From Evolutionary Biology to Interior-Point Methods

Transcription

1 From Evolutionary Biology to Interior-Point Methods Paul Tseng Mathematics, University of Washington Seattle WCOM, UBC Kelowna October 27, 2007 Joint work with Immanuel Bomze and Werner Schachinger (Univ. Vienna) Abstract This is a talk given at WCOM Kelowna, October 2007.

2 EVOLUTIONARY BIOLOGY TO IP METHODS 1 Talk Outline Affine-Scaling Method for Linearly Constrained Optimization Replicator Dynamics in Evolutionary Biology 1st-Order Interior-Point Methods for Linearly Constrained Optimization Convergence & Convergence Rate Numerical Tests Conclusions & Open Questions

3 EVOLUTIONARY BIOLOGY TO IP METHODS 2 Linearly Constrained Smooth Optimization (P) max x f(x) s.t. Ax = b, x 0 f : R n R is continuously diff., A R m n of rank m, b R m Seek a stationary pt: a feasible x (Ax = b, x 0) with x f(x) A λ 0 for some λ R m. Primal Nondegeneracy: For any feasible x, the columns of A corresponding to {j x j 0} have rank m.

4 EVOLUTIONARY BIOLOGY TO IP METHODS 3 Affine-Scaling Method for (P) Given Ax 0 = b, x 0 > 0, generate for t = 0, 1,... d t = { arg max f(x t ) d Ad = 0, (X t ) 1 d 2 1 } x t+1 = d x t + α t d t > 0 with X t = diag(x t ), α t > 0 suitably chosen Dikin 67, 72

5 EVOLUTIONARY BIOLOGY TO IP METHODS 4 The AS method is simple, fairly efficient in practice, but difficult to analyze. Barnes, Bonnans, Dikin, Gonzaga, Mascarenhas, Monma, Monteiro, Roos, Saigal, J. Sun, T, Tsuchiya, Vanderbei, Ye,... When f is linear, {x t } and {f(x t )} converge linearly; Luo, T 92 the limit x attains maximum if α t is not too large ; Tsuchiya 91, Tsuchiya & Muramatsu 95 x may fail to attain maximum if α t is too large. Mascarenhas 97, Terlaky & Tsuchiya 99 When f is concave or convex and assuming primal nondeg, every cluster pt of {x t } is a stationary pt of (P). Gonzaga & Carlos 90, Monteiro & Wang 98 What about general f? And convergence rate?

6 EVOLUTIONARY BIOLOGY TO IP METHODS 5 Replicator Dynamics in Evolutionary Biology x t j = fraction of jth genotype in pop. at time t, j = 1,..., n Initially, x 0 j > 0 and j x0 j = 1. xt = (x t j )n j=1 evolves according to x t+1 = Xt Qx t (x t ) Qxt, t = 0, 1,... with adaptation coefficients Q ii > 0 and Q ij 0 for all i, j..., Haldane 32,..., Moran 62,... {x t } converges and its limit x satisfies x j ((Q x) j λ) = 0 for all j, with λ = max j (Q x) j ; convergence rate is linear iff (Q x) j < λ whenever x j = 0; otherwise x x t = O(1/ t). Lyubich et al. 80

7 EVOLUTIONARY BIOLOGY TO IP METHODS 6 Connecting RD with AS Rewrite RD iteration as with g t = Qx t, e = (1,..., 1). x t+1 x t = Xt [g t e (x t ) g t ] (x t ) g t Thus x t+1 = x t + α t d t with d t = X t r(x t ) where α t = 1/(x t ) g t and r(x) = f(x) e x f(x)

8 EVOLUTIONARY BIOLOGY TO IP METHODS 7 Solve for d t in AS iteration when A = e yields ( d t (X t ) 2 g t e e (X t ) 2 g t ) with g t = f(x t ), e = (1,..., 1). x t 2 2 Thus where α t > 0 and x t+1 = x t + α t d t with d t = (X t ) 2 r(x t ) r(x) = f(x) e e X 2 f(x) x 2

