mthsc810-lecture10-1x2

mthsc810-lecture10-1x2 - MthSc 810: Mathematical...

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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 10 Pietro Belotti Dept. of Mathematical Sciences Clemson University September 29, 2011 Reading for today: Sections 3.3, textbook Reading for Oct. 4: Sections 3.7 and 4.1-4.2, textbook Recap: the simplex method Reduce problem to standard form Add slack variables Get initial dictionary If infeasible, do Phase I If no feasible solution, STOP problem infeasible Repeat pivoting until unbounded optimum Recap: all possible situations Degeneracy: the objective doesnt improve; keep pivoting. Cycling: very rare apply Blands rule. Infeasibility: apply Phase I If, at optimum of Phase I, y 6 = infeasible Otherwise, we have a feasible dictionary. Unboundedness: feasible direction d , i.e., at least one candidate for entering 1 has only non-negative coefficients in the remaining rows of the dictionary. 1 i.e., its coefficient in the obj. row of the dictionary is < 0. Inefficiencies in the simplex method For any B and N , the dictionary x B = B 1 b B 1 N x N z = c B B 1 b + ( c N c B B 1 N ) x N tells us if B is optimal the problem is unbounded Simplex: Repeat If c N = c N c B B 1 N stop, otherwise find entering x j Find leaving variable x B B \ { } { j } .. But we need to compute B 1 (computationally expensive) The revised simplex method To compute c N = c N c B B 1 N and N = B 1 N , find first c B B 1 . ( ) Dont invert B . Find vector y such that y B = c B ....
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mthsc810-lecture10-1x2 - MthSc 810: Mathematical...

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