L18_anaipm

L18_anaipm - Lecture 18: Interior-Point Method March 28,...

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Lecture 18: Interior-Point Method March 28, 2007
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Lecture 18 Outline Barrier-type interior point method Phase I method Bound on number of Newton’s iteration Limitations of interior point method Convex Optimization 1
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Lecture 18 Inequality Constrained Minimization minimize f ( x ) subject to g j ( x ) 0 , j = 1 , . . . , m Ax = b Assumptions : The functions f and g j are convex and twice continuously differ. The matrix A R p × n has rank p The optimal value f * is finite and attained The Slater condition holds: ˜ x dom f, g x ) 0 , A ˜ x = b Thus, no duality gap and a dual optimal ( μ * , λ * ) R m × R p exists Interior Point Method solves a sequence of Penalized Problems : t > 0 minimize f ( x ) - 1 t m j =1 ln( - g j ( x )) subject to Ax = b The objectives of penalized problems are convex and twice cont. differ. A solution of the penalized problem has to exist for each t > 0 For each penalized problem there is no dualiy gap (same ˜ x works) Thus, a dual optimal ˆ λ ( t ) exists for each penalized problem Convex Optimization 2
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Lecture 18 KKT Conditions Original problem : minimize f ( x ) subject to g ( x ) ± 0 , Ax = b x * is optimal iff there is ( μ * , λ * ) Ax * = b, g ( x * ) ± 0 , μ * ² 0 f ( x * ) + m j =1 μ * j g j ( x * ) + A T λ * = 0 ( μ * ) T g ( x * ) = 0 Penalized problem : minimize f ( x ) - 1 t m j =1 ln( - g j ( x )) subject to Ax = b x * t is optimal iff there is ˆ λ : Ax * t = b , g ( x * t ) 0 f ( x * t ) + m j =1 1 - tg j ( x * t ) g j ( x * t ) + A T ˆ λ = 0 Letting ˆ μ j = 1 - tg j ( x * t ) for all j , we see that x * t and μ, ˆ λ ) satisfy KKTs for the original problem with approx- imate CS condition ˆ μ T g ( x * t ) = - m t Furthermore f * f ( x * t ) f * + m t Convex Optimization 3
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Lecture 18 Interior-Point Method: Barrier Method Given a strictly feasible x , t := t 0 > 0 , β > 1 , and a tolerance ± > 0 . Repeat 1. Centering step. Compute x * ( t ) solving min Az = b { tf ( z ) + φ ( z ) } 2. Update. Set x := x * ( t ) . 3. Stopping criterion. Quit if m/t < ± . 4. Increase penalty. Set t := βt . Terminates with f ( x ) - f * ± [follows from f ( x * ( t )) - f * m/t ] Centering steps are viewed as outer iterations Inner iterations : computing x * ( t ) using Newton’s method starting at current x Choice of β involves a trade-off: large β means fewer outer iterations,
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L18_anaipm - Lecture 18: Interior-Point Method March 28,...

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