9 EVOLUTIONARY BIOLOGY TO IP METHODS 8 1st-Order Interior-Point Methods for (P) Given Ax 0 = b, x 0 > 0, generate for t = 0, 1,... x t+1 = x t + α t d t with d t = (X t ) 2γ r γ (x t ) where γ > 0, r γ (x) = f(x) A (AX 2γ A ) 1 AX 2γ f(x) and α t is the largest α {α0(β) t k } k=0,1,... satisfying f(x t + αd t ) f(x t ) + σα(g t ) d t, Armijo-type inexact LS where 0 < β, σ < 1, g t = f(x t ), and 0 < α t 0 < { if d t 0; else. 1 min j d t j /xt j

10 EVOLUTIONARY BIOLOGY TO IP METHODS 9 γ = 1/2 = RD γ = 1 = AS This method is simple, suited for large problems (n 10000). Convergence of {x t }? Convergence rate of {f(x t )}, {x t }? Choosing γ?

11 EVOLUTIONARY BIOLOGY TO IP METHODS 10 Convergence Results: Assume {x feasible f(x) f(x 0 )} is bounded. Then x t > 0, {f(x t )}, and {x t }, {d t } are bounded. (a) Assume primal nondeg, f is concave or convex, and we choose inf t α t 0 > 0, sup t α t 0 <. Then every cluster pt of {x t } is a stationary pt of (P). (b) Assume f is quadratic and we choose inf t α0 t > 0. Then ( υ f(x t ) = O 1/t 1/ max{γ,2γ 1}), with υ = lim t f(x t ). Assume in addition we choose γ < 1. Then {x t } converges and its limit x satisfies ) x x t = O (1/t 1 γ 2γ. Under primal nondeg, x is a stationary pt of (P). Moreover, if γ 1 2 and x r γ ( x) > 0, then {f(x t )} converges Q-linearly and { x x t } converges R-linearly. If instead sup t α t 0 <, γ 1 2 and x r γ( x) 0, then { x x t } cannot converge linearly. Thus, γ < 1 seems preferrable.

12 EVOLUTIONARY BIOLOGY TO IP METHODS 11 Numerical Tests: Implement 1st-order IP method in Matlab. For Armijo LS, use β =.5, σ =.1, { }} α0 t = min 0.95αfeas, t max {10 5, αt 1 β 2, αfeas t 1 = min j (d t j /xt j ), with α 1 =. Numerical tests on max x f(x) s.t. e x = 1, x 0 with f from Moré-Garbow-Hillstrom set (least square), and n = Initial x 0 = e/n. Terminate when resid := min{x t, r γ (x t )} tol.

13 EVOLUTIONARY BIOLOGY TO IP METHODS 12 f(x) γ #iter #f-eval cpu (sec) obj resid BAL BT DBV EPS ER LR VD Table 1: Performance of 1st-order IP method. Quit due to Armijo ascent condition not met when α < 10 20

14 EVOLUTIONARY BIOLOGY TO IP METHODS 13 Conclusions & Open questions 1. Theory and practice suggest γ < 1 is preferrable to γ The method and its analysis readily extends to 0 x u by replacing X t with min{x t, U X t }. 3. Convergence of {x t } when γ 1 or when f is not quadratic? 4. Linear convergence of {f(x t )} and {x t } when γ > 1 2 quadratic? or when f is not 5. Convergence of {x t } for 2nd-order AS method?

15 EVOLUTIONARY BIOLOGY TO IP METHODS 14 Convergence Proof Ideas (a) f(x t+1 ) f(x t ) σα t η t 2 with η t = (X t ) γ r(x t ). (b) If f is quadratic, use linearity of KT condition to show t := υ f(x t ) = O ( η t min{ 2 1+γ, 1 γ }). (c) If f is quadratic and γ < 1, then O x t+1 x t = α t (X t ) γ η t = O( η t ) = O ( t ) t 1+γ 2 dt = O t+1 ) (( t ) 1 γ 2 ( t+1 ) 1 γ 2 ( η t 2 ) η t = O ( ) t t+1 ( t ) 1+γ 2 = (d) Under primal nondegeneracy, r γ is continuous on the feasible set.