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Lecture 15 - Properties of the Simplex Method

Lecture 15 - Properties of the Simplex Method - Recap the...

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Lecture 15 Properties of the Simplex Method UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Friday, February 13th, 2009 Recap: the simplex method The simplex method moves from one vertex to another while reducing the objective function. Each iteration requires the solution of two systems of equations: A T k λ k = c and A k p k = e s where A k is the nonsingular working-set matrix. UCSD Center for Computational Mathematics Slide 2/46, Friday, February 13th, 2009 Example: minimize x R n c T x subject to Ax b where c = 2 1 ! , A = 1 1 0 1 1 0 - 1 0 - 1 1 , b = 1 0 0 - 2 - 2 UCSD Center for Computational Mathematics Slide 3/46, Friday, February 13th, 2009 x 1 x 2 x * #3 #1 #2 #5 #4 x 0 x 1 = x 0 + α 0 p 0
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In this example, the simplex method moves to an adjacent vertex . We must show that A k +1 satisfies the properties of a working set. Result If A k is nonsingular, then A k +1 is nonsingular. Proof: At the end of iteration k , row s of A k is replaced by a T t . A k +1 = a T w 1 a T w 2 . . . a T t . . . a T w n row s UCSD Center for Computational Mathematics Slide 5/46, Friday, February 13th, 2009 The vectors a w 1 , a w 2 , . . . , a w s - 1 , a w s +1 , . . . , a w n are linearly independent. If A k +1 is singular, then row a t must be dependent on a w 1 , a w 2 , . . . , a w s - 1 , a w s +1 , . . . , a w n , i.e., a t = X i 6 = s a w i y i for some { y i } not all zero Multiplying by p T k gives a T t p k = X i 6 = s ( a T w i p k ) y i UCSD Center for Computational Mathematics Slide 6/46, Friday, February 13th, 2009 From the previous slide: a T t p k = X i 6 = s ( a T w i p k ) y i But p k is defined so that A k p k = e s , i.e., a T w i p k = 0 for all i 6 = s a T w s p k = 1 a T t p k = 0. UCSD Center for Computational Mathematics Slide 7/46, Friday, February 13th, 2009 If a T t p k = 0 then a T t x b t is not decreasing along p k a T t x b t cannot be a blocking constraint a contradiction. A k +1 is nonsingular. UCSD Center for Computational Mathematics Slide 8/46, Friday, February 13th, 2009
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Two-dimensional example a 1 a 2 x k p k a T 2 p k = 0 a T 2 x b 2 is never blocking In general, there may be many working-set constraints with negative multipliers. Which index do we choose for s ? c T p k = d d α c T ( x k + α p k ) α =0 = initial rate of decrease of ( x ) = ( A T k λ k ) T p k = ( λ T k A k ) p k = λ T k ( A k p k ) = λ T k e s = ( λ k ) s < 0 UCSD Center for Computational Mathematics Slide 10/46, Friday, February 13th, 2009 From the previous slide c T p k = ( λ k ) s < 0 Choose the most negative ( λ k ) i This is called the “Dantzig rule” . c T x k +1 = c T x k + α k c T p k | {z } biggest change in per unit step Each choice of s gives a different direction p k .
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