17. Path-following methods

Transcription

1 L. Vandenberghe EE236C (Spring 2016) 17. Path-following methods central path short-step barrier method predictor-corrector method 17-1

2 Introduction Primal-dual pair of conic LPs minimize subject to c T x Ax b maximize b T z subject to A T z + c = 0 z 0 A R m n with rank(a) = n inequalities are with respect to proper cone K and its dual cone K we will assume primal and dual problem are strictly feasible This lecture feasible methods that follow the central path to find the solution complexity analysis based on theory of self-concordant functions Path-following methods 17-2

3 Outline central path short-step barrier method predictor-corrector method

4 Barrier for the feasible set Definition: as a barrier function for the feasible set we will use ψ(x) = φ(b Ax) where φ is a θ-normal barrier for K Notation (in this lecture): v x = (v T 2 ψ(x) 1 v) 1/2 Properties ψ is self-concordant with domain {x Ax b} Newton decrement of ψ is bounded by θ, i.e., ψ(x) 2 x = ψ(x)t 2 ψ(x) 1 ψ(x) θ x dom ψ (proof on next page) Path-following methods 17-3

5 Proof of bound on Newton decrement: gradient and Hessian of ψ are (with s = b Ax) ψ(x) = A T φ(s), 2 ψ(x) = A T 2 φ(s)a from page 16-24, φ(s) T 2 φ(s) 1 φ(s) = θ; therefore ψ(x) T 2 ψ(x) 1 ψ(x) = sup v = sup v sup w ( v T 2 ψ(x)v + 2 ψ(x) T v) ( (Av) T 2 φ(s)(av) 2 φ(s) T Av) ( w T 2 φ(s)w + 2 φ(s) T w) = φ(s) T 2 φ(s) 1 φ(s) = θ Path-following methods 17-4

6 Central path Definition: the set of minimizers x (t), for t > 0, of tc T x + ψ(x) = tc T x + φ(b Ax) Optimality conditions A T φ(s) = tc, s = b Ax implies that z = (1/t) φ(s) is strictly dual feasible by weak duality, c T x (t) p c T x + b T z = z T s = θ t hence, c T x (t) p as t Path-following methods 17-5

7 Existence and uniqueness Centering problem minimize subject to tc T x + φ(s) Ax + s = b Lagrange dual (with dual cone barrier φ of page 16-27) maximize tb T z φ (z) + θ log t subject to A T z + c = 0 strictly feasible z for dual conic LP is feasible for dual centering problem if dual conic LP is strictly feasible, tc T x + φ(b Ax) is bounded below from self-concordance theory (page 16-12), x (t) exists and is unique Path-following methods 17-6

8 Dual points in neighborhood of central path Newton step x for tc T x + ψ(x) = tc T x + φ(b Ax) satisfies Newton equation A T 2 φ(s)a x = tc + A T φ(s), s = b Ax Newton decrement is λ t (x) = ( x T 2 ψ(x) x ) 1/2 Dual feasible point: define z = 1 t ( φ(s) 2 φ(s)a x ) satisfies A T z + c = 0 by definition satisfies z 0 if λ t (x) < 1 (see next page) Path-following methods 17-7

9 Proof. z 0 follows from Dikin ellipsoid theorem Newton decrement is λ t (x) 2 = x T 2 ψ(x) x = x T A T 2 φ(s)a x = v T 2 φ(s) 1 v where v = 2 φ(s)a x define u = φ(s); then 2 φ (u) = 2 φ(s) 1 (see page 16-28) and λ t (x) 2 = v T 2 φ (u)v by Dikin ellipsoid theorem λ t (x) < 1 implies u + v = φ(s) + 2 φ(s)a x 0 Path-following methods 17-8

10 Duality gap in neighborhood of central path c T x p ( 1 + λ t(x) θ ) θ t if λ t (x) < 1 from weak duality, using the dual point z on page 17-7 s T z = 1 t 1 t ( θ s T 2 φ(s)a x ) (θ + 2 φ(s) 1/2 s 2 2 φ(s) 1/2 A x 2 ) = θ + θ λ t (x) t implies c T x p 2θ/t, since θ 1 holds for any θ-normal barrier φ (φ is unbounded below, so its Newton decrement θ 1 everywhere) Path-following methods 17-9

