12. Interior-point methods
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1 12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity analysis via self-concordance generalized inequalities 12 1 Inequality constrained minimization f (x) subject to f i (x), i = 1,..., m (1) f i convex, twice continuously differentiable A R p n with rank A = p we assume p is finite and attained we assume problem is strictly feasible: there exists x with x dom f, f i ( x) <, i = 1,..., m, A x = b hence, strong duality holds and dual optimum is attained Interior-point methods 12 2
2 Examples LP, QP, QCQP, GP entropy maximization with linear inequality constraints subject to n x i log x i F x g with dom f = R n ++ differentiability may require reformulating the problem, e.g., piecewise-linear minimization or l -norm approximation via LP SDPs and SOCPs are better handled as problems with generalized inequalities (see later) Interior-point methods 12 3 Logarithmic barrier reformulation of (1) via indicator function: subject to f (x) + m I (f i (x)) where I (u) = if u, I (u) = otherwise (indicator function of R ) approximation via logarithmic barrier f (x) (1/t) m log( f i(x)) subject to an equality constrained problem for t >, (1/t) log( u) is a smooth approximation of I approximation improves as t u Interior-point methods 12 4
3 logarithmic barrier function φ(x) = log( f i (x)), dom φ = {x f 1 (x) <,..., f m (x) < } convex (follows from composition rules) twice continuously differentiable, with derivatives φ(x) = 2 φ(x) = 1 f i (x) f i(x) 1 f i (x) 2 f i(x) f i (x) T + 1 f i (x) 2 f i (x) Interior-point methods 12 5 Central path for t >, define x (t) as the solution of subject to tf (x) + φ(x) (for now, assume x (t) exists and is unique for each t > ) central path is {x (t) t > } example: central path for an LP c T x subject to a T i x b i, i = 1,..., 6 hyperplane c T x = c T x (t) is tangent to level curve of φ through x (t) c x x (1) Interior-point methods 12 6
4 Dual points on central path x = x (t) if there exists a w such that t f (x) + 1 f i (x) f i(x) + A T w =, therefore, x (t) s the Lagrangian L(x, λ (t), ν (t)) = f (x) + λ i (t)f i (x) + ν (t) T (Ax b) where we define λ i (t) = 1/( tf i(x (t)) and ν (t) = w/t this confirms the intuitive idea that f (x (t)) p if t : p g(λ (t), ν (t)) = L(x (t), λ (t), ν (t)) = f (x (t)) m/t Interior-point methods 12 7 Interpretation via KKT conditions x = x (t), λ = λ (t), ν = ν (t) satisfy 1. primal constraints: f i (x), i = 1,..., m, 2. dual constraints: λ 3. approximate complementary slackness: λ i f i (x) = 1/t, i = 1,..., m 4. gradient of Lagrangian with respect to x vanishes: f (x) + λ i f i (x) + A T ν = difference with KKT is that condition 3 replaces λ i f i (x) = Interior-point methods 12 8
5 Force field interpretation centering problem (for problem with no equality constraints) tf (x) m log( f i(x)) force field interpretation tf (x) is potential of force field F (x) = t f (x) log( f i (x)) is potential of force field F i (x) = (1/f i (x)) f i (x) the forces balance at x (t): F (x (t)) + F i (x (t)) = Interior-point methods 12 9 example c T x subject to a T i x b i, i = 1,..., m objective force field is constant: F (x) = tc constraint force field decays as inverse distance to constraint hyperplane: F i (x) = where H i = {x a T i x = b i} a i b i a T i x, F i(x) 2 = 1 dist(x, H i ) c 3c t = 1 t = 3 Interior-point methods 12 1
