Lecture 14 Barrier method

Transcription

1 L. Vandenberghe EE236A (Fall ) Lecture 14 Barrier method centering problem Newton decrement local convergence of Newton method short-step barrier method global convergence of Newton method predictor-corrector method 14 1

2 Centering problem centering problem (with notation and assumptions of page 13 12) minimize f t (x) = tc T x+φ(x) φ(x) = m i=1 log(b i a T i x) is logarithmic barrier of Ax b minimizer x is point x (t) on central path minimizer is m/t-suboptimal solution of LP: c T x (t) p m t barrier method(s): use Newton s method to (approximately) minimize f t, for a sequence of t Barrier method 14 2

3 gradient and Hessian Properties of centering cost function f t (x) = tc+a T d x, 2 f t (x) = A T diag(d x ) 2 A d x = ( 1/(b 1 a T 1x),...,1/(b m a T mx) ) is a positive m-vector (strict) convexity and its consequences (see pages 13 6 and 13 9) 2 f t (x) is positive definite for all x P first order condition: f t (y) > f t (x)+ f t (x) T (y x) for all x,y P with x y x minimizes f t if and only if f t (x) = 0; if minimizer exists, it is unique Barrier method 14 3

4 Lower bound for centering problem centering problem is bounded below if dual LP is strictly feasible: f t (y) tb T z + m logz i +mlogt+m for all y P i=1 here z can be any strictly dual feasible point (A T z +c = 0, z > 0) proof: difference of left- and right-hand sides is m t(c T y +b T z) log(t(b i a T i y)z i ) m i=1 = t(b Ay) T z m log(t(b i a T i y)z i ) m i=1 0 (since u logu 1 0) Barrier method 14 4

5 Newton method for centering problem Newton step for f t at x P = {y Ay < b} x nt = 2 f t (x) 1 f t (x) = 2 φ(x) 1 (tc+ φ(x)) = ( A T diag(d x ) 2 A ) 1( tc+a T d x ) Newton iteration: choose suitable stepsize α and make update x := x+α x nt we will show that Newton method converges if f t is bounded below Barrier method 14 5

6 Outline centering problem Newton decrement local convergence of Newton method short-step barrier method global convergence of Newton method predictor-corrector method

7 Newton decrement the Newton decrement at x P is λ t (x) = ( x T nt 2 ) 1/2 φ(x) x nt = diag(d x )A x nt = x nt x λ t (x) 2 is the directional derivative of f t at x in the direction x nt : λ t (x) 2 = f t (x) T x nt = d dα f t(x+α x nt ) α=0 λ t (x) = 0 if and only if x = x (t) we will use λ t (x) to measure proximity of x to x (t) Barrier method 14 6

8 Dual feasible points near central path on central path (p ): strictly dual feasible point from x = x (t) z (t) = 1 t d x, z i(t) = 1 t(b i a T i x), i = 1,...,m near central path: for x P with λ t (x) < 1, define z = 1 t ( dx +diag(d x ) 2 A x nt ) x nt is the Newton step for f t at x z is strictly dual feasible (see next page); duality gap with x is (b Ax) T z = m+dt xa x nt t ( 1+ λ t(x) m ) m t Barrier method 14 7

9 proof: by definition, Newton step x nt at x satisfies A T diag(d x ) 2 A x nt = tc A T d x therefore z satisfies the equality constraints A T z +c = 0 z > 0 if and only if 1+diag(d x )A x nt > 0 a sufficient condition is λ t (x) = diag(d x )A x nt < 1 bound on duality gap follows from Cauchy-Schwarz inequality Barrier method 14 8

10 Lower bound for centering problem near central path substituting the dual z of p.14 7 in the lower bound of p.14 4 gives f t (y) f t (x) m (w i log(1+w i )) y P i=1 where w i = (a T i x nt)/(b i a T i x) this bound holds if λ t (x) < 1; a simpler bound holds if λ t (x) 0.68: f t (y) f t (x) m w 2 i u log(1 + u) u 2 i=1 0.5 = f t (x) λ t (x) (since u log(1 + u) u 2 for u 0.68) u Barrier method 14 9

11 Outline centering problem Newton decrement local convergence of Newton method short-step barrier method global convergence of Newton method predictor-corrector method

