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# lec6 - IE521 Advanced Optimization Lecture 6 Dr Zeliha Akca...

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IE521 Advanced Optimization Lecture 6 Dr. Zeliha Akc ¸a November 2011 1 / 12

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Reading I Bertsimas 3.3, 3.5. 2 / 12
Recall I Feasible and Improving Direction: d R n is feasible direction if ˆ x + θ d ∈ P for some θ R + . d R n is an improving direction if c > d < 0 . I Constructing Feasible Search Directions: d B = - B - 1 A j , d j = 1 , d i = 0 for all i in nonbasic variable index set. I Reduced cost of variable j : ¯ c j = c j - c > B B - 1 A j I Optimality Conditions Theorem Let ˆ x be a basic feasible solution and ¯ c be the corresponding vector of reduced costs. I If ¯ c 0 , then ˆ x is optimal . I If ˆ x is optimal and nondegenerate, then ¯ c 0 . 3 / 12

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Improving the Efficiency of Simplex I Matrix B - 1 plays a central role in the simplex method. I If we had B - 1 at each iteration, we could easily compute everything we need: direction vector, reduced costs, values of basic variables, step length, new basic variables, current objective, etc. I How can we update B - 1 at each iteration? Example B - 1 = 1 2 3 - 2 3 1 4 - 3 - 2 , u = - 4 2 2 Assume that we want to remove 3 rd basic variable, therefore obtain ( 0 , 0 , 1 ) from u (simplex tableau column of entering variable, d B ) using row operations. Apply each row operation to obtain ( 0 , 0 , 1 ) from u to B - 1 to get new ¯ B - 1 .
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lec6 - IE521 Advanced Optimization Lecture 6 Dr Zeliha Akca...

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