( ) ( ) OPTIS 2006 p.1/17
,, etc. ( ) [Marti & Stoeckel 04] [Lloyd Smith, Chuang & Munro 90], [Staat & Heitzer 03] worst-case detection [Elishakoff, Haftka & Fang 94] 2 [Cheng et al. 02], [Craig et al. 03] OPTIS 2006 p.2/17
,, etc. ( ) [Marti & Stoeckel 04] [Lloyd Smith, Chuang & Munro 90], [Staat & Heitzer 03] ( ) worst-case detection [Elishakoff, Haftka & Fang 94] 2 [Cheng et al. 02], [Craig et al. 03] & ( ) ( ) OPTIS 2006 p.2/17
ζ ζ ζ λ f g λ' f g+ζ OPTIS 2006 p.3/17
0-1 min c T x s.t. Ax b, R n x 0, x i {0, 1}, i =1,...,p : x 1,...,x n 0-1 0 x i 1 LP : A, b, c p<n OPTIS 2006 p.4/17
0-1 mixed 0-1 program 1 [Lang & Doig 60], [Held & Karp 70,71], etc. Gomory s cut [Gomory 58] lift-and-projection cut [Lov asz & Schrijver 91], [Sherali & Adams 90] (disjunctive) [Balas 74, 98] OPTIS 2006 p.5/17
0-1 mixed 0-1 program 1 [Lang & Doig 60], [Held & Karp 70,71], etc. Gomory s cut [Gomory 58] lift-and-projection cut [Lov asz & Schrijver 91], [Sherali & Adams 90] (disjunctive) [Balas 74, 98] ( ) ( ) CPLEX, Xpress-MP, bc-opt, etc. OPTIS 2006 p.5/17
unknown-but-bounded OPTIS 2006 p.6/17
unknown-but-bounded convex model [Ben-Haim & Elishakoff 90] interval analysis [Alefeld & Mayer 00], [Chen et al. 02], etc. LP, QP, SDP [Ben-Tal & Nemirovski 02] [Ben-Haim 01, 06] OPTIS 2006 p.6/17
( ) λ (f D ) = max{λ :(λ, q) Q(f D )} (LP) λ,q ( ) OPTIS 2006 p.7/17
( ) λ (f D ) = max{λ :(λ, q) Q(f D )} (LP) λ,q (λ, q) Q(f D ) Bq = f D + λf R ( ) q i q y i 0 ( ) q f R, q y λ (f D ) ( ) OPTIS 2006 p.7/17
( ) λ (f D ) = max{λ :(λ, q) Q(f D )} (LP) λ,q (λ, q) Q(f D ) Bq = f D + λf R ( ) q i q y i 0 ( ) q f R, q y λ (f D ) ( ) : f D f D OPTIS 2006 p.7/17
f D = f D + T ζ (LP) f D T : : ζ ζ Z(α) OPTIS 2006 p.8/17
f D = f D + T ζ (LP) f D T : : ζ α ζ j, j =1,...,m α : f D F D (α) α OPTIS 2006 p.8/17
( ) f D λ (f D ) f λ min (α)= min {λ (f D ):f D F D (α)} ( ) D : : ( ), λ min (α) = min α Tu 1 f T Du +(q y ) T z s.t. f T R u =1 z i b T i u u,z OPTIS 2006 p.9/17
( ) f D λ (f D ) f λ min (α)= min {λ (f D ):f D F D (α)} ( ) D : : ( ) ( ) ( ) ( ) OPTIS 2006 p.9/17
0-1 λ min (α)= min α1 T γ f T Du +(q y ) T z s.t. f T R u =1, z i b T i u γ T T u M(1 τ ) γ T T u Mτ τ {0, 1} m u,z,fl,fi ( ) ( ) ( ) f D OPTIS 2006 p.10/17
0-1 λ min (α)= min α1 T γ f T Du +(q y ) T z s.t. f T R u =1, z i b T i u γ T T u M(1 τ ) γ T T u Mτ τ {0, 1} m ( ) u,z,fl,fi 0-1 0 τ 1 LP τ j := 0 or τ j := 1 OPTIS 2006 p.10/17
LP OPTIS 2006 p.11/17
OPTIS 2006 p.11/17
OPTIS 2006 p.11/17
K : LP ( ) ( ) : MIP, LP ( ) : P j (K) = cl conv {x K τ j {0, 1}} P j (K) OPTIS 2006 p.12/17
α T x β LP ˆx x P j (K) ˆx OPTIS 2006 p.13/17
α T x β LP ˆx x P j (K) ˆx min s.t. β α T ˆx (α,β) Pj (K) α I 1 (LP) P j (K) : P j (K) I = {i {1,...,m} ˆx i =0} OPTIS 2006 p.13/17
( ) LP j 1 LP LP OPTIS 2006 p.14/17
(28, 42 ) f D y fd (e) f D (f) f R (d) H f R (c) H H (a) (b) x W W W OPTIS 2006 p.15/17
(28, 42 ) : λ ( f D )=48.4 reference disturbance load OPTIS 2006 p.15/17
(28, 42 ) : λ min (α 1 )=37.0 α 1 =40.0 kn reference disturbance load OPTIS 2006 p.15/17
(28, 42 ) : λ min (α 2 )=44.4 α 2 =20.0 kn reference disturbance load OPTIS 2006 p.15/17
(28, 42 ) : λ (βf D1) --: λ (2βf D2),, : α 50 48 (A) 46 (B) λ 44 42 (D) 40 (E) 38 (C) 36 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 β OPTIS 2006 p.15/17
(40, 68 ) f D f R y (h) fd f D fd (i) (j) (k) H f R (g) f R (f) H H H (a) (b) (c) (d) (e) W W W W x OPTIS 2006 p.16/17
(40, 68 ) : λ ( f D )=14.3 reference disturbance load{ OPTIS 2006 p.16/17
(40, 68 ) : λ min (α 1 )=7.73 :34 B&B : 9 LPs α 1 =40.0 kn OPTIS 2006 p.16/17
(40, 68 ) 20 18 16 14 14.2650 λ 12 10 8 7.7296 6 0 500 1000 1500 2000 2500 3000 sample number OPTIS 2006 p.16/17
( ) ( ) 0-1 deepest cut LP OPTIS 2006 p.17/17