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Lecture 23 - Non-Simplex Steps

# Lecture 23 - Non-Simplex Steps - Lecture 23 Non-Simplex...

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Unformatted text preview: Lecture 23 Non-Simplex Steps UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Monday, March 9th, 2009 Recap: Starting at a non-vertex A simple modification of the simplex method allows us to: 1 Start at a feasible point that is not necessarily a vertex. 2 Solve problems that may not have a vertex solution. The modification is to add temporary bound constraints to define a “fake” vertex. At an arbitrary (feasible) x we add temporary bound constraints: x i ≥ ≤ ( x ) i i.e., e T i x ≥ ≤ ( x ) i UCSD Center for Computational Mathematics Slide 2/44, Monday, March 9th, 2009 Example 1: minimize 2 x 1 + x 2 subject to the constraints: constraint #1: x 1 + x 2 ≥ 1 constraint #2: x 2 ≥ constraint #3: x 1 ≥ constraint #4:- x 1 ≥ - 2 constraint #5:- x 1 + x 2 ≥ - 2 In all-inequality form, we have min c T x subject to Ax ≥ b , where c = 2 1 ! , A = 1 1 1 1- 1- 1 1 , b = 1- 2- 2 UCSD Center for Computational Mathematics Slide 3/44, Monday, March 9th, 2009 x 1 x 2 x * #3 #1 #2 #5 #4 Apply the simplex method for all-inequality form. Start at the interior point x = 3 2 1 2 ! . At x , we have A ( x ) = ∅ , so no constraints are active. Add the temporary bounds: constraint #6: x 1 ≥ ≤ 3 2 constraint #7: x 2 ≥ ≤ 1 2 If W = { 6 , 7 } , then A = 1 1 ! , b = 3 2 1 2 ! UCSD Center for Computational Mathematics Slide 5/44, Monday, March 9th, 2009 x 1 x 2 x * #3 #1 #2 #5 #4 x First iteration Step 1: Check for optimality A T λ = c ⇒ 1 0 0 1 ! λ = 2 1 ! ⇒ λ = 2 1 ! ← s = 1 W = { 6 ↑ ws =6 , 7 } ( λ ) 1 = 2 is associated with temporary constraint x 1 ≥ ≤ 3 2 . UCSD Center for Computational Mathematics Slide 7/44, Monday, March 9th, 2009 First iteration Step 2: Compute the search direction Compute the search direction A p = ± e 1 ( λ ) s > 0, so we choose- e 1 for the right-hand side. A p =- e 1 ⇒ 1 0 0 1 ! p =- 1 ! ⇒ p =- 1 ! UCSD Center for Computational Mathematics Slide 8/44, Monday, March 9th, 2009 x 1 x 2 x * #3 #1 #2 #5 #4 x p First iteration Step 3: Take a step Find the constraint residuals: r = Ax- b = 1 1 1 1- 1- 1 1 3 2 1 2 !- 1- 2- 2 = 1 1 2 3 2 1 2 1 UCSD Center for Computational Mathematics Slide 10/44, Monday, March 9th, 2009 First iteration Step 3: Take a step Take a step....
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Lecture 23 - Non-Simplex Steps - Lecture 23 Non-Simplex...

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