12 Quadratic convergence of Newton s method theorem: if λ t (x) < 1 and x nt is Newton step of f t at x, then x + = x+ x nt P, λ t (x + ) λ t (x) 2 Newton method (with unit stepsize) x (k+1) = x (k) 2 f t (x (k) ) 1 f t (x (k) ) if λ t (x (0) ) 1/2, Newton decrement after k iterations is λ t (x (k) ) (1/2) 2k decreases very quickly: (1/2) 25 = , (1/2) 26 = ,... λ t (x (k) ) very small after a few iterations Barrier method 14 10

13 proof of quadratic convergence result feasibility of x + : follows from λ t (x) = x nt x and result on p.13 7 quadratic convergence: define D = diag(d x ), D + = diag(d x +) λ t (x + ) 2 = D + A x + nt 2 D + A x + nt 2 + (I D 1 + D)DA x nt +D + A x + nt 2 = (I D 1 + D)DA x nt 2 = (I D 1 + D) (I D 1 + D)1 4 = DA x nt 2 = λ t (x) 4 Barrier method 14 11

14 on lines 4 and 6 we used DA x nt = D(b Ax b+ax + ) = (I D 1 + D)1 line 3 follows from A T D + ( D+ A x + nt+(i D 1 + D)DA x nt ) = A T D 2 +A x + nt A T D 2 A x nt +A T D + DA x nt = tc A T D + 1+tc+A T D1+A T D + (I D 1 + D)1 = 0 line 5 follows from ( i y4 i ) ( i y2 i )2 Barrier method 14 12

15 Outline centering problem Newton decrement local convergence of Newton method short-step barrier method global convergence of Newton method predictor-corrector method

16 Short-step barrier method simplifying assumptions a central point x (t 0 ) is given x (t) is computed exactly algorithm: define tolerance ǫ (0, 1) and parameter µ = m starting at t = t 0, repeat until m/t ǫ: compute x (µt) by Newton s method with unit step started at x (t) set t := µt Barrier method 14 13

17 Newton decrement after update of t gradient of f t + at x = x (t) for new value t + = µt: f t +(x) = µtc+a T d x = (µ 1)A T d x Newton decrement for new value t + is λ t +(x) = ( f t +(x) T 2 φ(x) 1 f t +(x) ) 1/2 = (µ 1) ( 1 T B(B T B) 1 B T 1 ) 1/2 (µ 1) m = 1/2 (with B = diag(d x )A) line 3 follows because maximum eigenvalue of B(B T B) 1 B T is one x (t) is in region of quadratic convergence of Newton s method for f µt Barrier method 14 14

18 Iteration complexity Newton iterations per outer iteration: a small constant number of outer iterations: we reach t (k) = µ k t 0 m/ǫ when k log(m/(ǫt 0)) logµ cumulative number of Newton iterations ( ( )) mlog m O ǫt 0 (we used logµ = log(1+1/(2 m)) (log2)/(2 m)) multiply by flops per Newton iteration to get polynomial complexity m dependence is lowest known complexity for interior-point methods Barrier method 14 15

19 Outline centering problem Newton decrement local convergence of Newton method short-step barrier method global convergence of Newton method predictor-corrector method

20 Maximum stepsize to boundary for x P and arbitrary v 0, define σ x (v) = 0 if Av 0 a T i v b i a T i x otherwise max i=1,...,m point x+αv (α 0) is in P if and only if ασ x (v) x + σ 1 v x + v x x+αv P for α [0,1/σ] if σ > 0 α [0, ) if σ = 0 Barrier method 14 16

21 Upper bound on centering cost function arbitrary direction: for x P and arbitrary v 0 if σ = σ x (v) > 0 and α [0,1/σ): f t (x+αv) f t (x)+α f t (x) T v v 2 x (ασ +log(1 ασ)) σ2 if σ = σ x (v) = 0 and α [0, ): f t (x+αv) f t (x)+α f t (x) T v + α2 2 v 2 x on the right-hand sides, v x = (v T 2 f t (x)v) 1/2 Newton direction: for v = x nt, substitute f t (x) T v = v 2 x = λ t (x) 2 Barrier method 14 17

