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ROBUST OPTIMIZATION

Lecturer : Majid Rafiee

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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

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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

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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

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Robust Mixed Integer Linear Optimization.

Conclusion

Lecturer : Majid Rafiee

END

Lecturer : Majid Rafiee