ROBUST OPTIMIZATION Lecturer : Majid Rafiee
ROBUST OPTIMIZATION
Lecturer : Majid Rafiee
؟
Lecturer : Majid Rafiee
Uncertainty :
Very often, the realistic data are subject to uncertainty due
to their random nature, measurement errors, or other
reasons.
Robust optimization belongs to an important methodology
for dealing with optimization problems with data
uncertainty.
One major motivation for studying robust optimization is
that in many applications the data set is an appropriate
notion of parameter uncertainty, e.g., for applications in
which infeasibility cannot be accepted at all and for those
cases that the parameter uncertainty is not stochastic, or if
no distributional information is available.
INTRODUCTION
Lecturer : Majid Rafiee
a
b
ˆ ˆ, 1,1a a a a
ˆ ˆ,a a a a
ROBUST OPTIMIZATION
The random variables are distributed in the range [-1, 1]
nominal values
perturbation
ˆ 1,1a a a
𝑎
𝑎
𝜉
Lecturer : Majid Rafiee
min
. .
0
j j
ij j i
j
j
c x
s t
a x b
x
min
. .
0
iij j
j
j
cx
s t
a x b
x
𝑐𝑗 𝑎𝑖𝑗 𝑏𝑖
𝑥𝑗
ˆij ij ij ija a a
0ˆ
i i i ib b b
0ˆ
i i i jc c c
ROBUST OPTIMIZATION
Lecturer : Majid Rafiee
1. In the first stage of this type of method, a deterministic data set is
defined within the uncertain space.
2. in the second stage the best solution which is feasible for any
realization of the data uncertainty in the given set is obtained. The
corresponding second stage optimization problem is also called robust
counterpart optimization problem.
OPTIMIZATION STEPS ROBUST
Lecturer : Majid Rafiee
Step 1
Assume that the left-hand side (LHS) constraint coefficients
The random variables are distributed in the range [-1, 1]
nominal values
parameter uncertainty
perturbation
Lecturer : Majid Rafiee
???
where are predefined uncertainty sets for (ξ11, ξ12) and (ξ21, ξ22), respectively.
Step 1
1 2,U U
1 2,U U
Lecturer : Majid Rafiee
𝜉11
𝜉12
11
9
18 22 𝑎12
𝑎11 𝑎 11
𝑎 12
𝜉11, 𝜉12 تعریف مجموعه ی فرمول بندی کردن
U
Step 1
Lecturer : Majid Rafiee
Soyster 1973
Bertsimas and Sim
2004
Ben-Tal, Nemirovski
1998
History Robust optimization
Lecturer : Majid Rafiee
1. In the first stage of this type of method, a deterministic data set is
defined within the uncertain space.
2. in the second stage the best solution which is feasible for any
realization of the data uncertainty in the given set is obtained. The
corresponding second stage optimization problem is also called robust
counterpart optimization problem.
OPTIMIZATION STEPS ROBUST
U𝜉11, 𝜉12, …… .
Lecturer : Majid Rafiee
If the set U is the box uncertainty set, then the corresponding robust counterpart constraint is equivalent to the following constraint
Property
Step 2 : Soyster 1973
Lecturer : Majid Rafiee
If the set U is the ellipsoidal uncertainty set, then the corresponding robust counterpart constraint is equivalent to the following constraint
Property
Step 2 : Ben-Tal, Nemirovski 1998
Lecturer : Majid Rafiee
If the set U is the “box+polyhedral” uncertainty set, then the corresponding robust counterpart constraint is equivalent to the following constraint
Property
Step 2 : Bertsimas and Sim 2004
Lecturer : Majid Rafiee
ˆij ij ij ija a a
0ˆ
i i i ib b b
min
. .
0,1
iij j
j
j
cx
s t
a x b
x
0
min
. .
ˆ ˆmax
0,1
i
i
ij j U i i ij ij j
j j J
j
cx
s t
a x b a x b
x
min
. .
0,1
iij j
j
j
cx
s t
a x b
x
min
. .
0,1
iij j
j
j
z
s t
cx z
a x b
x
0ˆ
i i i jc c c
A VARIETY OF PARAMETERS UNCERTAINTY
Lecturer : Majid Rafiee
Robust Linear Optimization
Conclusion
Lecturer : Majid Rafiee
Robust Mixed Integer Linear Optimization.
Conclusion
Lecturer : Majid Rafiee
END
Lecturer : Majid Rafiee