Semidefinite Programming Relaxation vs Polyhedral Homotopy ...

Katsura n system of polynomial equations; n = 8 case0 = −x 1 + 2x 2 9 + 2x 2 8 + 2x 2 7 + · · · + 2x 2 2 + x 2 1,0 = −x 2 + 2x 9 x 8 + 2x 8 x 7 + 2x 7 x 6 + · · · + 2x 3 x 2 + 2x 2 x 1 ,· · · · · · not c-sparse0 = −x 8 + 2x 9 x 2 + 2x 8 x 1 + 2x 7 x 2 + 2x 6 x 3 + 2x 5 x 4 ,1 = 2x 9 + 2x 8 + 2x 7 + 2x 6 + 2x 5 + 2x 4 + 2x 3 + 2x 2 + x 1 .Numerical results on SparsePOP (WKKM 2004)n obj.funct. relax. order r cpu∑8 xi ↑ 1 0.08∑8 x2i ↓ 2 7.1∑11 xi ↑ 1 0.14∑11 x2i ↓ 2 101.3A formulation∑in terms of a POPmax ni=1 x i or min ∑ ni=1 x2 isub.to Katsura n system , −5 ≤ x i ≤ 5 (i = 1,...,n).Different objective functions ⇒ different solutions.Workshop on Advances in Optimization, April 19-21, 2007 – p.13/24