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17ipm - EE236C (Spring 2008-09) 17. Primal-dual...

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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 171 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 172 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 173 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 174 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 175 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 176 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...

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