linear2 - Ellipsoid Algorithm 15-853:Algorithms in the Real...

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1 15-853:Algorithms in the Real World g Linear and Integer Programming II – Ellipsoid algorithm – Interior point methods 15-853 Page1 Ellipsoid Algorithm First polynomial-time algorithm for linear programming (Khachian 79) Solves find x subject to Ax b i.e find a feasible solution Run Time: O(n 4 ), where L = #bits to represent A and b 15-853 Page2 ( L), L p Problem in practice : always takes this much time. Reduction from general case To solve: maximize: z = c T x subject to: Ax b, x 0 Convert to: find: x, y subject to: Ax b, -x 0 -yA –c 15-853 Page3 -y 0 -cx +by 0 Ellipsoid Algorithm Consider a sequence of smaller and smaller ellipsoids each with the feasible region inside. For iteration k: c k = center of E k Eventually c k has to be inside of F, and we are done. Feasible region 15-853 Page4 c k F
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2 Ellipsoid Algorithm To find the next smaller ellipsoid: find most violated constraint a find smallest ellipsoid that includes constraint k Feasible region 15-853 Page5 c k F a k ) 1 2 /( 1 1 2 1 ) ( ) ( + + = n k k E Vol E Vol Interior Point Methods Travel through the interior with a combination of x 2 1. An optimization term (moves toward objective) 2. A centering term (keeps away from boundary) Used since 50s for nonlinear 15-853 Page6 f programming. Karmakar proved a variant is polynomial time in 1984 x 1 Methods Affine scaling: simplest, but no known time bounds Potential reduction : O(nL) iterations Central trajectory : O(n 1/2 L) iterations The time for each iteration involves solving a linear system so it takes polynomial time. The “real world” time depends heavily on the matrix tructure 15-853 Page7 structure. Example times fuel continent car initial ize (K) 13x31K x57K 43x107K 1 x12K sze (K) 3x3 K 9x57K 3x 07K 9x K non-zeros 186K 189K 183K 80K iterations 66 64 53 58 time (sec) 2364 771 645 9252 Cholesky non-zeros 1.2M .3M .2M 6.7M 15-853 Page8 Central trajectory method (Lustic, Marsten, Shanno 94) Time depends on Cholesky non-zeros (i.e. the “fill”)
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3 Assumptions We are trying to solve the problem: minimize z = c T x subject to Ax = b x 0 15-853 Page9 Outline 1. Centering Methods Overview
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This note was uploaded on 01/26/2010 for the course COMPUTER S 15-853 taught by Professor Guyblelloch during the Fall '09 term at Carnegie Mellon.

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linear2 - Ellipsoid Algorithm 15-853:Algorithms in the Real...

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