Semidefinite Programming Relaxation vs Polyhedral Homotopy ...
Semidefinite Programming Relaxation vs Polyhedral Homotopy ...
Semidefinite Programming Relaxation vs Polyhedral Homotopy ...
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Discretization of Mimura’s ODE with 2 unknowns u, v : [0, 5] → Ru xx = −(20/9)(35 + 16u − u 2 )u + 20uv,v xx = (1/4)((1 + (2/5)v)v − uv),u x (0) = u x (5) = v x (0) = v x (5) = 0,Discretize:x i = i∆x (i = 0, 1, 2,... ), u x (x i ) ≈ (u(x i+1 ) − u(x i−1 ))/(2∆x).Discretized system of polynomials with ∆x = 1:f 1 (u,v) = 76.8u 1 + u 3 + 35.6u 2 1 − 20.0u 1 v 1 − 2.22u 3 2,f 2 (u,v) = −1.25v 1 + v 2 + 0.25u 1 v 1 − 0.1v1,2f 3 (u,v) = u 1 + 75.8u 2 + u 3 + 35.6u 2 2 − 20.0u 2v 2 − 2.22u 3 2 ,f 4 (u,v) = v 1 − 2.25v 2 + v 3 + 0.25u 2 v 2 − 0.1v2,2f 5 (u,v) = u 2 + 75.8u 3 + u 4 + 35.6u 2 3 − 20.0u 3 v 3 − 2.22u 2 3,f 6 (u,v) = v 2 − 2.25v 3 + v 4 + 0.25u 3 v 3 − 0.1v3,2f 7 (u,v) = u 3 + 76.8u 4 + 35.6u 2 4 − 20.0u 4 v 4 − 2.22u 3 4,f 8 (u,v) = v 3 − 1.25v 4 + 0.25u 4 v 4 − 0.1v4.2Here u i = u(x i ), v i = v(x i ) (i = 0, 1, 2, 3, 4, 5),u 0 = u 1 , u 5 = u 4 , v 0 = v 1 and v 5 = v 4 .⇒ c-sparseWorkshop on Advances in Optimization, April 19-21, 2007 – p.16/24