[Boyd]_cvxslides

Complexity analysis via self-concordance

same assumptions as on page 12–2, plus:

•sublevel sets (of f0, on the feasible set) are bounded

•tf0 + φ is self-concordant with closed sublevel sets

second condition

•holds for LP, QP, QCQP

•may require reformulating the problem, E.G.,

•needed for complexity analysis; barrier method works even when self-concordance assumption does not apply

Newton iterations per centering step: from self-concordance theory

#Newton iterations ≤ µtf0(x) + φ(x) − µtf0(x+) − φ(x+) + c

γ

•bound on e ort of computing x+ = x (µt) starting at x = x (t)

•γ, c are constants (depend only on Newton algorithm parameters)

•from duality (with λ = λ (t), ν = ν (t)):

µtf0(x) + φ(x) − µtf0(x+) − φ(x+)

Xm

i=1

≤ µtf0(x) − µtg(λ, ν) − m − m log µ = m(µ − 1 − log µ)

total number of Newton iterations (excluding first centering step)

•confirms trade-o in choice of µ

•in practice, #iterations is in the tens; not very sensitive for µ ≥ 10

polynomial-time complexity of barrier method

• for µ = 1 + 1/√m:

N = O √m log m/t(0)

ǫ

•number of Newton iterations for fixed gap reduction is O(√m)

•multiply with cost of one Newton iteration (a polynomial function of problem dimensions), to get bound on number of flops

this choice of µ optimizes worst-case complexity; in practice we choose µ fixed (µ = 10, . . . , 20)

Generalized inequalities

•f0 convex, fi : Rn → Rki, i = 1, . . . , m, convex with respect to proper cones Ki Rki

•fi twice continuously di erentiable

•A Rp×n with rank A = p

•we assume p is finite and attained

•we assume problem is strictly feasible; hence strong duality holds and dual optimum is attained

examples of greatest interest: SOCP, SDP

Generalized logarithm for proper cone

ψ : Rq → R is generalized logarithm for proper cone K Rq if:

• dom ψ = int K and 2ψ(y) 0 for y K 0

• ψ(sy) = ψ(y) + θ log s for y K 0, s > 0 (θ is the degree of ψ)

examples

• second-order cone K = {y Rn+1 | (y12 + · · · + yn2 )1/2 ≤ yn+1}:

•second-order cone K = {y Rn+1 | (y12 + · · · + yn2 )1/2 ≤ yn+1}:

Logarithmic barrier and central path

logarithmic barrier for f1(x) K1 0, . . . , fm(x) Km 0:

Xm

φ(x) = − ψi(−fi(x)), dom φ = {x | fi(x) Ki 0, i = 1, . . . , m}

i=1

•ψi is generalized logarithm for Ki, with degree θi

•φ is convex, twice continuously di erentiable

central path: {x (t) | t > 0} where x (t) solves

Dual points on central path

x = x (t) if there exists w Rp,

Xm

t f0(x) + Dfi(x)T ψi(−fi(x)) + AT w = 0

i=1

(Dfi(x) Rki×n is derivative matrix of fi)

• therefore, x (t) minimizes Lagrangian L(x, λ (t), ν (t)), where

i=1

example: semidefinite programming (with Fi Sp)

•logarithmic barrier: φ(x) = log det(−F (x)−1)

•central path: x (t) minimizes tcT x − log det(−F (x)); hence

tci − tr(FiF (x (t))−1) = 0, i = 1, . . . , n

• dual point on central path: Z (t) = −(1/t)F (x (t))−1 is feasible for

Z0

•duality gap on central path: cT x (t) − tr(GZ (t)) = p/t