On comparison of different
approaches to the stability
radius calculation
Olga Karelkina
Department of Mathematics
University of Turku
MCDM 2011
Outline
 Preliminaries
 Problem statement
 Exact method for calculation stability radius
proposed by Chakravarti and Wagelmans
 NSGA-II adaptation for calculation stability
radius
 Illustration
and
comparison
of
two
approaches
Slide 2 of 25
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
Slide 3 of 25
Shortest path problem (SP)
Given a directed graph G = (V,E),|V|= m and |E|= n
c – a nonnegative cost associated with each edge
i
e E
i
Problem: find a directed path from a source node s
to a distinguished terminal node t , with the minimum
total cost.
The feasible set is the set of all sequences P =(ei1 ,…,eik ) ,
these sequences are directed paths from s to t in G.
Slide 4 of 25
SP as LP
Vector of ordered edges costs
C= c1,c2 ,…,cn  Rn+ , x =  x1,x2 ,…,xn  En
1, if ei  P,
xi = 
0 otherwise
 c x  min
eiE
i i
1, if j = s,

x i -  x i = -1, if j = t,

e:ei  j
e:ei  j
0 otherwise

Slide 5 of 25
Perturbation of the problem
We define norms
d N
l1
y 1 =  yi ,
iNd
and
l
y
in

Rd
for any finite dimension
= max  y i |i  Nd ,
y =(y1,y2 ,…,yd)T Rd,Nd =1,2,…,d.
The perturbation of the problem parameters is modeled
by adding to the cost vector C perturbing vector
C = c1,c2 ,…,cn  Rn, C  < ε, ε > 0.
The set of the perturbing vectors is denoted by
Ω(ε).
Slide 6 of 25
Stability radius
En
Let X  2
be the set of feasible solutions to the shortest
path problem
Let Xopt(C) be the set of optimal solutions to the shortest
path problem with cost vector C .
An optimal solution
x  Xopt(C) is called stable if
ε > 0 C Ω(ε) x  Xopt(C+C).
Stability radius of an optimal solution x  Xopt(C)

supΘ, if Θ  ,
ρ(x,C) = 
if Θ = .
0,
Θ = ε > 0 | C  Ω(ε)
x X

(C + C) .
opt
Slide 7 of 25
Stability radius
V. A. Emelichev, D.P. Podkopaev, Quantitative stability
analysis for vector problems of 0 – 1 programming,
Discrete Optimization. 7 (2010) 48 – 63
ρ(x,C) = min
 c (x - x )
iNn
x X\{x}
The largest
i
i
i
x - x
1
ρ such that for ci  ρ, i Nn
 c +c x   c +c x,
iNn
(1)
i
i
i
iNn
i
i
i
x  X
Slide 8 of 25
Calculating the stability radii of an optimal
solution to the linear problem of 0-1
programming
Cx  min
(2)
xX
Theorem Let x be an optimal solution to (2). The stability
radius of x is the maximum number ρ satisfying the
following inequality :


min    ci - ρdi  xi     ci + ρ x i
x X\x
 iNn
 iNn
(3)
if x i = 0,
1,
di = 
-1, if x i =1.
Slide 9 of 25
ρ(x,C)
is the maximal
ρ
ρ  min
x X\{x}
satisfying the inequality :
 C (x - x )
iNn
i
i
x - x
i
1
From here taking into account
xi - x i = x + di xi , i  Nn
x - x 1 =  xi - x i =   x + di xi 
iNn
iNn
we get


min    ci - ρdi  xi     -ci + ρ x i
xX\x
iNn
 iNn
Slide 10 of 25
Let us denote


v  ρ  = min    ci - ρdi  x i 
x X\x
 iNn

D. Gusfield, Parametric combinatorial computing and a problem of
program module distribution, J. Assoc. Comput. Mach. 30 (1983) 551 –
563
v ρ
is a continuous, piecewise linear and concave
function of ρ
Lemma The number of linear pieces of
v ρ is Οn
Slide 11 of 25
Chakravarti and Wagelmans polinomial
algorithm
Construction of
v ρ
 
on [0,

C ]

