A multi-objective strategy for Pareto set refinement
Adriano Lisboa . Fellipe Santos . Douglas Vieira . Rodney Saldanha . Marcus Lobato
1
2
Contents
1
2
3
4
5
Why refining the Pareto set?
A supporting theorem for refinement
Quasi-convex function
Theorem: refinement for quasi-convex functions
Geometrical interpretation of theorem
Refinement algorithm
Performance in practice
Concluding remarks
Why refining the Pareto set?
3
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Why refining the Pareto set?
custom sampling of Pareto set
uniform
by curvature
refine where you desire to refine
good sampling of Pareto set
avoid repetition of samples
promote enough distance between samples
f2(x)
f2(x)
f1(x)
f2(x)
f1(x)
f1(x)
4
A supporting theorem for
refinement
5
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Quasi-convex function
sublevel sets are convex
1
x2
4
2
3
x1
6
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Quasi-convex function
f (c) < max{f (a), f (b)}, ∀c ∈ (a, b)
f(x)
f(b)
f(a)
max{f(a), f(b)}
a
c
b
x
7
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Theorem: refinement for quasi-convex functions
Theorem (quasi-convex Pareto)
For any two strictly quasiconvex functions f1 and f2 and for any
two Pareto optimal points a and b where f1 (a) < f1 (b), any
c = a + λ(b − a), λ ∈ (0, 1), is dominated only by Pareto optimal
points d where f1 (d) ∈ (f1 (a), f1 (b)) and f2 (d) ∈ (f2 (b), f2 (a)).
x2
f2(x)
a
a
c dominating
point here
c
b
x1
b
f1(x)
8
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Theorem: refinement for quasi-convex functions
Proof.
Because f1 and f2 are strictly quasiconvex functions, any point
c = a + λ(b − a), λ ∈ (0, 1), must lie in the cone
{x | f1 (x ) < f1 (b)} ∩ {x | f2 (x ) < f2 (a)} since f1 (a) < f1 (b) and
f2 (b) < f2 (a). Because a and b are Pareto optimal points, no
feasible point can lie in {x | f1 (x ) ≤ f1 (a), f2 (x ) ≤ f2 (a), f (x ) 6=
f (a)} ∪ {x | f1 (x ) ≤ f1 (b), f2 (x ) ≤ f2 (b), f (x ) 6= f (b)}. Hence, the
remaining space for points dominating c to lie is
f1 (x ) ∈ (f1 (a), f1 (b)) and f2 (x ) ∈ (f2 (b), f2 (a)), including Pareto
optimal points x = d because f (d) ≤ f (c), f (d) 6= f (c).
9
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Geometrical interpretation of theorem
5
5
4
2
f (x)
f ,f
1 2
4
3
2
2
1
-2
3
a
c
a 0
x
b
2
b
1
1
2
3
f1(x)
4
10
Refinement algorithm
11
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Refinement algorithm
1
2
find optimal Pareto vertices
loop
1
2
3
normalize objetive functions
elect a segment to refine
solve with a dominating optimization algorithm
f2(x)
f2(x)
f1(x)
f2(x)
f1(x)
f1(x)
12
Performance in practice
13
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
two quadratic functions
one streched, displaced and rotated
20
15
x2
10
5
0
−5
−10
−10
−5
0
5
x1
10
15
20
14
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
epsilon constrained (left)
Pareto refinement (right)
20
20
15
15
f2
25
f2
25
10
10
5
5
0
0
0.1
0.2
0.3
f1
0.4
0.5
0
0
0.1
0.2
0.3
0.4
0.5
f1
15
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
two quasiconvex functions
one streched, displaced and rotated
2
1.5
x2
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
x1
1
1.5
2
16
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
epsilon constrained (left)
Pareto refinement (right)
0.8
0.8
0.6
0.6
f2
1
f2
1
0.4
0.4
0.2
0.2
0
0
0.05
0.1
0.15
0.2
f1
0.25
0.3
0.35
0
0
0.05
0.1
0.15
0.2
f1
0.25
0.3
0.35
17
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
refinement is about 10% faster epsilon constrained
5
8
x 10
epsilon−constrained
refinement
problem evaluatinos, k
7
6
5
4
3
2
1
0
2
4
6
dimension, n
8
10
18
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
test on seasonalizing of energy problem
income versus risk
formulation
minimize f1 (x , x0 ) = − mean
j
f2 (x , x0 ) = − mean
j∈W
subject to
n
X
n
X
i=1
n
X
xi (pi,j ηi,j + si ) + x0 δi ri,j [b + pi,j (ηi,j − 1)]
xi (pi,j ηi,j + si ) + x0 δi ri,j [b + pi,j (ηi,j − 1)]
i=1
xi ≤ G
i=1
x ≥0
where
ηi,j = min 1,
hi,j
xi + x0 δi + gi
19
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
5% worst income mean, E − ePaR (millions R$)
−635
−640
−645
−650
−655
−660
−665
139.8
139.9
140
140.1
140.2
income mean, E (millions R$)
140.3
20
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Performance in practice
5% worst income mean, E − ePaR (millions R$)
−635
−640
−645
−650
−655
−660
−665
139.8
139.9
140
140.1
140.2
income mean, E (millions R$)
140.3
21
Concluding remarks
22
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Concluding remarks
the refinement works for many variables
the refinement typically works faster than scalarization
methods
the refinement generalizes for more than 2 objetive functions
23
Why refining the Pareto set? A supporting theorem for refinement Refinement algorithm Performance in practice Concluding rem
Concluding remarks
presentation available at www.enacom.com.br
Thanks
24
© 2012
ENACOM . CEMIG . UFMG
25