Robust Optimization for Risk Control in Enterprise-wide Optimization

Transcription

1 Robust Optimization for Risk Control in Enterprise-wide Optimization Juan Pablo Vielma Department of Industrial Engineering University of Pittsburgh EWO Seminar, 011 Pittsburgh, PA

2 Uncertainty in Optimization Problems x R n Decision Variables L x (ω) :Ω R Random Variable Average or Expected Value E (L x ) No Risk Control Add Risk Probabilistic Constraint P (L x 1) 0.95 Hard to Solve Make Easy Risk Measures Equivalent Robust Constraint W (L x ) L x (ω) 1 ω R

3 Overview Risk Measures: Portfolio Optimization Example Robust and Probabilistic Constraints Meaning of Risk Measures and Robust Constraints Conclusions 3/7

4 Risk Measures Single Period Portfolio Optimization min E 1 0 ω i x i 0 possible assets x i : millions in asset i=1 s.t. ω i : (random) rate of return 0 x i =1 of asset i i=1 Pessimistic: Minimize x i 0 average loss 0 σ i=1 ω i x i k Risk control: Variance Constraint 4/7

5 Risk Measures Variance OK, but Penalizes Gains Average... Variance Constrained Average /7

6 Risk Measures Uncertain Rate of Return = Scenarios Data Driven Uncertainty: ω ω 1, ω,...,ω 100 R 0, P(ω = ω s )= Loss is Random Variable: L x (ω) =1 0 i=1 ω i x i {L x 1,...,L x 100} L x (ω) Scalar Representation of :, L x s =1 0 i=1 ω s i x i E (L x )= s=1 Lx s 6/7

7 Risk Measures Including Risk in Scalar Representation Loss is Random Variable: L x {L x 1,...,L x 100} Pessimistic approach = Worst Case: W 1 (L x ):= 100 max i=1 Lx i Less pessimistic = k-th Worst Case: Only k scenarios are equal or worse. 7/7

8 Risk Measures k-th Worst Case for k=10 L x i W 10 (L x )= Sort i /7

9 Risk Measures A More Conservative k-th Worst Case Average: E (L x ):= s=1 Lx s k-th worst case: 100 max i=1 Lx i = W 1 (L x ) > W (L x ) >...>W 100 (L x 100 )= min What about average of k-th worst cases: i=1 Lx i A k (L x ):= 1 k k s=1 W k (L x ) We have: A k (L x ) W k (L x ) 9/7

10 Risk Measures Two New Families of Problems min f(x) :=W k (L x ) s.t. min g(x) :=A k (L x ) s.t. 0 i=1 x i =1 x i 0 0 i=1 x i =1 x i 0 10/7

11 Risk Measures Behavior of Optimal Solutions Variance Constrained E (L x ) W 1 (L x ) W 5 (L x ) W 10 (L x ) W 15 (L x ) Variance Constrained E (L x ) A 1 (L x ) A 5 (L x ) A 10 (L x ) A 15 (L x ) 11/7

12 Risk Measures k-worst v/s average k-worst W 15 (L x ) A 15 (L x ) /7

13 Risk Measures Probabilistic Meaning k-worst = Value at Risk: W k (L x )=inf =: t t : P(L x t) 1 k 1 (L x ) 100 Average k-worst = Average Value at Risk Also known as Conditional Value at Risk E L x L x t (L x ) 13/7

14 Risk Measures Shape of Objective Function for Assets g(x) :=A (L x ) f(x) :=W (L x ) x 1 14/7

15 Risk Measures Convexity and Coherence Average k-worst is a: Coherent Risk Measure Monotonic, Translation Invariant, Subadditive, Positive Homogeneous. Distortion/Spectral Risk Measure Coherent + Co-monotonicity and Law Invariance 15/7