22 proof: define w i = (a T i v)/(b i a T i x) and note that w = v x if σ = max i w i > 0: f t (x+αv) f t (x) α f t (x) T v m = (αw i +log(1 αw i )) i=1 w i >0 w i >0 m i=1 (αw i +log(1 αw i ))+ w i 0 α 2 w 2 i 2 wi 2 (ασ)2 σ2(ασ +log(1 ασ))+ 2 wi 2 σ2(ασ +log(1 ασ)) w i 0 w 2 i σ 2 ( ) if σ = 0, upper bound follows from ( ) Barrier method 14 18

23 Damped Newton iteration x + = x+ 1 1+σ x ( x nt ) x nt theorem: damped Newton iteration at any x P decreases cost f t (x + ) f t (x) λ t (x)+log(1+λ t (x)) graph shows u log(1+u) if λ t (x) 0.5 f t (x + ) f t (x) 0.09 u log(1 + u) u Barrier method 14 19

24 proof: apply upper bounds on page with v = x nt if σ > 0, the value of the upper bound f t (x) αλ t (x) 2 λ t(x) 2 σ 2 (ασ +log(1 ασ)) at α = 1/(1+σ) is f t (x) λ t(x) 2 σ 2 (σ log(1+σ)) f t (x) λ t (x)+log(1+λ t (x)) if σ = 0, value of upper bound at α = 1 is f t (x) αλ t (x) 2 + α2 2 λ t(x) 2 f t (x) λ t(x) 2 2 f t (x) λ t (x)+log(1+λ t (x)) Barrier method 14 20

25 Summary: Newton algorithm for centering centering problem minimize f t (x) = tc T x+φ(x) algorithm given: tolerance δ (0,1), starting point x := x (0) P repeat: 1. compute Newton step x nt at x and Newton decrement λ t (x) 2. if λ t (x) δ, return x 3. otherwise, set x := x+α x nt with α = 1 1+σ x ( x nt ) if λ t (x) > 1/2 α = 1 if λ t (x) 1/2 Barrier method 14 21

26 Convergence theorem: if δ < 1/2 and f t (x) is bounded below, algorithm takes at most f t (x (0) ) min y f t (y) log 2 log 2 (1/δ) iterations (1) proof: combine theorems on pages and if λ t (x (k) ) > 1/2, iteration k decreases the function value by at least λ t (x (k) ) log(1 λ t (x (k) )) 0.09 the first term in (1) bounds the number of iterations with λ t (x) > 1/2 if λ t (x (l) ) 1/2, quadratic convergence yields λ t (x (k) ) δ after k = l+log 2 log 2 (1/δ) iterations Barrier method 14 22

27 Computable bound on #iterations replace unknown min y f t (y) in (1) by the lower bound from page 14 4: f t (x) min y f t (y) V t (x,z) where z is a strictly dual feasible point and V t (x,z) = f t (x)+tb T z = t(b Ax) T z m logz i mlogt m i=1 m log(t(b i a T i x)z i ) m i=1 number of Newton iterations to minimize f t starting at x is bounded by 10.6V t (x,z)+log 2 log 2 (1/δ) Barrier method 14 23

28 Outline centering problem Newton decrement local convergence of Newton method short-step barrier method global convergence of Newton method predictor-corrector method

29 Predictor-corrector methods short-step methods stay in narrow neighborhood of central path, defined by limit on λ t make small, fixed increases t + = µt quite slow in practice predictor-corrector methods select new t using a linear approximation to central path ( predictor ) recenter with new t ( corrector ) can make faster and adaptive increases in t Barrier method 14 24

30 Tangent to central path central path equation [ 0 s (t) ] = [ 0 A T A 0 ][ x (t) z (t) ] + [ c b ] s i(t)z i(t) = 1 t, i = 1,...,m derivatives: ẋ = dx (t)/dt, ṡ = ds /dt, ż = dz (t)/dt satisfy [ 0 ṡ ] = [ 0 A T A 0 ][ ẋ ż ] s i(t)ż i +zi(t)ṡ i = 1 t2, i = 1,...,m tangent direction: defined as x tg = tẋ, s tg = tṡ, z tg = tż Barrier method 14 25

