On comparison of different approaches to the stability radius calculation

Olga KarelkinaDepartment of MathematicsUniversity of Turku

MCDM 2011

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Outline

PreliminariesProblem statementExact method for calculation stability

radius proposed by Chakravarti and WagelmansNSGA-II adaptation for calculation stability

radiusIllustration and comparison of two

approaches

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Two major directions of investigation can be single out

quantitative

•bounds for feasible changes in initial data, which preserve some pre-assigned properties of optimal solutions•deriving algorithms for the bounds

calculation

qualitative

•conditions under which the set of optimal solutions of the problem possesses a certain pre-assigned property of invariance to external influence on initial data of the problem

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Shortest path problem (SP)

Given a directed graph and

– a nonnegative cost associated with each edge

G =(V,E),| V| =m | E| =n

cie Ei

Problem: find a directed path from a source node to a distinguished terminal node , with the minimum total cost.

st

The feasible set is the set of all sequences , these sequences are directed paths from to in .

s t G1 ki iP =(e ,…,e )

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SP as LP

Vector of ordered edges costs

n n1 2 n + 1 2 nC = c ,c ,…,c , x= x ,x ,…,xR E

ii

1, if e P,x =

0 otherwise

i

i ie E

c x min

i i

i ie:e j e:e j

1, if j=s,

x - x = -1, if j=t,

0 otherwise

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Perturbation of the problem

We define norms and in for any finite dimension

1ldR

d N

d

i1i N

y = y ,

T d1 2 d dy=(y ,y ,…,y ) ,N =1,2,…,d.R

The perturbation of the problem parameters is modeled by adding to the cost vector perturbing vector

The set of the perturbing vectors is denoted by

C

n

1 2 nC = c ,c ,…,c , C <ε,ε>0.R

Ω(ε).

l

i dy =max y | i N ,

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Stability radius

Let be the set of feasible solutions to the shortest path problem

nEX 2

Let be the set of optimal solutions to the shortest path problem with cost vector .

optX (C)C

An optimal solution is called stable if optx X (C)

optε>0 C Ω(ε) x X (C +C ).

supΘ, if Θ ,ρ(x,C)=

0, if Θ = .

Stability radius of an optimal solution

optΘ = ε>0 | C Ω(ε) x X (C +C ) .

optx X (C)

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Stability radius

n

i i ii N

x X\ {x}1

c (x - x)

ρ(x,C)=minx - x

(1)

n n

i i i i i ii N i N

c +c x c +c x , x X

The largest such that forρ i nc ρ, i N

V. A. Emelichev, D.P. Podkopaev, Quantitative stability analysis for vector problems of 0 – 1 programming, Discrete Optimization. 7 (2010) 48 – 63

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Calculating the stability radii of an optimal solution to the linear problem of 0-1 programming

Theorem Let be an optimal solution to (2). The stability radius of is the maximum number satisfying the following inequality :

x XCx min (2

)

xx ρ

n n

i i i i ix X\ xi N i N

min c -ρd x c +ρ x

ii

i

1, if x =0,d =

-1, if x =1.

(3)

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ρρ(x,C) is the maximal satisfying the inequality :

n

i i ii N

x X\ {x}1

C (x - x)

ρ minx - x

From here taking into account

n n

i i i i n

i i i i1i N i N

x - x =x+d x, i N

x - x = x - x = x+d x

we get

n n

i i i i ix X\ xi N i N

min c -ρd x -c +ρ x

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Let us denote

n

i i ix X\ xi N

v ρ = min c -ρd x

v ρ is a continuous, piecewise linear and concave function of ρ

Lemma The number of linear pieces of is v ρ Ο n

D. Gusfield, Parametric combinatorial computing and a problem of program module distribution, J. Assoc. Comput. Mach. 30 (1983) 551 – 563

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Chakravarti and Wagelmans polinomial algorithm

Construction of on v ρ

[0, C ]

Compute andThe optimal solutions associated with these values

each defines a linear function onIf these functions are identical, then is simply

this linear functionOtherwise, we have two linear functions which

intersect at a unique value If coincides with the intersection point,

then is the concave lower envelope of the two linear functionsOtherwise, the optimal solution associated with

defines a third linear function which intersects each of the other linear functions on

v C

ρ [0, C ]

ρ,v ρ

ρ

v 0

[0, C ]

v ρ

v ρ

[0, C ]

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Chakravarti and Wagelmans polinomial algorithm

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A fast and elitist multi-objective genetic algorithm: NSGA-IIModules

A. A fast non-dominated sorting approachB. Diversity presentation

•Density estimation•Crowded comparison operator

C. The main loop

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Begin

Initialize Population

gen=0

Evaluation Assign Fitness

Cond?