16 Risk Measures Monotonic L x1 i L x i i {1,...,100} A k L x1 A k L x 16/7

17 Risk Measures Sub-additive A k L x1 + L x A k L x1 + A k L x = Co-monotonicity (Distortion) 17/7

18 Risk Measures Formulation for Average Worst Case min A k (L x ) s.t. 0 i=1 x i =1 x i 0 t = W k L x min t + 1 k s.t. 100 s=1 α s 0 i=1 x i =1 x i 0 h (x, ω s ) t α s α s 0 t R h (x, ω s 0 ):=1 i=1 ωs i x i 18/7

19 Risk Measures Constraining with Risk Measures min A k (L x ) s.t. 0 x i =1 i=1 x i 0 min s.t. 0 z i=1 x i =1 x i 0 A k (L x ) z A k (L x ) 1 A k (L x ) z z =1 19/7

20 Robust Constraints Probabilistic Constraints ω ω 1, ω,...,ω 6 R, P(ω = ω s )= 1 6 P(ω 1 x 1 + ω x 4) 5 6 : Satisfy 5 of 6 inequalities: {ω s 1x 1 + ω s x 4} 6 s=1 W (ω 1 x 1 + ω x ) 4 A (ω 1 x 1 + ω x ) W (ω 1 x 1 + ω x ) so: Conservative approximation: A (ω 1 x 1 + ω x ) 4 0/7

21 Robust Constraints Satisfy 5 out of 6 Constraints ω 1 1 ω 6 ω ω 5 ω 3 4 ω 4 3 W (ω 1 x 1 + ω x ) /7

22 Robust Constraints No Guarantee: Violate Each Constraint ω 1 1 ω 6 ω ω 5 ω 3 4 ω 4 3 W (ω 1 x 1 + ω x ) /7

23 Robust Constraints Guarantee to Satisfy All Constraints 3 6 A 1 (ω 1 x 1 + ω x ) 4 4 ω 1 1 ω 6 ω ω 5 ω 3 3 ω 1 x 1 + ω x 4 ω R 1 R 1 = conv ω 4 {ω s } 6 s= /7

24 Robust Constraints Weaker Guarantee: All -Averages 6 3 A (ω 1 x 1 + ω x ) 4 4 ω 1 ω 6 ω ω 5 ω ω 1 x 1 + ω x 4 ω R R = conv ω ωs + 1 ωt 6 s,t=1 4/7

25 Robust Constraints Coherent: Worst Case Expectation Direct from previous slide: A (ω 1 x 1 + ω x )=sup P P E P (ω 1 x 1 + ω x ) P := i=j P : P(ω = ω i )=P(ω = ω j )= 1 All Coherent Risk Measures are of this form 5/7

26 Robust Constraints Distortion: Average of Worst Case Average Worst Case: A k (L x ):= 1 k k s=1 W k (L x ) General Distortion Risk Measure: D p (L x ):= S s=1 p iw k (L x ) p R S S + : p i =1,p 1 p... p S s=1 6/7

27 Conclusions Conclusions Robust Constraints = Risk Measures Add risk control Easy to solve conservative approximation of hard to solve probabilistic constraints Very easy to implement for scenario based Do not need a priori scenario probabilities 7/7

28 References Everything in this talk: D. Bertsimas and D. B. Brown, 010. Constructing Uncertainty Sets for Robust Linear Optimization, Operations Research, 58, Books: A. Ben-Tal, L. El Ghaoui and A. Nemirovski, 010. Robust Optimization, Princeton University Press. A. Shapiro, D. Dentcheva and A. Ruszczyski, 010. Lectures on Stochastic Programming: Modeling and Theory, SIAM. 8/7

29 Translation Invariance m A k (L x + m) =A k (L x )+m A k (L x m) =A k (L x ) m 9/7

30 Positive Homogeneous A k (λl x )=λa k (L x ) λ > 0 30/7

31 Co-Monotonicity L x1 (ω) L x1 (ω ) L x (ω) L x (ω ) 0 ω, ω A k L x1 + L x = A k L x1 + A k L x 31/7

32 Law Invariance (for ) P ω = ω i = P ω = ω j L x 1 ω i = L x ω π(i) π permutation A k (L x 1 )=A k (L x ) 3/7