lec6 - IE521 Advanced Optimization Lecture 6 Dr. Zeliha Akc...

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Unformatted text preview: IE521 Advanced Optimization Lecture 6 Dr. Zeliha Akc ¸a November 2011 1 / 12 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 < . I Constructing Feasible Search Directions: d B =- B- 1 A j , d j = 1 , d i = 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 ≥ , then ˆ x is optimal . I If ˆ x is optimal and nondegenerate, then ¯ c ≥ . 3 / 12 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 ( , , 1 ) from u (simplex tableau column of entering variable, d B ) using row operations....
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This note was uploaded on 01/09/2012 for the course IE 521 taught by Professor Zelihaakça during the Fall '11 term at Fatih Üniversitesi.

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lec6 - IE521 Advanced Optimization Lecture 6 Dr. Zeliha Akc...

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