Semidefinite Programming Relaxation vs Polyhedral Homotopy ...

POP min. f 0 (x) s.t. f j (x) ≥ 0 or = 0 (j = 1,...,m).Example:f 0 (x) = ∑ nk=1 (−x2 k )f j (x) = 1 − x 2 j − 2x 2 j+1 − x 2 n (j = 1,...,n − 1).Hf 0 (x) : the n × n Hessian mat. of f 0 (x),Jf ∗ (x) : the m × n Jacob. mat. of f ∗ (x) = (f 1 (x),...,f m (x)) T ,R : the csp matrix, the n × n density pattern matrix ofI + Hf 0 (x) + Jf ∗ (x) T Jf ∗ (x) (no cancellation in ’+’).[Jf ∗ (x) T Jf ∗ (x)] ij ≠ 0 iff x i and x j are in a common constraint.Workshop on Advances in Optimization, April 19-21, 2007 – p.7/24