IOE510+Lecture+02+08+10+Mon

IOE510+Lecture+02+08+10+Mon - IOE 510 Linear Programming I...

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1 IOE 510 Linear Programming I Monday February 8, 2010 Professor Amy Cohn

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3 Re-cap from Last Time Proved optimality of extreme points Motivating the simplex method for solving linear programs

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4 Optimality of Extreme Points
5 Optimal Points vs Extreme Points If a problem has an extreme point, does it have an optimal extreme point? Do unbounded problems have extreme points? Claim: If a problem has an optimal solution and has an EP , then it has an optimal EP If a problem has an EP, then it either has an optimal EP or is unbounded (i.e. has no optimal solution)

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6 Algorithmic approach Enumerate all BFS’s and pick the one with the best objective value Problem 1: n-choose-m possible BFS’s Problem 2: Unbounded problems

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8 Degeneracy A basic feasible solution is said to be degenerate if it has more than n active constraints Can these all be linearly independent? What does this look like?
9 Adjacency Two basic solutions are adjacent if they share n – 1 linearly independent active constraints Two adjacent basic feasible solutions are connected by a line which is called an edge of the polyhedron

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10 Optimality Conditions
11 Feasible Directions

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