Full-Newton step polynomial-time methods for LO based on locally ...

Linear optimization via self-dual embedding (2)We associate to any vector x ∈ R n its slack vector s(x) according tos(x) = Mx + q.In the sequel we simply denote s(x) as s, and s will always have this meaning. Since M isskew-symmetric we have z T Mz = 0 <strong>for</strong> every vector z ∈ R n . Hence we haveq T x = (s − Mx) T x = s T x − x T Mx = s T x.There<strong>for</strong>e, if x is feasible, then x is optimal if and only if s T x = 0. Since x and s arenonnegative this holds if and only if x i s i = 0 <strong>for</strong> each i. This shows that x is optimalif and only if the vectors x and s are complementary vectors. We say that x is a strictlycomplementary solution if moreover x i + s i > 0 <strong>for</strong> each i.Summarizing these facts, we have that x is feasible if x ≥ 0 and s ≥ 0. A feasible x isoptimal if xs = 0, and x is a strictly complementary solution if moreover x + s > 0. Thuswe need to solve the systems = Mx + q, x ≥ 0, s ≥ 0,xs = 0x + s > 0.15