6 Barrier method given strictly feasible x, t := t () >, µ > 1, tolerance ɛ >. repeat 1. Centering step. Compute x (t) by minimizing tf + φ, subject to. 2. Update. x := x (t). 3. Stopping criterion. quit if m/t < ɛ. 4. Increase t. t := µt. terminates with f (x) p ɛ (stopping criterion follows from f (x (t)) p m/t) centering usually done using Newton s method, starting at current x choice of µ involves a trade-off: large µ means fewer outer iterations, more inner (Newton) iterations; typical values: µ = 1 2 several heuristics for choice of t () Interior-point methods Convergence analysis number of outer (centering) iterations: exactly log(m/(ɛt () )) log µ plus the initial centering step (to compute x (t () )) centering problem tf (x) + φ(x) see convergence analysis of Newton s method tf + φ must have closed sublevel sets for t t () classical analysis requires strong convexity, Lipschitz condition analysis via self-concordance requires self-concordance of tf + φ Interior-point methods 12 12
7 duality gap Examples inequality form LP (m = 1 inequalities, n = 5 variables) µ = 5 µ = 15 µ = µ starts with x on central path (t () = 1, duality gap 1) terminates when t = 1 8 (gap 1 6 ) centering uses Newton s method with backtracking total number of not very sensitive for µ 1 Interior-point methods geometric program (m = 1 inequalities and n = 5 variables) subject to ( 5 ) log k=1 exp(at k x + b k) ( 5 ) log k=1 exp(at ik x + b ik), i = 1,..., m 1 2 duality gap µ = 15 µ = 5 µ = Interior-point methods 12 14
8 family of standard LPs (A R m 2m ) c T x subject to, x m = 1,..., 1; for each m, solve 1 randomly generated instances m number of iterations grows very slowly as m ranges over a 1 : 1 ratio Interior-point methods Feasibility and phase I methods feasibility problem: find x such that f i (x), i = 1,..., m, (2) phase I: computes strictly feasible starting point for barrier method basic phase I method (over x, s) s subject to f i (x) s, i = 1,..., m (3) if x, s feasible, with s <, then x is strictly feasible for (2) if optimal value p of (3) is positive, then problem (2) is infeasible if p = and attained, then problem (2) is feasible (but not strictly); if p = and not attained, then problem (2) is infeasible Interior-point methods 12 16
9 sum of infeasibilities phase I method 1 T s b i a T i x max subject to s, f i (x) s i, i = 1,..., m for infeasible problems, produces a solution that satisfies many more inequalities than basic phase I method example (infeasible set of 1 linear inequalities in 5 variables) 6 6 number 4 2 number b i a T i x max b i a T i x sum left: basic γ phase I solution; satisfies 39 inequalities γ right: sum of infeasibilities Newton phase I iterations solution; satisfies 79 solutions Infeasible Feasible 1 Interior-point methods example: family of linear inequalities Ax b + γ b 1 data chosen to be strictly feasible for γ >, infeasible for γ 2 use basic phase I, terminate when s < or dual objective is positive Infeasible γ Infeasible Feasible Feasible γ γ 2 1 number of iterations roughly proportional to log(1/ γ ) Interior-point methods 12 18
10 Complexity analysis via self-concordance same assumptions as on page 12 2, plus: sublevel sets (of f, on the feasible set) are bounded tf + φ is self-concordant with closed sublevel sets second condition holds for LP, QP, QCQP may require reformulating the problem, e.g., subject to n x i log x i F x g n x i log x i subject to F x g, x needed for complexity analysis; barrier method works even when self-concordance assumption does not apply Interior-point methods per centering step: from self-concordance theory # µtf (x) + φ(x) µtf (x + ) φ(x + ) γ bound on effort 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)): + c µtf (x) + φ(x) µtf (x + ) φ(x + ) = µtf (x) µtf (x + ) + log( µtλ i f i (x + )) m log µ µtf (x) µtf (x + ) µt λ i f i (x + ) m m log µ µtf (x) µtg(λ, ν) m m log µ = m(µ 1 log µ) Interior-point methods 12 2
11 total number of (excluding first centering step) log(m/(t () ( ɛ)) m(µ 1 log µ) # N = log µ γ ) + c N figure shows N for typical values of γ, c, m = 1, m t () ɛ = µ confirms trade-off in choice of µ in practice, #iterations is in the tens; not very sensitive for µ 1 Interior-point methods polynomial-time complexity of barrier method for µ = 1 + 1/ m: ( ( )) m m/t () N = O log ɛ number of 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 = 1,..., 2) Interior-point methods 12 22
