Interior-point methods Optimization Geoff Gordon Ryan Tibshirani

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1 Interior-point methods Optimization Geoff Gordon Ryan Tibshirani

2 Analytic center Review force field interpretation Newton s method: y = 1./(Ax+b) A T Y 2 A Δx = A T y Dikin ellipsoid unit ball of Hessian norm for log barrier contained in feasible region Dikin ellipsoid at analytic center: scale up by m, contains feasible region 2

3 Central path: force field Review Newton: A T Y 2 A Δx = A T y tc trading centering v. optimality Affine invariance Constraint form v. penalty form of central path Primal-dual correspondence for central path duality gap m/t t 0 objective t 3

4 Primal-dual constraint form Primal-dual pair: min c T x st Ax + b 0 max b T y st A T y = c y 0 KKT: Ax + b 0 y 0 A T y = c c T x + b T y 0 (primal feasibility) (dual feasibility) (strong duality) or, c T x + b T y λ (relaxed strong duality) 4

5 Analytic center of relaxed KKT Relaxed KKT conditions: Ax + b = s 0 y 0 A T y = c c T x + b T y λ Central path = {analytic centers of relaxed KKT} 5

6 A simple algorithm t := 1, y := 1m, x := 0 n Repeat: Use infeasible-start Newton to find point y on dual central path (and corresponding multipliers x) t := αt (α > 1) After any outer iteration: Multipliers x are primal feasible; gap c T x + b T y = m/t or, recover w/ duality: s = 1./ty x = A\(s b) 6

7 Example ty gap 10 4 m/t 10 2 duality gap m = 50 m = 500 m = Newton iterations 7

8 An algorithm and proof Feasible for KKT conditions: Ax + b = s 0 y 0 A T y = c Optimal for KKT conditions: c T x + b T y 0 or s T y 0 A potential combining feasibility & optimality: p(s,y) = (m+k) ln y T s ln yi ln si [Kojima, Mizuno, Yoshise, Math. Prog., 1991] 8

9 Potential reduction Potential: p(s,y) = (m+k) ln y T s ln yi ln si = k ln y T s + [m ln y T s ln yi ln si] Algorithm strategy: start w/ strictly feasible (x, y, s) update by (Δx, Δy, Δs): reduce p(s,y) by at least δ per iter: 9

10 Potential reduction strategy Upper bound p(s,y) locally with a quadratic will look like Hessian from Newton s method analyze upper bound: reduce by at least δ / iter p(s,y) = (m+k) ln y T s ln yi ln si p 1(s,y) 10

11 Upper bound, cont d p2(s,y) = ln yi ln si ln(x) bound

12 Algorithm: repeat Choose (Δx, Δy, Δs) to minimize p1 + p2 st A T Δy=0 Δs=AΔx Δy T Y -2 Δy + Δs T S -2 Δs (2/3) 2 stronger than box constraint Step along (Δx, Δy, Δs) while keeping y>0, s>0 Claim: can always decrease potential by δ=1/4 per iteration 12

13 Intuition Suppose s = y = 1m _ p1 = p1 + [(m+k)/m] [1 T Δy + 1 T Δs] _ p2 = p2 1 T Δy 1 T Δs + Δy T Δy + Δs T Δs Δp How much decrease is possible? 13

14 The simple case range(s\a) null(a Y)

15 Farther from equilibrium range(s\a) null(a Y)

16 In general 16

17 Bounding g g = (m+k)y s/y T s 1 min g T Δu + g T Δv + Δu T Δu + Δv T Δv s.t. A T YΔu = 0 Δv = S -1 AΔx π(g,g) g T Δu+g T Δv feasible (Δu,Δv) 1 17

18 (g,g) k/ m step size: decrease: Step size 18

19 Algorithm summary Pick parameters k>0, τ>1 and feasible (x, y, s) Repeat until yt s is small enough: choose (Δx, Δy, Δs) to minimize ((m+k)s/y T s 1/y) T Δy + ((m+k)y/y T s 1/s) T Δs + τδy T Y -2 Δy/2 + τδs T S -2 Δs/2 Δs = AΔx A T Δy = 0 Δy T Y -2 Δy+Δs T S -2 Δs f(τ) quadratic w/ linear constrs looks like Newton line search for best step length with s>0, y>0 update (x, y, s) with our direction and step length 19

20 Example Infeasible initializer k= m τ=2 A R

21 Diagnostics 1: step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot

22 Example Same initializer k=.999m τ=1.95 A R

23 Diagnostics 1: step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot : step , mean gap 10^ , pot

24 When is IP useful? Newton: naively cubic in min(n,m) unless we can take advantage of structure, limited to 1000s of variables but structure often present! Convergence rate is on a different level from firstorder methods: ln(1/ϵ) vs. (at best) 1/ ϵ and the latter requires more smoothness so, great if accuracy requirements high / bad condition Intuition from IP/duality can help algorithm design ln(x) sqrt(x) x 10 5

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