ROBUST OPTIMIZATION. Lecturer : Majid Rafiee

Transcription

1 ROBUST OPTIMIZATION

2 Uncertainty :

3 INTRODUCTION Very often, the realistic data are subect 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 maor 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.

4 ROBUST OPTIMIZATION a aˆ, a aˆ b a nominal values perturbation a a The random variables are distributed in the range [-1, 1] ξ ˆ ˆ a a, a a 1,1 a a ˆ a 1,1

5 ROBUST OPTIMIZATION min cx st.. x a x 0 b i i c a i b i x c c cˆ i i i 0 a a aˆ i i i i b b bˆ i i i 0i min cx st.. x a x i 0 b i

6 OPTIMIZATION STEPS ROBUST 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.

7 Step 1 parameter uncertainty Assume that the left-hand side (LHS) constraint coefficients nominal values The random variables are distributed in the range [-1, 1] perturbation

8 Step 1 where U1, U2 are predefined uncertainty sets for (ξ11, ξ12) and (ξ21, ξ22), respectively. U1, U2???

9 Step 1 ξ 12 ξ 11, ξ 12 تعریف مجموعه ی U فرمول بندی کردن ξ a 11 a a a 12

10 History Robust optimization Soyster 1973 Ben-Tal, Nemirovski 1998 Bertsimas and Sim 2004

11 OPTIMIZATION STEPS ROBUST 1. In the first stage of this type of method, a deterministic data set is defined within the uncertain space. ξ 11, ξ 12,. 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. U

12 Step 2 : Soyster 1973 Property If the set U is the box uncertainty set, then the corresponding robust counterpart constraint is equivalent to the following constraint

13 Step 2 : Ben-Tal, Nemirovski 1998 Property If the set U is the ellipsoidal uncertainty set, then the corresponding robust counterpart constraint is equivalent to the following constraint

14 Step 2 : Bertsimas and Sim 2004 Property If the set U is the box+polyhedral uncertainty set, then the corresponding robust counterpart constraint is equivalent to the following constraint

15 A VARIETY OF PARAMETERS UNCERTAINTY min cx st.. x a x i 0,1 b i c c cˆ i i i 0 min z st.. cx z x a x i 0,1 b i min cx st.. x a x i 0,1 b i a a aˆ i i i i b b bˆ i i i 0i min cx st.. x max ˆ a ˆ ix U i 0bi iaix b Ji 0,1 i

16 Conclusion Robust Linear Optimization

17 Conclusion Robust Mixed Integer Linear Optimization.

18 END