y y i LB k+1 f(x, y k+1 ) Ax + By k+1 b,

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1 a b b a b

3 min x,y f(x, y) g j (x, y) 0 j = 1,... l, Ax + By b, x R n, y Z m. f, g 1,..., g l : R n R m R y x

4 { (x i, y i ) } k i=0 k y k+1 min x,y,µ µ [ ] f(x i, y i ) + f(x i, y i ) T x x i y y i g j (x i, y i ) + g j (x i, y i ) T [ x x i y y i Ax + By b, µ i = 1,..., k, ] 0 i = 1,... k, j A i, x R n, y Z m, µ R. A i (x i, y i ) LB k+1 y k+1 x min x f(x, y k+1 ) g j (x, y k+1 ) 0 j = 1,... l, Ax + By k+1 b, x R n. x k+1 UB k+1 x y l min x,r r g j (x, y k+1 ) r j = 1,... l, Ax + By k+1 b, x R n, r R +.

5 x k+1 (x k+1, y k+1 ) [ ] f(x k+1, y k+1 ) + f(x k+1, y k+1 ) T x x k+1 y y k+1 µ, g j (x k+1, y k+1 ) + g j (x k+1, y k+1 ) T [ x x k+1 y y k+1 ] 0 j A k+1. y k+1 6x y 0.3(x 8) (y 6) e 2x y /x + 1/y x 0.5 y x 5y 1 1 x 20, 1 y 20, x R, y Z. x 0 = 5.29 y 0 = 3 (x 0, y 0 )

6 ϵ 0 x, ỹ x, ỹ k = 1 UB 0 = inf LB 0 = inf UB k 1 LB k 1 ϵ y k LB k y k x k x k UB k = UB k 1 x k, y k x k, y k UB k = min{f(x k, y k ), UB k 1 } k = k + 1 x, ȳ f( x, ȳ) x, ȳ LB 1

7 f LB 1 f f( x, ȳ) k ˆ f k = (1 α)f( x, ȳ) + αlbk, α (0, 1] x, ȳ LB k α α f ˆ k y k+1 x, ȳ f ˆ k min x,y,µ x x y ȳ 2 µ ˆ f k f(x i, y i ) + f(x i, y i ) T [ x x i y y i g j (x i, y i ) + g j (x i, y i ) T [ x x i y y i Ax + By b, x R n, y Z m, µ R, ] µ i = 1,..., k, ] 0 i = 1,... k, j A i,

8 (x 0, y 0 ) y k+1 f ˆ k fˆ k

9 ϵ 0 α (0, 1] x, ȳ x, ȳ k = 1 LB 0 = inf f( x, ȳ) LB k 1 ϵ LB k f ˆ k y k y k x k x k x k, y k x k, y k f(x k, y k ) f( x, ȳ) x, ȳ = x k, y k k = k + 1 x, ȳ α = 0.4 LB 4 = f( x, ȳ) x x 2 y ȳ r k, r k

10 µ f ˆ k x, y, µ r k = x 2 x y ȳ r k x, y, µ r k > x 2 x y ȳ. µ µ µ f ˆ k x, y, µ x, y, µ

11 L : R n R m R l R L(x, y, λ) = f(x, y) + l λ j g j (x, y), λ j 0 j f, g 1,..., g j x, y [ ] L( x, ȳ, λ) + x,y L( x, ȳ, λ) T x + 1 y 2 j=1 [ ] T x 2 y x,yl( x, ȳ, λ) [ ] x, y

12 x,y L x, y 2 x,y x = x x y = y ȳ 2 x,y λ 0 x, ȳ ϵ ϵ ϵ µ f( x, ȳ) ϵ [ ] f(x i, y i ) + f(x i, y i ) T x x i y y i µ i = 1,..., k, x, ȳ ϵ x, ȳ λ f ˆ k µ ˆ f k f(x i, y i ) + f(x i, y i ) T [ x x i y y i ] µ i = 1,..., k. f ˆ k ˆ f k

13 y k+1 min x,y,µ [ ] x,y L( x, ȳ, λ) T x + 1 y 2 µ ˆ f k f(x i, y i ) + f(x i, y i ) T [ x x i y y i g j (x i, y i ) + g j (x i, y i ) T [ x x i y y i Ax + By b, [ ] T x 2 y x,yl( x, ȳ, λ) [ ] x y ] µ i = 1,..., k ] 0 i = 1,... k, j A i, x R n, y Z m, µ R, x = x x y = y ȳ x, ȳ λ x λ 2 x,y α α = 1 α

14 ϵ 0 α ]0, 1] x, ȳ λ x, ȳ k = 1 LB 0 = inf f( x, ȳ) LB k 1 ϵ LB k f ˆ k y k y k x k λ k x k x k, y k x k, y k f(x k, y k ) f( x, ȳ) x, ȳ, λ = x k, y k, λ k k = k + 1 x, ȳ (x 0, y 0 ) α = 0.5 LB 3 = f( x, ȳ)

15 µ f ˆ k

16 ϕ(x, y) [ ] ϕ(x, y) ϕ(x 0, y 0 ) + ϕ(x 0, y 0 ) T x x 0 y y 0 (x, y), (x 0, y 0 ) D ϕ, D ϕ y k y k f ˆ k f ˆ k < f( x, ȳ) ˆx i, ŷ i ˆ f k < f( x, ȳ) f(ˆxi, ŷ i ) i. ˆx i, ŷ i [ ] f(ˆx i, ŷ i ) + f(ˆx i, ŷ i ) T x ˆx i y ŷ i µ. µ ˆµ i ˆ f k < f( x, ȳ) ˆµi i.

17 µ f ˆ k [ ] f(ˆx i, ŷ i ) T x ˆx i y ŷ i < 0 i. x, ỹ {ˆx i, ŷ i } x x x x f( x, ỹ) + g j ( x, ỹ) 0 l λ j x g j ( x, ỹ) + A T γ = 0 j=1 j = 1,..., l A x + Bỹ b λ, γ 0 λ j g j ( x, ỹ) = 0 j = 1,..., l (A x + Bỹ b) γ = 0, x x γ x, ỹ [ ] g j ( x, ỹ) + g j ( x, ỹ) T x x 0 j λ y ỹ j 0. g j ( x, ỹ) = 0 x λ j x g j ( x, ỹ) T x 0 j = 1,..., l. x γ γ T A x 0. x f( x, ỹ) T x < 0. l x f( x, ỹ) T x + λ j x g j ( x, ỹ) T x + γ T A x < 0. j=1

18 x l x f( x, ỹ) T x + λ j x g j ( x, ỹ) T x + γ T A x = 0, j=1 x y

19 n nonlin n + m > 0.5, n nonlin m + n

20 α 10 9 i 2 x,yl(i, i) := 2 x,yl(i, i) + λ min, λ min ϵ

21 ϵ rel f( x, ȳ) LB ϵ f( x, ȳ) LB f( x, ȳ) ϵ rel LB ϵ = 10 5 ϵ rel = s

23 τ τ τ 3 τ iter 1.5

24 τ t τ iter

25 α = 0.5 α α

26 α

29 m int m bin n n + m n nonlin n+m n nonlin n+m m int m bin n n + m

32 LB UB / LB LB UB

Nonlinear Programming

Nonlinear Programming Kees Roos e-mail: C.Roos@ewi.tudelft.nl URL: http://www.isa.ewi.tudelft.nl/ roos LNMB Course De Uithof, Utrecht February 6 - May 8, A.D. 2006 Optimization Group 1 Outline for week