Reproduction

Crossover

Mutation

gen=gen+1 Stop

No

Yes

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n

i i i 1i N

21

c (x - x)=f(x,C) min

x - x =f (x,C) max

Implementation of NSGA-II into calculation stability radius

Pareto set

2

1 1 2 2

1 1 2 2

|

, , , ,

, , , ,

P C x X x X

f x C f x C f x C f x C

f x C f x C f x C f x C

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Representation

Graph is represented by costs matrix (vector)Every variable (feasible solution) is coded in a fixed

length binary string

InitializationBreadth First Search

EvaluationA fast non-dominated sorting approach

find-nondominated-front(P) include first member in

for each take one soltion at a time

include in temporarily for each compare with other members of if , then if dominates a member of , delete it

else if , then if is dominated by other members of ,

do not include in

p P p P

P = P = 1 P

P =P U p q P q p

Pp

p q P =P \ q

q p P =P \ p

Ppp P

p PP

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Assign fitness

Density estimation

Crowding distance is an estimate of the size of the largest cuboid enclosing the point without including any other point in the population

Crowded comparison operator

n rank rank rank rank distance distancei j i <j i =j i >j

distanceii

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Reproduction

The tournament selection scheme

The strings with minimum front number and minimum value of ratios

are selected to the mating pool.

1

2

f(x,C)f (x,C)

A directed graph on 10 nodes

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Crossovers

One-Node crossover5 is a common node for both parents

One-Edge crossoverEdges (2,3) and (1,7) are used as links

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One-Node-Two-Edges crossover

Nodes 4 and 8 do not belong to any of the parents, subpaths ((3,4),(4,6)) and ((6,8),(8,7)) are used as links

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Mutation

The search of genetic algorithm is mainly guided by crossover operators, even though mutationoperator is also used to maintain diversity in the population.

Scheme of two mutation types

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Pareto fronts5 generations

10 generations

20 generations

15 generations

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Simulation resultsWe consider a family of randomly generated directed graphs on 100 verticesand with approximately 5000 edges. Weight range is [1, 50].

Calculation results were compared with solutions obtained by exact method proposed by N. Chakravarti and A. P. M. Wagelmans in Calculation of stabilityradii for combinatorial optimization problems, OR Letters. 23 (1998) 1 – 7.

The population size is set to 100 (number of vertices), while the probabilities of the one-node, one-edge and one-node-two-edges crossovers are 0.2, 0.3 and 0.5 correspondingly, mutation probability increases with the number of generations. Tests show that in average NSGA-II converges in 80% cases and gives the exact solution after 5 – 20 generations.

Complexity of the exact method is

2n n nLog n

NSGA-II complexity is

2kn

Here n is the number of vertices and k is the number of generations.

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References

1. V.A. Emelichev, V.N. Krichko, D.P. Podkopaev, On the radius of stability of a vector problem of linear Boolean programming, Discrete Math. Appl. 10 (2000) 103 – 108

2. N. Chakravarti, A. P.M. Wagelmans, Calculation of stability radii for combinatorial optimization problems, OR Letters. 23 (1998) 1 – 7

3. D. Gusfield, Parametric combinatorial computing and a problem of program module distribution, J. Assoc. Comput. Mach. 30 (1983) 551 – 563

4. V. A. Emelichev, D.P. Podkopaev, Quantitative stability analysis for vector problems of 0 – 1 programming, Discrete Optimization. 7 (2010) 48 – 63

5. K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multi-objective genetic algorithm: NSGA-II, Evolutionary Computation. 6 (2) (2002), 182 – 197

Thank You for Your interest