18ipm2

18ipm2 - EE236C(Spring 2008-09 18 Primal-dual...

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Unformatted text preview: EE236C (Spring 2008-09) 18. Primal-dual interior-point methods II • self-dual embedding • path-following algorithm 18–1 Initialization and infeasibility detection barrier method (EE236B) • assumes problem is primal and dual feasible • requires phase I to find initial primal feasible point primal-dual path-following method (last lecture) • assumes problem is primal and dual feasible • allows infeasible starting points methods based on self-dual embedding • can detect primal and dual infeasibility • embed cone program in slightly larger problem that is always feasible • from solution of embedded problem, extract solution of original problem, or certificates of primal or dual infeasibility Primal-dual interior-point methods II 18–2 Infeasibility primal infeasibility: a solution y , z of A T y + G T z = 0 , h T z + b T y = − 1 , z followsequal is a certificate of infeasibility of Gx precedesequal h , Ax = b dual infeasibility: a solution x of Gx precedesequal , Ax = 0 , c T x = − 1 is a certificate of infeasibility of A T y + G T z + c = 0 , z followsequal these are strong alternatives if a constraint qualification holds Primal-dual interior-point methods II 18–3 Self-dual embedding minimize subject to s κ = A T G T c − A b − G h − c T − b T − h T x y z τ ( s, κ, z, τ ) followsequal • problem has a trivial solution (all variables zero) • equality constraint implies s T z + κτ = 0 at feasible points • problem is not strictly feasible (hence, central path does not exist) Primal-dual interior-point methods II 18–4 Optimality condition for embedded problem s κ = A T G T c − A b − G h − c T − b T − h T x y z τ ( s, κ, z, τ ) followsequal , z T s + τκ = 0 • follows from self-dual property • a (mixed) linear complementarity problem Primal-dual interior-point methods II 18–5 Classification of nonzero solution let s, κ, x, y, z, τ be a nonzero solution of the self-dual embedding case 1: τ > , κ = 0 ˆ s = s/τ, ˆ x = x/τ, ˆ y = y/τ, ˆ z = z/τ are primal and dual solutions of the cone program, i.e. , satisfy ˆ s = A T G T − A − G ˆ x ˆ y ˆ z + c b h (ˆ s,...
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This note was uploaded on 01/25/2010 for the course EE 236 taught by Professor Staff during the Spring '08 term at UCLA.

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18ipm2 - EE236C(Spring 2008-09 18 Primal-dual...

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