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AMS 540 / MBA 540 (Fall, 2008) Estie Arkin Important facts I should know (and remember) Given two math programs z 1 = min { f ( x ) | x P } , z 2 = min { f ( x ) | x Q } , with P Q then z 1 z 2 . If x 1 and x 2 are both feasible and optimal to an LP, then so is λx 1 + (1 - λ ) x 2 for every 0 λ 1. (See HW 2 problem 1.) Consider an LP: min { cx | Ax = b, x 0 } , A is an m by n matrix of linearly indep rows. Defnitions Basic Solution: Pick m cols of A that are linearly indep , call these cols B . Set the variables not corresponding to these col x N = 0 and solve for the remaining (basic variables) x B = B 1 b . Basic Feasible Solution (BFS): Basic Solution in which x B 0. Degenerate BFS: BFS in which one or more basic variable(s) is = 0.
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Unformatted text preview: Some important theorems • If ∃ a ±nite optimal solution to the LP then ∃ an optimal BFS to the LP. • x ′ optimal to LP does not necessarily imply x ′ is a BFS. • number of BFS’s ≤ number of BS’s ≤ ( n m ) < ∞ . • Simplex applied to an LP in which all BFS are non degenerate implies BFS are not repeated implies Simplex will terminate in a ±nite number of steps. • Simplex applied to an LP in which some BFS are degenerate may result in a BFS being repeated (cycling)....
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