12 Generalized inequalities f (x) subject to f i (x) Ki, i = 1,..., m f convex, f i : R n R k i, i = 1,..., m, convex with respect to proper cones K i R k i f i twice continuously differentiable A R p 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 Interior-point methods Generalized logarithm for proper cone ψ : R q R is generalized logarithm for proper cone K R q if: dom ψ = int K and 2 ψ(y) for y K ψ(sy) = ψ(y) + θ log s for y K, s > (θ is the degree of ψ) examples nonnegative orthant K = R n +: ψ(y) = n log y i, with degree θ = n positive semidefinite cone K = S n +: ψ(y ) = log det Y (θ = n) second-order cone K = {y R n+1 (y y 2 n) 1/2 y n+1 }: ψ(y) = log(y 2 n+1 y 2 1 y 2 n) (θ = 2) Interior-point methods 12 24
13 properties (without proof): for y K, ψ(y) K, y T ψ(y) = θ nonnegative orthant R n +: ψ(y) = n log y i ψ(y) = (1/y 1,..., 1/y n ), y T ψ(y) = n positive semidefinite cone S n +: ψ(y ) = log det Y ψ(y ) = Y 1, tr(y ψ(y )) = n second-order cone K = {y R n+1 (y yn) 2 1/2 y n+1 }: y 1 2 ψ(y) = yn+1 2 y2 1. y2 n y n, yt ψ(y) = 2 y n+1 Interior-point methods Logarithmic barrier and central path logarithmic barrier for f 1 (x) K1,..., f m (x) Km : φ(x) = ψ i ( f i (x)), dom φ = {x f i (x) Ki, i = 1,..., m} ψ i is generalized logarithm for K i, with degree θ i φ is convex, twice continuously differentiable central path: {x (t) t > } where x (t) solves subject to tf (x) + φ(x) Interior-point methods 12 26
14 Dual points on central path x = x (t) if there exists w R p, t f (x) + Df i (x) T ψ i ( f i (x)) + A T w = (Df i (x) R ki n is derivative matrix of f i ) therefore, x (t) s Lagrangian L(x, λ (t), ν (t)), where λ i (t) = 1 t ψ i( f i (x (t))), ν (t) = w t from properties of ψ i : λ i (t) K i, with duality gap f (x (t)) g(λ (t), ν (t)) = (1/t) θ i Interior-point methods example: semidefinite programming (with F i S p ) subject to c T x F (x) = n x if i + G logarithmic barrier: φ(x) = log det( F (x) 1 ) central path: x (t) s tc T x log det( F (x)); hence tc i tr(f i F (x (t)) 1 ) =, i = 1,..., n dual point on central path: Z (t) = (1/t)F (x (t)) 1 is feasible for maximize tr(gz) subject to tr(f i Z) + c i =, i = 1,..., n Z duality gap on central path: c T x (t) tr(gz (t)) = p/t Interior-point methods 12 28
15 Barrier method given strictly feasible x, t := t () >, µ > 1, tolerance ɛ >. repeat 1. Centering step. Compute x (t) by minimizing tf + φ, subject to. 2. Update. x := x (t). 3. Stopping criterion. quit if ( P i θ i)/t < ɛ. 4. Increase t. t := µt. only difference is duality gap m/t on central path is replaced by i θ i/t number of outer iterations: log(( i θ i)/(ɛt () )) log µ complexity analysis via self-concordance applies to SDP, SOCP Interior-point methods duality gap 1 2 Examples second-order cone program (5 variables, 5 SOC constraints in R 6 ) duality gap duality gap µ = 5 µ = 2 µ = µ semidefinite program (1 variables, LMI constraint in S 1 ) µ = 15 µ = 5 µ = µ Interior-point methods 12 3
16 family of SDPs (A S n, x R n ) 1 T x subject to A + diag(x) n = 1,..., 1, for each n solve 1 randomly generated instances n 1 3 Interior-point methods Primal-dual interior-point methods more efficient than barrier method when high accuracy is needed update primal and dual variables at each iteration; no distinction between inner and outer iterations often exhibit superlinear asymptotic convergence search directions can be interpreted as Newton directions for modified KKT conditions can start at infeasible points cost per iteration same as barrier method Interior-point methods 12 32
12. Interior-point methods
12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity
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