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Unformatted text preview: EE236A (Fall 200708) Lecture 14 Primaldual interiorpoint methods primaldual pathfollowing Mehrotras corrector step computing the search directions 141 Central path and complementary slackness s + Ax b = 0 A T z + c = 0 z i s i = 1 /t, i = 1 , . . . , m z , s continuous deformation of optimality conditions defines central path: solution is x = x * ( t ) , s = b Ax * ( t ) , z i = 1 /ts i m + n linear and m nonlinear equations in the variables s R m , x R n , z R m Primaldual interiorpoint methods 142 Interpretation of barrier method apply Newtons method to s + Ax b = 0 , A T z + c = 0 , z i 1 / ( ts i ) = 0 , i = 1 , . . . , m i.e. , linearize around current x , z , s : A I A T X X 1 /t z x s = ( Ax + s b ) ( A T z + c ) 1 /t Xz where X = diag ( s ) solution (for s + Ax b = 0 , A T z + c = 0 ): determine x from A T X 2 A x = tc A T X 1 1 i.e. , x is the Newton direction used in barrier method substitute to obtain s , z Primaldual interiorpoint methods 143 Primaldual pathfollowing methods modifications to the barrier method: different linearization of central path update both x and z after each Newton step allow infeasible iterates very aggressive step size selection (99% or 99.9% of step to the boundary) update t after each Newton step (hence distinction between outer & inner iteration disappears) linear or polynomial approximation to the central path limited theory, fewer convergence results work better in practice (faster and more reliable) Primaldual interiorpoint methods 144 Primaldual linearization apply Newtons method to s + Ax b = 0 A T z + c = 0 z i s i 1 /t = 0 , i = 1 , . . . , m i.e. , linearize around s , x , z : A I A T X Z z x s =...
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 Spring '07
 Dr.Vandenber

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