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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 • Mehrotra’s corrector step • computing the search directions 14–1 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 14–2 Interpretation of barrier method apply Newton’s 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 Δ z Δ x Δ 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 Δ x from A T X- 2 A Δ x = − tc − A T X- 1 1 i.e. , Δ x is the Newton direction used in barrier method • substitute to obtain Δ s , Δ z Primal-dual interior-point methods 14–3 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 14–4 Primal-dual linearization apply Newton’s 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 Δ z Δ x Δ s = −...
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mpc - EE236A(Fall 2007-08 Lecture 14 Primal-dual...

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