 Compute v 0 and v C

 The optimal solutions associated with these values each
defines a linear function on [0, C  ]
 If these functions are identical, then v ρ is simply this
linear function
 Otherwise, we have two linear functions which intersect at
a unique value ρ [0, C  ]
 If
ρ,v ρ coincides with the intersection point, then
v ρ is the concave lower envelope of the two linear
functions
 Otherwise, the optimal solution associated with ρ defines
a third linear function which intersects each of the other
linear functions on [0, C ]
 

 
 

Slide 12 of 25
Chakravarti and Wagelmans polinomial
algorithm
Slide 13 of 25
A fast and elitist multi-objective genetic
algorithm: NSGA-II
Modules
A. A fast non-dominated sorting approach
B. Diversity presentation
• Density estimation
• Crowded comparison operator
C. The main loop
Slide 14 of 25
Begin
Initialize Population
gen=0
Evaluation
Cond
?
Yes
gen=gen+1
Assign Fitness
No
Reproduction
Stop
Crossover
Mutation
Slide 15 of 25
Implementation of NSGA-II into calculation
stability radius
 c (x - x ) = f(x,C)  min
iNn
i
i
i
1
x - x 1 = f2(x,C)  max
Pareto set
P2  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   
1
1
1
2
1
2
2
2
Slide 16 of 25
Representation
 Graph is represented by costs matrix (vector)
 Every variable (feasible solution) is coded in a fixed
length binary string
Initialization
 Breadth First Search
Evaluation
 A fast non-dominated sorting approach
P = find-nondominated-front(P)
P = 1
p  P  p  P
P =PUp
for each q  P  q  p
if p q , then P =P \ q
for each
else if
q p, then P =P \ p
include first member in P
take one soltion at a time
include p in P temporarily
P
compare p with other members of
if p dominates a member of P, delete it
if p is dominated by other members of P,
do not include P in
Slide 17 of 25
Assign fitness
 Density estimation
Crowding distance idistance is an estimate of the size of the
largest cuboid enclosing the point i without including
any other point in the population
 Crowded comparison operator
i
n
j   irank < jrank     irank = jrank    idistance > jdistance  
Slide 18 of 25
Reproduction
 The tournament selection scheme
The strings with minimum front number and minimum
value of ratios
f(x,C)
1
f2(x,C)
are selected to the mating pool.
A directed graph on 10
nodes
Slide 19 of 25
Crossovers
 One-Node crossover
5 is a common node for both parents
 One-Edge crossover
Edges (2,3) and (1,7) are used as links
Slide 20 of 25
 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
Slide 21 of 25
Mutation
The search of genetic algorithm is mainly guided by
crossover operators, even though mutation
operator is also used to maintain diversity in the population.
 Scheme of two mutation types
Slide 22 of 25
Pareto fronts
5 generations
10 generations
15 generations
20 generations
Slide 23 of 25
Simulation results
We consider a family of randomly generated directed graphs on 100 vertices
and with approximately 5000 edges. Weight range is [1, 50].
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.
Calculation results were compared with solutions obtained by exact method
proposed by N. Chakravarti and A. P. M. Wagelmans in Calculation of stability
radii for combinatorial optimization problems, OR Letters. 23 (1998) 1 – 7.
Complexity of the exact method is
 
 n n2  nLog  n

NSGA-II complexity is

 kn2

Here n is the number of vertices and k is the number of generations.
Slide 24 of 25
References
1.
2.
3.
4.
5.
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
N. Chakravarti, A. P.M. Wagelmans, Calculation of stability radii for
combinatorial optimization problems, OR Letters. 23 (1998) 1 – 7
D. Gusfield, Parametric combinatorial computing and a problem of
program module distribution, J. Assoc. Comput. Mach. 30 (1983) 551
– 563
V. A. Emelichev, D.P. Podkopaev, Quantitative stability analysis for
vector problems of 0 – 1 programming, Discrete Optimization. 7
(2010) 48 – 63
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, Evolutionary Computation. 6
(2) (2002), 182 – 197
Slide 25 of 25
Thank You for Your interest