Probabilistic Approach to Robust Optimization

HTML
DOWNLOAD

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Probabilistic Approach to Robust Optimization"

Φώτιος Αλεβίζος
6 χρόνια πριν
Προβολές:

1 Probabilistic Approach to Robust Optimization Akiko Takeda Department of Mathematical & Computing Sciences Graduate School of Information Science and Engineering Tokyo Institute of Technology Tokyo , Japan u f(x, u) 0, u ( ) ( ) N N Calafiore & Campi N Keywords: Ben-Tal&Nemirovski 998 Tal&Nemirovski,. 70 Soyster [9] Ben- El Ghaoui&Lebret [4]

2 Ben-Tal&Nemirovski[2] [2] Calafiore &Campi[3] N N N 2 min x X c x s.t. f(x, u) 0, u () f(x, u) x X, minx X max u g(x, u) () t g(x, u) t min x X,t t s.t. g(x, u) t, u () g i (x, u) 0, i =,...,m, f(x, u) =max i=,...,m g i (x, u) 0 () () ()

3 : 2: max u f(x, u) 0 min x X c x s.t. f(x, u) 0, u. (2) =[a 0 ā, a 0 + ā] : (Soyster [9]) ā 0 (2) max a a x = a 0 x + ā x b (3) y a 0 x + ā y b, y x y, y 0 ( ) (2) 2 (3)

4 = {a 0 + Au : u } : (Ben-Tal&Nemirovski [2]) a u (2) max (a 0 + Au) x b u: u = a 0 x + max u: u (A x) u 2 u = A x A x a 0 x + A x b (2) min x c x s.t. a 0 x + A x b [2, 5] 3 ( ) Calafiore&Campi P P u u (C-C ) x V ( x) =P {u : f( x, u) > 0} (K-T ) f max ( x) =max u f( x, u) V ( x) f max ( x) x N () N N (P N ) (P N ) min x X c x s.t. f(x, u i ) 0, i=...,n, V ( x N ) (P N ) x N Calafiore&Campi () x N

5 3: f(x,) V ( x N ) ɛ ɛ η (n x ) N N(ɛ, η) = 2 ɛ log η +2n + 2n ɛ log 2 ɛ (Calafiore&Campi [3]) ɛ (0, ), η (0, ) N>N(ɛ, η) (P N ) x N η V ( x N ) ɛ ɛ ɛ 0 ( N(ɛ, η) (P N ) ) f max ( x N ) 2(Kanamori&Takeda[6])ɛ (0,q(B)), η (0, ) N>N(ɛ, η) (P N ) x N η V ( x N ) ɛ f max ( x N ) q (ɛ) 2 q(δ), 0 δ B, f(x, ) ( 3 ) { } p(δ, x) :=P max f(x, u) δ<f(x,) u 0 q(δ) p(δ, x), 0 δ B, ( x X) (4) () f(x, u) u f(x, u) f(x, v) L u v, x X, u, v (2) ( ) [6]

6 (3) P ( ) q(δ) B [6] q(δ) B L :=max u,v u v [6] q(δ) B [6] 2 q (ɛ) ɛ ɛ (P N ) N 2 ɛ N(ɛ, η) (P N ) 3(Kanamori&Takeda[6]) N (P N ) x N δ (0,B], η (0, ) M( ln η ) ln( q(δ)) ũ,...,ũ M f max ( x N ) } P {f M max ( x N ) < max f( x N, ũ i )+δ i=,...,m η (P N ) x N ũ,...,ũ M M f( x N, ũ i ) () N (P N ) x N 3 max i=,...,m f( x N, ũ i )+δ 0 0 x N () 0 N (P N ) 4 Kouvelis&Yu [7] u minx X f(x, u) 3

