Barrierb - EE236A(Fall 2007-08 Lecture 13 Convergence analysis of the barrier method • complexity analysis of the barrier method – convergence

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 13 Convergence analysis of the barrier method • complexity analysis of the barrier method – convergence analysis of Newton’s method – choice of update parameter μ – bound on the total number of Newton iterations • initialization 13–1 Complexity analysis we’ll analyze the method of page 12–21 with • update t + = μt • starting point x ∗ ( t (0) ) on the central path main result : #Newton iters is bounded by O ( √ m log( ǫ (0) /ǫ )) (where ǫ (0) = m/t (0) ) caveats: • methods with good worst-case complexity don’t necessarily work better in practice • we’re not interested in the numerical values for the bound—only in the exponent of m and n • doesn’t include initialization • insights obtained from analysis are more valuable than the bound itself Convergence analysis of the barrier method 13–2 Outline 1. convergence analysis of Newton’s method for ϕ ( x ) = tc T x − m summationdisplay i =1 log( b i − a T i x ) (will give us a bound on the number of Newton steps per outer iteration) 2. effect of μ on total number of Newton iterations to compute x ∗ ( μt ) from x ∗ ( t ) 3. combine 1 and 2 to obtain the total number of Newton steps, starting at x ∗ ( t (0) ) Convergence analysis of the barrier method 13–3 The Newton decrement Newton step at x : v = −∇ 2 ϕ ( x ) − 1 ∇ ϕ ( x ) = − ( A T diag ( d ) 2 A ) − 1 ( tc + A T d ) where d = (1 / ( b 1 − a T 1 x ) , . . . , 1 / ( b m − a T m x )) Newton decrement at x : λ ( x ) = radicalBig ∇ ϕ ( x ) T ∇ 2 ϕ ( x ) − 1 ∇ ϕ ( x ) = radicalBig v T ∇ 2 ϕ ( x ) v = parenleftBigg m summationdisplay i =1 parenleftbigg a T i v b i − a T i x parenrightbigg 2 parenrightBigg 1 / 2 = bardbl diag ( d ) Av bardbl Convergence analysis of the barrier method 13–4 theorem. if λ = λ ( x ) < 1 , then ϕ is bounded below and ϕ ( x ) ≤ ϕ ( x ∗ ( t )) − λ − log(1 − λ ) 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 4 • if λ ≤ . 81 , then ϕ ( x...
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This note was uploaded on 09/26/2009 for the course CAAM 236 taught by Professor Dr.vandenber during the Spring '07 term at Monmouth IL.

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Barrierb - EE236A(Fall 2007-08 Lecture 13 Convergence analysis of the barrier method • complexity analysis of the barrier method – convergence

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