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Lecture 16 - Finding a Feasible Point

# Lecture 16 - Finding a Feasible Point - Lecture 15 Finding...

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Unformatted text preview: Lecture 15 Finding a Feasible Point UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Wednesday, February 18th, 2009 Recap: the simplex method One step moves from a vertex to an adjacent vertex . Each step requires the solution of two sets of equations: A T k λ k = c and A k p k = e s where A k is the nonsingular working-set matrix. If there is a tie in the choice of constraint to enter or leave the working set, the simplex method may stall at a vertex. The method cycles infinitely if there is a repeat of a sequence of constraint changes at a stalled vertex. UCSD Center for Computational Mathematics Slide 2/44, Wednesday, February 18th, 2009 Example minimize- 2 . 3 x 1- 2 . 15 x 2 + 13 . 55 x 3 + 0 . 4 x 4 subject to- . 4 x 1- . 20 x 2 + 1 . 40 x 3 + 0 . 2 x 4 ≥ 7 . 8 x 1 + 1 . 40 x 2- 7 . 80 x 3- . 4 x 4 ≥ x i ≥ , i = 1 , 2 , 3 , 4 , UCSD Center for Computational Mathematics Slide 3/44, Wednesday, February 18th, 2009 A = - . 4- . 2 1 . 4 . 2 7 . 8 1 . 4- 7 . 8- . 4 1 1 1 1 , b = , c = - 2 . 30- 2 . 15 13 . 55 . 40 This LP is unbounded, but has a degenerate vertex at x = 0. i.e., an “unbounded solution” indication means that we have broken free of the degenerate vertex. UCSD Center for Computational Mathematics Slide 4/44, Wednesday, February 18th, 2009 First we apply the simplex method with standard tie-breaking rules ( λ k ) s = min 1 ≤ i ≤ n ( λ k ) i (Dantzig rule) t = min { a T i p k < 0 : σ i = α k } UCSD Center for Computational Mathematics Slide 5/44, Wednesday, February 18th, 2009 The simplex method with “standard” rules cycles Itn Objective min LM Del Add Step 0.0000000e+00-2.30e+00 3 1 0.00e+00 1 0.0000000e+00-1.00e+00 4 2 0.00e+00 2 0.0000000e+00-2.30e+00 5 3 0.00e+00 3 0.0000000e+00-1.00e+00 6 4 0.00e+00 4 0.0000000e+00-2.30e+00 1 5 0.00e+00 5 0.0000000e+00-1.00e+00 2 6 0.00e+00 6 0.0000000e+00-2.30e+00 3 1 0.00e+00 7 0.0000000e+00-1.00e+00 4 2 0.00e+00 8 0.0000000e+00-2.30e+00 5 3 0.00e+00 9 0.0000000e+00-1.00e+00 6 4 0.00e+00 10 0.0000000e+00-2.30e+00 1 5 0.00e+00 11 0.0000000e+00-1.00e+00 2 6 0.00e+00 12 0.0000000e+00-2.30e+00 3 1 0.00e+00 13 0.0000000e+00-1.00e+00 4 2 0.00e+00 14 0.0000000e+00-2.30e+00 5 3 0.00e+00 15 0.0000000e+00-1.00e+00 6 4 0.00e+00 . . . . . . . . . Next we apply the simplex method with Bland’s least-index rules w s = min { w i : ( λ k ) i < } t = min { j : σ j = α k } UCSD Center for Computational Mathematics Slide 7/44, Wednesday, February 18th, 2009 The simplex method with Bland’s least-index rule does not cycle Itn Objective min LM Del Add Step 0.0000000e+00-2.30e+00 3 1 0.00e+00 1 0.0000000e+00-1.00e+00 4 2 0.00e+00 2 0.0000000e+00-2.30e+00 5 3 0.00e+00 3 0.0000000e+00-7.50e-01 2 Unbounded step Unfortunately, Bland’s rule is not useful...
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Lecture 16 - Finding a Feasible Point - Lecture 15 Finding...

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