7 (Absolute Robustness) min max f(x, u). (5) x X u (Deviation Robustness) (RD) f (u) =minx X f(x, u). (Relative Robustness) f (u) =minx X f(x, u). min max x X u {f(x, u) f (u)}, (6) min max x X u f(x, u) f (u), f (u) f (u) (5) () (6) u (6) f (u) (5) (6) [2] [, ] Deviation Robustness (6) f(x, u) x f (u) u (6) (RD) (RD N ) min x X,t s.t. t f(x, u i ) f (u i ) t, i =,...,N. f(x, u) x u,...,u N f (u i )=min x X f(x, u i)=f(x i, u i ), i =,...,N, (7) f (u i ) (RD N ) (RD N ) (CRP N ) min x X,t t s.t. f(x, u) f(x, u) t, u... f(x, u) f(x N, u) t, u

8 x i f (u i ) i f(x, u) f(x i, u) t, u, u = u i f(x, u i ) f (x i ) t (RD N ) i (RD N ) (CRP N ) (CRP N ) (RD N ) 2 4 ɛ (0,q(B)), η (0, ) N N(ɛ, η). opt(rd) (CRP N ) (RD N ) opt(crp N ) opt(rd N ) η opt(rd) opt(crp N ) opt(rd) opt(rd N ) q (ɛ) Proof (RD N ) ( x N, t N ) 2 (RD N ) η { } max f( xn, u) f (u) t N q (ɛ) (8) u t N (RD N ) max u {f( x N, u) f (u)} (8) min max x X u {f(x, u) f (u)} min max {f(x, u i) f (u i )} q (ɛ) x X i=,...,n opt(rd) opt(rd N ) q (ɛ) opt(rd N ) opt(crp N ) N (CRP N ) (x N,t N ) η (0, ) δ ln η (0,B] M ln( q(δ)) ũ,...,ũ M β M = max i=,...,m {f(x N, ũ i) f (ũ i ) t N } η opt(rd) opt(crp N ) <β M + δ (9)

9 : (CRP N ) (CRP N ) (CRP N ) no? yes : 4: 5 4 x N ζ + δ t N η N = N 0 (CRP N ) (x N,t N ) V = {i : f(x N, ũ i ) f (ũ i ) t N > 0} N>N 0 (CRP N ) 5 μ x r max f x X x Qx, Q r f 0 Krishnamurthy[8] μ l μ = μ 0 + u j μ j, u := {u : u } j= min x X max u {f (u) f(x, u)} (0)

10 .2. proposed relaxation (CRP N ) robust deviation (RD).2. sampled relaxation (RD N ) robust deviation (RD) Optimal Value Optimal Value Num. of Samples (N) Num. of Samples (N) 0 3 5: (CRP N ) (RD N ) 00 proposed relaxation (CRP N ) sampled relaxation (RD N ) Average Comp. Time [sec] Num. of Samples (N) 0 3 6: (CRP N )and(rd N ) X := {x : x 0, e x =}, f(x, u) = (μ 0 + l j= u jμ j r f e) x, and f (u) =max x Qx x X (μ 0 + l j= u jμ j r f e) x. x Qx f (u) f(x, u) (RD N ) (CRP N ) f (u i ) f(x, u i ) t (i =,...,N) f(x i, u) f(x, u) t, u (i =,...,N) [8] (RD N ) (RD N ) f (u i ) i =,...,N (RD N ) Takeda,Taguchi-Tanaka[0] (0) (CRP N ) f (u i ) i =,...,N (CRP N ) 20 (x IR 20 ) l =4(u IR 4 ) (CRP N ) (RD N ) /. μ j, j =