11 Outline central path short-step barrier method predictor-corrector method

12 Short-step methods General idea: keep the iterates in the region of quadratic convergence for tc T x + ψ(x), by limiting the rate at which t is increased (hence, short-step ) Quadratic convergence results (from self-concordance theory) if λ t (x) 1/4, a full Newton step gives λ t (x + ) 2λ t (x) 2 started at a point with λ t (x) 1/4, an accuracy ɛ cent is reached in log 2 log 2 (1/ɛ cent ) iterations for practical purposes this is a constant (4 6 for ɛ cent ) Path-following methods 17-10

13 Short-step method with exact centering simplifying assumptions: x (t) is computed exactly a central point x (t 0 ) is given Algorithm: define a tolerance ɛ (0, 1) and parameter starting at t = t 0, repeat until θ/t ɛ: µ = θ compute x (µt) by Newton s method started at x (t) set t := µt Path-following methods 17-11

14 Newton iterations for recentering Newton decrement at x = x (t) for new value t + = µt is λ t +(x) = µtc + ψ(x) x = µ(tc + ψ(x)) (µ 1) ψ(x) x = (µ 1) ψ(x) x (µ 1) θ = 1/4 line 3 follows because tc + ψ(x) = 0 for x = x (t) line 4 follows from ψ(x) x θ (see page 17-3) Conclusion number of iterations to compute x (t + ) from x (t) is bounded by a small constant Path-following methods 17-12

15 Iteration complexity Number of outer iterations: t (k) = µ k t 0 θ/ɛ when k log(θ/(ɛt 0)) log µ Cumulative number of Newton iterations ( ( )) θ θ O log ɛt 0 (we used log µ (log 2)/(4 θ) by concavity of log(1 + u)) multiply by flops per iteration to get polynomial worst-case complexity θ dependence is lowest known complexity for interior-point methods Path-following methods 17-13

16 Short-step method with inexact centering Improvements of short-step method with exact centering keep iterates in region of quadratic region, but avoid complete centering at each iteration: make small increase in t, followed by one Newton step Algorithm: define a tolerance ɛ (0, 1) and parameters select x and t with λ t (x) β repeat until 2θ/t ɛ: β = 1 8, µ = θ t := µt, x := x 2 ψ(x) 1 (tc + ψ(x)) Path-following methods 17-14

17 Newton decrement after update we first show that λ t (x) β at the end of each iteration if λ t (x) β and t + = µt, then λ t +(x) = t + c + ψ(x)) x = µ(tc + ψ(x)) (µ 1) ψ(x) x µ tc + ψ(x) x + (µ 1) ψ(x) x µβ + (µ 1) θ = 1 4 from theory of Newton s method for self-concordant functions (page 16-16) λ t +(x + ) 2λ t +(x) = β Path-following methods 17-15

18 Iteration complexity from page 17-9, stopping criterion implies c T x p ɛ stopping criterion is satisified when t (k) t 0 = µ k 2θ, k log(2θ/(ɛt 0)) ɛt 0 log µ taking the logarithm on both sides gives an upper bound of ( ( )) θ θ O log ɛt 0 iterations (using log µ log 2/(1 + 8 θ)) Path-following methods 17-16

19 Outline central path short-step barrier method predictor-corrector method

20 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 as a result, quite slow in practice Predictor-corrector method select new t using a linear approximation to central path ( predictor ) recenter with new t ( corrector ) allows faster and adaptive increases in t Path-following methods 17-17

21 Global convergence bound for centering problem minimize f t (x) = tc T x + φ(b Ax) Convergence result (damped Newton algorithm of page started at x) #iterations f t(x) inf u f t (u) ω(η) + log 2 log 2 (1/ɛ cent ) ɛ cent is accuracy in centering ω(η) = η log(1 + η) and η (0, 1/4] for practical purposes, second term is a small constant first term depends on unknown optimal value inf u f t (u) Path-following methods 17-18

22 Bound from duality Dual centering problem (see page 17-6) maximize tb T z φ (z) + θ log t subject to A T z + c = 0 strictly feasible z provides lower bound on inf u f t (u): inf u f t(u) tb T z φ (z) + θ log t Bound on centering cost: f t (x) inf u f t (u) V t (x, s, z) where V t (x, s, z) = t(c T x + b T z) + φ(s) + φ (z) θ log t = ts T z + φ(s) + φ (z) θ log t Path-following methods 17-19

