theorem • (Complementary slackness conditions) Let x and y be primal and dual feasible solutions, respectively. Then, x and y are both optimal iff all of the following conditions are satisfied: Primal complementary slackness conditions Dual complementary slackness conditions

  15. Algorithm design techniques • LP-rounding • Primal-dual schema

  16. Thank you.

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