11 : opt(rd) opt(crp N ) (η =0.0) ɛ N(ɛ, η) q (ɛ) R.err [%] : opt(rd) opt(crp N ) (η =0.0) x N (N = 00) x N 2 (N 2 = 000) δ M β M A.err R.err [%] M β M A.err R.err [%] x N 3 (N 3 = 0000) δ M β M A.err R.err [%] ,,...,4 (CRP N ) [0] u,...,u N 00 (CRP N ) (RD N ) (00 ) 5 (RD) N =5 0 4 (CRP N ) (t N =.060) (RD) 5 (CRP N) (RD N ) f (u i ) i =,...,N 6 (CRP N ) (CRP N ) N(ɛ, η) η ɛ q (ɛ) R.err R.err := q (ɛ) opt(rd) N =0 3 q (ɛ) = % 5 N =0 3 (CRP N ) 2 N =0 2, 0 3, 0 4 (CRP N ) β M = max {f (ũ i ) f(x N, ũ i) t N } i=,...,m

12 3: (N 0 =0 η =0.0 ζ =0.0) δ =0.5 ( ζ + δ 0.6 ) itr. V.rate (V/M) β M t N 42/ / δ =0.0 ( ζ + δ 0. ) itr. V.rate (V/M) β M t N 248/ / δ =0.05 ( ζ + δ 0.06 ) itr. V.rate (V/M) β M t N 3653/ / A.err := β M + δ A.err := A.err 00 opt(rd). N =02 (CRP N ) δ =0.05 A.err = 0.08 q (ɛ) =.397 δ M β M x N (CRP N ) (CRP N ) ζ 4 η N 0 δ η opt(rd) opt(crp N ) ζ + δ (RD) opt(crp N ) N 0 =0 η =0.0 ζ =0.0 (0) 3 δ ũ,...,ũ M V.rate (V/M) M V (x N,t N ) M V 3 δ (0) 2 δ =0.05 N =5 0 4 (CRP N ) (t N =.060) N =0 (CRP N ) N = 3663 (CRP N ) ( ζ + δ 0.06) t N =.060

13 6 N N Calafiore & Campi [3] (Relative Robustness) [] I.D Aron and P.V. Hentenryck: On the complexity of the robust spanning tree problem with interval data, Operations Research Letters, 32 (2004) [2] A. Ben-Tal and A. Nemirovski: Robust convex optimization, Mathematics of Operations Research, 23 (998) [3] G. Calafiore and M.C. Campi: A new bound on the generalization rate of sampled convex programs, 43rd IEEE Conference on Decision and Control (CDC04), 5 (2004) [4] L. El Ghaoui and H. Lebret: Robust solutions to least-squares problems with uncertain data, SIAM Journal on Matrix Analysis and Applications, 8 (997) [5] D. Goldfarb and G. Iyengar: Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003) -38. [6] T. Kanamori and A. Takeda: Worst-case violation of sampled convex programs for optimization with uncertainty, Research Report B-425, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology (2006). [7] P. Kouvelis and G. Yu: Robust discrete optimization and its applications, (Kluwer Academic Publishers, Norwell, 997). [8] V. Krishnamurthy: Robust optimization in finance, (Second Summer Paper for the Doctoral Program, Tepper School of Business, Carnegie Mellon niversity, 2004). [9] A.L. Soyster: Convex programming with set-inclusive constraints and applications to inexact linear programming, Operations Research, 2 (973)

14 [0] A. Takeda, S. Taguchi and T. Tanaka: Relaxation methods for robust deviation and relative robust decision models, Research Report B-427, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology (2006). [] H. Yaman, O.E. Karaşan and M.Ç. Pinar: The robust spanning tree problem with interval data, Operations Research Letters, 29 (200) [2] G. Yu and J. Yang: On the robust shortest path problem, Computers & Operations Research, 25 (998)

Σχετικά έγγραφα

Βασίλειος Μαχαιράς Πολιτικός Μηχανικός Ph.D.

Βασίλειος Μαχαιράς Πολιτικός Μηχανικός Ph.D. Κλασικές Τεχνικές Βελτιστοποίησης Πανεπιστήμιο Θεσσαλίας Σχολή Θετικών Επιστημών ΤμήμαΠληροφορικής Διάλεξη 2 η /2017 Μαθηματική Βελτιστοποίηση Η «Μαθηματική