mpc - EE236A (Fall 2007-08) Lecture 14 Primal-dual...

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 14 Primal-dual interior-point methods primal-dual path-following 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 Primal-dual interior-point 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 Primal-dual interior-point methods 143 Primal-dual path-following 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) Primal-dual interior-point methods 144 Primal-dual 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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mpc - EE236A (Fall 2007-08) Lecture 14 Primal-dual...

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