31 Predictor equations with x = x (t), s = s (t), z = z (t) (1/t)diag(s) 2 0 I 0 0 A T I A 0 s tg x tg z tg = z 0 0 (1) equivalent equation (using s i z i = 1/t) I 0 (1/t)diag(z) A T I A 0 s tg x tg z tg = s 0 0 (2) Barrier method 14 26

32 Properties of tangent direction from 2nd and 3rd block in (1): s T tg z tg = 0 take inner product with s on both sides of first block in (1): s T z tg +z T s tg = s T z hence, gap in tangent direction is (s+α s tg ) T (z +α z tg ) = (1 α)s T z take inner product with s tg on both sides of first block in (1): s T tgdiag(s) 2 s tg = tz T s tg similarly, from first block in (2): z T tgdiag(z) 2 z tg = ts T z tg Barrier method 14 27

33 Potential function definition: for strictly primal, dual feasible x, z, define Ψ(x,z) = mlog zt s m m i=1 = mlog ( i z is i )/m ( i z is i ) 1/m log(s i z i ) with s = b Ax = mlog arithmetic mean of z 1s 1,..., z m s m geometric mean of z 1 s 1,..., s m z m properties Ψ(x,z) is nonnegative for all strictly feasible x, z Ψ(x,z) = 0 only if x and z are centered, i.e., for some t > 0, z i s i = 1/t, i = 1,...,m Barrier method 14 28

34 Potential function and proximity to central path for any strictly feasible x, z, with s = b Ax: minimizing t is t = m/s T z Ψ(x,z) = min V t(x,z) (see page 14 23) t>0 ( ) m = min ts T z log(ts i z i ) m t>0 i=1 Ψ as global measure of proximity to central path V t (x,z) bounds the effort to compute x (t), starting at x (page 14 23) Ψ(x, z) bounds centering effort, without imposing a specific t Barrier method 14 29

35 Predictor-corrector method with exact centering simplifying assumptions: exact centering, a central point x (t 0 ) is given algorithm: define tolerance ǫ (0,1), parameter β > 0, and initial values t := t 0, x := x (t 0 ), z := z (t 0 ), s := b Ax (t 0 ) repeat until m/t ǫ: compute tangent direction x tg, s tg, z tg at x, s, z determine α by solving Ψ(x+α x tg,z +α z tg ) = β and take x := x+α x tg, z := z +α z tg, s := b Ax set t := m/(s T z) and use Newton s method to compute x := x (t), z := z (t), s := b Ax Barrier method 14 30

36 Iteration complexity bound on potential function in tangent direction (proof on next page): Ψ(x+α x tg,z +α z tg ) α m log(1 α m) lower bound on predictor step length α: α m γ with γ the solution of γ log(1 γ) = β reduction in duality gap after one predictor-corrector cycle: t γ = 1 α 1 exp( γ ) t + m m bound on total #Newton iterations to reach t (k) m/ǫ: ( ( )) mlog m O ǫt 0 Barrier method 14 31

37 proof of bound on Ψ: let s + = s+α s tg, z + = z +α z tg from definition of Ψ and (s + ) T z + = (1 α)s T z: Ψ(x +,z + ) Ψ(x,z) = mlog (s+ ) T z + z T s = mlog(1 α) m i=1 m i=1 (log s+ i s i +log z+ i z i ) (log s+ i s i +log z+ i z i ) define a (2m)-vector w = (diag(s) 1 s tg,diag(z) 1 z tg ) m (log(s + i /s i)+log(z + i /z i)) = 2m i=1 i=1 log(1+αw i ) α1 T w α w log(1 α w ) last inequality can be proved as on page Barrier method 14 32

38 from the properties on page and s = (1/t)z: 1 T w = t(s T z tg +z T s tg ) = ts T z = m w 2 = s T tgdiag(s) 2 s tg + ztgdiag(z) T 2 z tg = t(z T s tg +s T z tg ) = m substituting this in the upper bound on Ψ gives Ψ(x +,z + ) Ψ(x,z) mlog(1 α)+αm α m log(1 α m) α m log(1 α m) Barrier method 14 33

39 Conclusion: barrier methods started at x (t 0 ), find ǫ-suboptimal point after O ( ( )) mlog m ǫt 0 Newton iterations analysis can be modified to account for inexact centering end-to-end complexity analysis must include the cost of phase I parameters were chosen to simplify analysis, not for efficiency in practice Barrier method 14 34