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Lecture 8 Chapter 5

Lecture 8 Chapter 5. Def 5.1 Every local optimum for a LP is a global optimum. Def 5.3 If x is an extreme point of a convex set, then there are no other points y and z in the set such that x lies on the line segment connecting y and z. Extreme Points. Unique LP Optimum.

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Lecture 8 Chapter 5

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  1. Lecture 8Chapter 5 • Def 5.1 Every local optimum for a LP is a global optimum. • Def 5.3 If x is an extreme pointof a convex set, then there are no other points y and z in the set such that x lies on the line segment connecting y and z. Extreme Points

  2. Unique LP Optimum • Def 5.5 If an LP has an optimum, then it has some extreme point that is an optimum. • (There may be other points that are optimal as well.) Unique Optimal Solution

  3. Infinite Number Of Optimal Solutions • Infinite number of optimal solutions. Do we have an extreme point that is an optimum?

  4. Interior Point That Is An Optimum • Please give an example of a LP that has an interior point as an optimum. Minimize cx What is c, so that red point is an optimum?

  5. Standard Notation For LPs • Minimize j cjxj • Subject to • j aijxj = bi, all i • xj> 0, for all j • Minimixe cx • Subject to • Ax = b • x > 0 Summation Notation Matrix Notation

  6. Converting To Standard Notation • Converting inequalities to equalities • 2x + y < 10, x > 0, y > 0 • Becomes • 2x + y + s = 10, x > 0, y > 0, s > 0 • Try it • x = 3, y = 3.5 • Implies that s must be 0.5 • How do you handle 2x + y > 10, x > 0, y > 0

  7. Solving Systems Of Equations • (Finding The Inverse By Inspection) • Example #1 • X1 + X2 = 10 • -X1 = -5 • -X2 – X3 = -3 • In matrix notation we have Bx = b b = B =

  8. B-1 Is Inverse Of B • BB-1 = I = • Find B-1 by inspection

  9. Matrix Multiply • B B-1 = I • Matrix Multiply: Row r of B time Col c of B-1 • Produces the r,c element of the result

  10. Determine 1st Row • B B-1 = I • Matrix Multiply: Row r of B time Col c of B-1 • Produces the r,c element of the result

  11. Determine 2nd Row • B B-1 = I • Matrix Multiply: Row r of B time Col c of B-1 • Produces the r,c element of the result

  12. Determine 3rd Row • B B-1 = I • Matrix Multiply: Row r of B time Col c of B-1 • Produces the r,c element of the result

  13. Solution To Equations • x = B-1b = = • Check The Solution X1 = 5, X2 = 5, X3 = -2 • X1 + X2 = 10 • -X1 = -5 • -X2 – X3 = -3

  14. Example 2 • X1 + X2 = 10 • -X1 = -2 • X3 = -3 • X4 = -1 • -X2 – X3 – X4 + X5 = 2 • Bx = b

  15. What Row Do We Find First? • B B-1 = I

  16. Row 3 • B B-1 = I

  17. Row 1 • B B-1 = I

  18. Row 2 • B B-1 = I

  19. Row 4 • B B-1 = I

  20. Row 5 • B B-1 = I

  21. Solution • Solution is given by x = B-1b =

  22. Check • X1 = 2, X2 = 8, X3 = -3, X4 = -1, X5 = 6 X1 + X2 = 10 -X1 = -2 X3 = -3 X4 = -1 -X2 – X3 – X4 + X5 = 2

  23. To Solve LPs • To solve linear programs, we have to solve a sequence of systems of equations. Actually, we solve • vB = cB and By = aj • for v any y at each iteration (step). • cB, B, and aj are all original data in the problem.

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