23 Potential function Definition (for strictly feasible x, s, z) Ψ(x, s, z) = inf t V t (x, s, z) = θ log st z θ (optimal t is t = argmin t V t (x, s, z) = θ/s T z) + φ(s) + φ (z) + θ Properties homogeneous of degree zero: Ψ(αx, αs, αz) = Ψ(x, s, z) for α > 0 nonnegative for all strictly feasible x, s, z zero only if x, s, z are centered can be used as a global measure of proximity to the central path Path-following methods 17-20

24 Tangent to central path Central path equation [ 0 s (t) ] = [ 0 A T A 0 ] [ x (t) z (t) ] + [ c b ] z (t) = 1 t φ(s (t)) Derivatives ẋ = dx (t)/dt, ṡ = ds /dt, ż = dz (t)/dt satisfy [ 0 ṡ ] = [ 0 A T A 0 ] [ ẋ ż ] ż = 1 t z (t) 1 t 2 φ(s (t))ṡ Tangent direction: derivatives scaled by t (to simplify notation) x t = tẋ, s t = tṡ, z t = tż Path-following methods 17-21

25 Predictor equations with x = x (t), s = s (t), z = z (t) (1/t) 2 φ(s) 0 I 0 0 A T I A 0 s t x t z t = z 0 0 (1) Equivalent equations I 0 (1/t) 2 φ (z) 0 0 A T I A 0 s t x t z t = s 0 0 (2) equivalence follows from primal-dual relations on central path z = 1 t φ(s), s = 1 t φ (z), 1 t 2 φ(s) = t 2 φ (z) 1 Path-following methods 17-22

26 Properties of tangent direction from 2nd and 3rd block in (1): s T t z t = 0 from first block in (1) and 2 φ(s)s = φ(s): s T z t + z T s t = s T z hence, gap in tangent direction is (s + α s t ) T (z + α z t ) = (1 α)s T z from first block in (1) s t 2 s = s T t 2 φ(s) s t = tz T s t similarly, from first block in (2) z t 2 z = z T t 2 φ (z) z t = ts T z t Path-following methods 17-23

27 Predictor-corrector method with exact centering Simplifying assumptions: exact centering, a central point x (t 0 ) is given Algorithm: define tolerance ɛ (0, 1), parameter β > 0, and set t := t 0, (x, s, z) := (x (t 0 ), s (t 0 ), z (t 0 )) repeat until θ/t ɛ: compute tangent direction ( x t, s t, z t ) at (x, s, z) set (x, s, z) := (x, s, z) + α( x t, s t, z t ) with α determined from Ψ(x + α x t, s + α s t, z + α z t ) = β set t := θ/(s T z) and compute (x, s, z) := (x (t), s (t), z (t)) Path-following methods 17-24

28 Iteration complexity Potential function in tangent direction (proof on next page) Ψ(x + α x t, s + α s t, z + α s t ) ω (α θ) = α θ log(1 α θ) Lower bound on predictor step length: since ω is an increasing function α γ/ θ where ω (γ) = β Reduction in duality gap after one predictor/corrector cycle t/t + = 1 α 1 γ/ θ exp( γ/ θ) Cumulative Newton iterations: t (k) θ/ɛ after O ( ( )) θ θ log t 0 ɛ Newton iterations Path-following methods 17-25

29 Proof of upper bound on Ψ (with s + = s + α s t, z + = z + α z t ) bounds on φ(s + ) and φ (z + ): from the inequality on page 16-8, φ(s + ) φ(s) α φ(s) T s t + ω (α s t s ) = αtz T s t + ω (α s t s ) φ (z + ) φ (z) α φ(z) T z t + ω (α z t z ) = αts T z t + ω (α z t z ) add the inequalities and use properties on page φ(s + ) φ(s) + φ (z + ) φ (z) αθ + ω (α s t s ) + ω (α z t z ) αθ + ω (α ( 1/2) s t 2 s + z t 2 z) = αθ + ω (α θ) since (s + ) T z + = (1 α)s T z, Ψ(x +, s +, z + ) θ log(1 α) + αθ + ω (α θ) ω (α θ) Path-following methods 17-26

30 References Yu. Nesterov, Introductory Lectures on Convex Optimization. A Basic Course (2004), chapter 4. Yu. Nesterov, Towards nonsymmetric conic optimization, Optimization Methods and Software (2012). Path-following methods 17-27