17ipm

17ipm - EE236C(Spring 2008-09 17 Primal-dual interior-point...

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Unformatted text preview: EE236C (Spring 2008-09) 17. Primal-dual interior-point methods • cone programming • logarithmic barrier function and central path • symmetrization • Nesterov-Todd scaling • path-following algorithm • quadratic cone program 17–1 Cone program primal problem minimize c T x subject to Gx + s = h Ax = b s followsequal s followsequal is inequality with respect to proper convex cone C dual problem maximize − h T z − b T y subject to A T y + G T z + c = 0 z followsequal * z followsequal * is generalized inequality with respect to dual cone C * Primal-dual interior-point methods 17–2 Primal-dual path-following methods closely related to barrier methods (EE236B) • methods follow central path to find approximate solution • steps are computed by linearizing central path equations incorporate several modifications to improve efficiency and robustness • symmetric treatment of primal and dual iterates • barrier parameter is updated after each Newton step; no distinction between outer and inner iterations • more aggressive step sizes • infeasible starting points • higher-order approximation of central path • (some algorithms) detect infeasibility Primal-dual interior-point methods 17–3 Three standard cones we will limit the discussion to three types of cones nonnegative orthant { u ∈ R p | u i ≥ , i = 1 , . . . , p } second-order cone { ( u , u 1 ) ∈ R × R p- 1 | bardbl u 1 bardbl 2 ≤ u } semidefinite cone { u ∈ R p ( p +1) / 2 | mat ( u ) followsequal } ( mat ( u ) is symmetric matrix of order p constructed from u ; see below) these cones are self-dual, so we can drop the subscript in z followsequal * Primal-dual interior-point methods 17–4 Notation for symmetric matrices symmetric matrix as a vector if U = U 11 U 21 ··· U p 1 U 21 U 22 ··· U p 2 . . . . . . . . . . . . U p 1 U p 2 ··· U pp ∈ S p , then vec ( U ) = √ 2 parenleftbigg U 11 √ 2 , U 21 , . . . , U p 1 , U 22 √ 2 , U 32 , . . . , U p 2 , . . . , U pp √ 2 parenrightbigg • vec ( U ) ∈ R p ( p +1) / 2 are lower triangular entries in column-major order • off-diagonal entries are scaled by √ 2 so that tr ( UV ) = vec ( U ) T vec ( V ) Primal-dual interior-point methods 17–5 vector as symmetric matrix if u = ( u 1 , u 2 , . . . , u p ( p +1) / 2 ) ∈ R p ( p +1) / 2 , then mat ( u ) = 1 √ 2 √ 2 u 1 u 2 ··· u p u 2 √ 2 u p +1 ··· u 2 p- 1 . . . . . . u p u 2 p- 1 ··· √ 2 u p ( p +1) / 2 • mat ( u ) ∈ S p is the symmetric matrix of order p constructed from u • off-diagonal entries are scaled by √ 2 so that u T v = tr ( mat ( u ) mat ( v )) Primal-dual interior-point methods 17–6 semidefinite constraints in vector notation • suppose C is the cone of vectorized positive semidefinite matrices C = { u ∈ R p ( p +1) / 2 | mat ( u ) followsequal } = { vec ( U ) | U ∈ S p + } • for G i ∈ S p , h ∈ S p , define G = bracketleftbig vec (...
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17ipm - EE236C(Spring 2008-09 17 Primal-dual interior-point...

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