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

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1 15-853 Page1 15-853:Algorithms in the Real World Linear and Integer Programming II – Ellipsoid algorithm – Interior point methods 15-853 Page2 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 Problem in practice : always takes this much time. 15-853 Page3 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 -y 0 -cx +by 0 15-853 Page4 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. c k F Feasible region
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2 15-853 Page5 Ellipsoid Algorithm - find smallest ellipsoid that includes the intersection of E k and the feasible side of the constraint. For an elipsoid E k to find the next smaller ellipsoid: - find most violated constraint a k c k F Feasible region a k ) 1 2 /( 1 1 2 1 ) ( ) ( + + = n k k E Vol E Vol 15-853 Page6 Interior Point Methods Travel through the interior with a combination of 1. An optimization term (moves toward objective) 2. A centering term (keeps away from boundary) Used since 50s for nonlinear programming. Karmakar proved a variant is polynomial time in 1984 x 1 x 2 15-853 Page7 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 Page8 Example times Central trajectory method (Lustic, Marsten, Shanno 94) Time depends on Cholesky non-zeros (i.e. the “fill”) 6.7M .2M .3M 1.2M Cholesky non-zeros 9252 645 771 2364 time (sec) 58 53 64 66 iterations 80K 183K 189K 186K non-zeros 19x12K 43x107K 9x57K 13x31K size (K) initial car continent fuel
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3 15-853 Page9 Assumptions We are trying to solve the problem: minimize z = c T x subject to Ax = b x
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linear2 - Ellipsoid Algorithm 15-853:Algorithms in the Real...

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