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

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1 15-853:Algorithms in the Real World 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 L), where L = #bits to represent A and b 15-853 Page2 Problem in practice : always takes this much time. Reduction from general case To solve: maximize: z = c T x bj t t A b 0 subject to: Ax b, x 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 k find smallest ellipsoid that includes constraint 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 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 structure 15-853 Page7 structure. Example times fuel continent car initial size (K) 13x31K 9x57K 43x107K 19x12K 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”)