On comparison of different approaches to the stability radius calculation. Olga Karelkina

Transcription

1 O compariso of differet approaches to the stability radius calculatio Olga Karelkia Uiversity of Turku 2011

2 Outlie Prelimiaries Problem statemet Exact method for calculatio stability radius proposed by Chakravarti ad Wagelmas NSGA-II adaptatio for calculatio stability radius Illustratio ad compariso of two approaches Slide 2 of 24

3 Two major directios of ivestigatio ca be sigle out quatitative bouds for feasible chages i iitial data, which preserve some pre-assiged properties of optimal solutios derivig algorithms for the bouds calculatio qualitative coditios uder which the set of optimal solutios of the problem possesses a certai pre-assiged property of ivariace to exteral ifluece o iitial data of the problem Slide 3 of 24

4 Shortest path problem (SP) Give a directed graph c i e i G ( V, E), V m ad E a oegative cost associated with each edge E Problem: fid a directed path from a source ode to a distiguished termial ode t, with the miimum total cost. The feasible set is the set of all sequeces, i1 i k these sequeces are directed paths from to i. P ( e, K, e ) s t G s Cost mappig c( P) k c i1 i Slide 4 of 24

5 SP as LP Vector of ordered edges costs,, K, R,,, K, C c c c x x x x E x i 1, if ei P, 0 otherwise ee i cx i i mi 1, if j s, x xi 1, if j t, i 0 otherwise i e: e j e: e j i Slide 5 of 24

6 Perturbatio of the problem l We defie orms ad i for ay fiite dimesio d N y y, 1 i in d T d y ( y, y, K, y ) R, N 1,2, K, d. d l d R d y max y i N, i d The perturbatio of the problem parameters is modeled by addig to the cost vector perturbig vector 1 2 C c, c, K, c R, C, 0. The set of the perturbig vectors is deoted by C ( ). Slide 6 of 24

7 Stability radius X E 2 Let be the set of feasible solutios to the shortest path problem X ( C) Let opt be the set of optimal solutios to the shortest path problem with cost vector. A optimal solutio C x X ( C) opt is called stable if 0 C( ) x Xopt( C C). x X ( C) Stability radius of a optimal solutio sup, if, ( xc, ) 0, if. 0 C( ) x X ( C C). opt opt Slide 7 of 24

8 Stability radius V.A. Emelichev, V.N. Krichko, D.P. Podkopaev, O the radius of stability of a vector problem of liear Boolea programmig, Discrete Math. Appl. 10 (2000) in c ( x x ) i i i ( xc, ) mi xx \{ x} x x 1 (1) The largest c i N such that for, i in, c c x c c x x X i i i i i i in Slide 8 of 24

9 Calculatig the stability radii of a optimal solutio to the liear problem of 0-1 programmig x Cx mi (2) xx Theorem Let be a optimal solutio to (2). The stability radius of x is the maximum umber satisfyig the followig iequality : mi xx \ x d i in c d x c x i i i i i in 1, if xi 0, 1, if xi 1. (3) Slide 9 of 24

10 ( xc, ) is the maximal satisfyig the iequality : mi xx \{ x} in From here takig ito accout we get 1 C ( x x ) i i i x x x x x d x, i N i i i i x x x x x d x mi xx \ x in in i i i i in c d x c x 1 i i i i i in Slide 10 of 24

11 Let us deote v mi ci di xi xx \ x i N D. Gusfield, Parametric combiatorial computig ad a problem of program module distributio, J. Assoc. Comput. Mach. 30 (1983) v is a cotiuous, piecewise liear ad cocave fuctio of Lemma The umber of liear pieces of v is 2 Slide 11 of 24

12 Chakravarti ad Wagelmas poliomial algorithm Costructio of 0 v o [0, ] C v v C Compute ad The optimal solutios associated with these values each defies a liear fuctio o[0, C ] If these fuctios are idetical, the v is simply this liear fuctio Otherwise, we have two liear fuctios which itersect at a uique value [0, C ] If,v coicides with the itersectio poit, the v is the cocave lower evelope of the two liear fuctios Otherwise, the optimal solutio associated with defies a third liear fuctio which itersects each of the other liear fuctios o [0, ] C Slide 12 of 24

13 A fast ad elitist multi-objective geetic algorithm: NSGA-II Modules A. A fast o-domiated sortig approach B. Diversity presetatio Desity estimatio Crowded compariso operator C. The mai loop Slide 13 of 24

14 Begi Iitialize Populatio ge=0 Evaluatio Assig Fitess ge=ge+1 Yes Cod? Stop No Reproductio Crossover Mutatio Slide 14 of 24

15 Implemetatio of NSGA-II ito calculatio stability radius in c ( x x ) f ( x, C) mi i i i x x f ( x, C) max 1 Pareto set 2 1,,,, 2 P C x X x X f 1 x C f1 x C f2 x C f2 x C,,,, f x C f x C f x C f x C Slide 15 of 24

16 Represetatio Graph is represeted by costs matrix (vector) Every variable (feasible solutio) is coded i a fixed legth biary strig Iitializatio Breadth First Search Evaluatio A fast o-domiated sortig approach P fid-odomiated-frot(p) P 1 for each pp pp P PU p iclude first member i take oe soltio at a time iclude i temporarily for each qp q p compare p with ppther members of P if, the if domiates a member of, delete it \ \ pp q P P q p P pp q P P p p else if, the if is domiated by other members of, do ot iclude i p P P P Slide 16 of 24 P

17 Assig fitess Desity estimatio i distace Crowdig distace is a estimate of the size of the largest cuboid eclosig the poit i without icludig ay pther poit i the populatio Crowded compariso operator i p j i j i j i j rak rak rak rak distace distace Slide 17 of 24

18 Reproductio The touramet selectio scheme The strigs with miimum frot umber ad miimum ratios f1( x, C) f ( x, C) 2 are selected to the matig pool. Slide 18 of 24

19 Crossovers Oe-Node crossover Chromosomes before crossover Chromosomes after crossover Slide 19 of 24

20 Oe-Edge crossover Chromosomes before crossover Chromosomes after crossover Slide 20 of 24

21 Oe-Node-Two-Edges crossover Chromosomes before crossover Chromosomes after crossover Slide 21 of 24

22 Mutatio Two mutatio types Slide 22 of 24

23 Simulatio results Slide 23 of 24

24 Refereces 1. V.A. Emelichev, V.N. Krichko, D.P. Podkopaev, O the radius of stability of a vector problem of liear Boolea programmig, Discrete Math. Appl. 10 (2000) N. Chakravarti, A. P.M. Wagelmas, Calculatio of stability radii for combiatorial optimizatio problems, OR Letters. 23 (1998) D. Gusfield, Parametric combiatorial computig ad a problem of program module distributio, J. Assoc. Comput. Mach. 30 (1983) V. A. Emelichev, D.P. Podkopaev, Quatitative stability aalysis for vector problems of 0 1 programmig, Discrete Optimizatio. 7 (2010) K. Deb, A. Pratap, S. Agarwal, T. Meyariva, A fast ad elitist multiobjective geetic algorithm: NSGA-II, Evolutioary Computatio. 6 (2) (2002), Slide 24 of 24

25 Thak You for Your iterest