Semidefinite Programming Relaxation vs Polyhedral Homotopy ...

A POP alkyl from globalibmin − 6.3x 5 x 8 + 5.04x 2 + 0.35x 3 + x 4 + 3.36x 6sub.to − 0.820x 2 + x 5 − 0.820x 6 = 0,0.98x 4 − x 7 (0.01x 5 x 10 + x 4 ) = 0, −x 2 x 9 + 10x 3 + x 6 = 0,x 5 x 12 − x 2 (1.12 + 0.132x 9 − 0.0067x 2 9) = 0,x 8 x 13 − 0.01x 9 (1.098 − 0.038x 9 ) − 0.325x 7 = 0.574,x 10 x 14 + 22.2x 11 = 35.82, x 1 x 11 − 3x 8 = −1.33,lbd i ≤ x i ≤ ubd i (i = 1, 2,...,14).14 variables, 7 poly. equality constraints with deg. 3.SparseDense (Lasserre)r ǫ obj ǫ feas cpu ǫ obj ǫ feas cpu2 1.0e-02 7.1e-01 1.8 7.2e-3 4.3e-2 14.43 5.6e-10 2.0e-08 23.0 out of memoryǫ obj = approx.opt.val. − lower bound for opt.val.ǫ feas = the maximum error in the equality constraintsWorkshop on Advances in Optimization, April 19-21, 2007 – p.11/24