Worst-case-expectation approach to optimization under uncertainty

Transcription

1 Worst-case-expectation approach to optimization under uncertainty Wajdi Tekaya Joint research with Alexander Shapiro, Murilo Pereira Soares and Joari Paulo da Costa : Cambridge Systems Associates; : Georgia Tech; : ONS, Brazil ICSP 13 Bergamo, ITALY July 8-12, 2013 July 10, 2013 Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

2 Outline 1 Motivation 2 Risk neutral approach Risk neutral formulation Methodology SDDP algorithm 3 Worst-case-expectation approach Worst-case-expectation formulation The (WCE) algorithm Analysis of the worst-case-expectation policy 4 Conclusion Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

3 Motivation: Hydrothermal operation planning problem Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

4 Motivation: Hydrothermal operation planning problem Demand constraints: + + = turbined thermal energy Load volume generation exchange Z t Y t F t D t Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

5 Motivation: Hydrothermal operation planning problem Demand constraints: + + = turbined thermal energy Load volume generation exchange Z t Y t F t D t } {{ } incurs costs c > 0 Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

6 Motivation: Hydrothermal operation planning problem Demand constraints: + + = turbined thermal energy Load volume generation exchange Z t Y t F t D t } {{ } incurs costs c > 0 Balance equation: + + = + stored turbined spillage stored Energy volume at t+1 volume volume volume at t inflows V t+1 Z t S t V t ξ t Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

7 Motivation: Hydrothermal operation planning problem The purpose is to define an operation strategy which, for each stage of the planning period, given the system state at the beginning of the stage, produces generation targets for each plant. Q t (V t, ξ t ) = min { c T [Y t, F t ] + E[Q t+1 (V t+1, ξ t+1 )] } s.t. Z t }{{} turbined volume + Y }{{} t + F }{{} t = D t (Demand constraints) thermal generation energy exchange V }{{} t+1 + Z }{{} t + S }{{} t = V t + ξ t (Balance equation) stored volume turbined volume spillage volume at begining t+1 0 Z t Z t, 0 V t+1 V t, Y t Y t Ȳ t, 0 F t F t (Capacity constraints) Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

8 Outline 1 Motivation 2 Risk neutral approach Risk neutral formulation Methodology SDDP algorithm 3 Worst-case-expectation approach Worst-case-expectation formulation The (WCE) algorithm Analysis of the worst-case-expectation policy 4 Conclusion Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

9 Risk neutral formulation Nested formulation: Min A 1x 1=b 1 x 1 0 c1 T x 1 +E ξ1 min B 2x 1+A 2x 2=b 2 x 2 0 c T 2 x 2 + E ξ[2] [ + E ξ[t 1] [ min B T x T 1 +A T x T =b T x T 0 ct T ] ] x T Components of vectors c t, b t and matrices A t, B t are modelled as random variables forming the stochastic data process ξ t = (c t, A t, B t, b t ), t = 2,..., T, with ξ 1 = (c 1, A 1, b 1 ) being deterministic. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

10 Risk neutral formulation Nested formulation: Min A 1x 1=b 1 x 1 0 c1 T x 1 +E ξ1 min B 2x 1+A 2x 2=b 2 x 2 0 c T 2 x 2 + E ξ[2] [ + E ξ[t 1] [ min B T x T 1 +A T x T =b T x T 0 ct T ] ] x T Components of vectors c t, b t and matrices A t, B t are modelled as random variables forming the stochastic data process ξ t = (c t, A t, B t, b t ), t = 2,..., T, with ξ 1 = (c 1, A 1, b 1 ) being deterministic. Key assumption : stagewise independence (vector ξ t+1 is independent of ξ [t] = (ξ 1,..., ξ t ) for t = 1,..., T 1.) Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

11 Risk neutral formulation Nested formulation: Min A 1x 1=b 1 x 1 0 c1 T x 1 +E ξ1 min B 2x 1+A 2x 2=b 2 x 2 0 c T 2 x 2 + E ξ[2] [ + E ξ[t 1] [ min B T x T 1 +A T x T =b T x T 0 ct T ] ] x T Components of vectors c t, b t and matrices A t, B t are modelled as random variables forming the stochastic data process ξ t = (c t, A t, B t, b t ), t = 2,..., T, with ξ 1 = (c 1, A 1, b 1 ) being deterministic. Key assumption : stagewise independence (vector ξ t+1 is independent of ξ [t] = (ξ 1,..., ξ t ) for t = 1,..., T 1.) Dynamic Programming equations: Q t (x t 1, ξ t ) = min x t R n t { c t x t + Q t+1 (x t ) : B t x t 1 + A t x t = b t, x t 0 where: Q t+1 (x t ) = E [Q t+1 (x t, ξ t+1 )] for 2 t T. such that: Q T +1 (.) 0 and B 1 = 0. } Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

12 Methodology Three levels of approximations: 1 Modelling: we assume we have a true problem to solve. 2 The true problem is approximated by the so-called Sample Average Approximation (SAA) problem : a sample ξ t 1,..., ξ t Nt, of size N t, from the distribution of the random vector ξ t, t = 1,..., T, is generated. These samples generate a scenarios tree with the total number of scenarios N = T t=1 N t, each with equal probability 1/N. 3 Total number of scenarios N quickly becomes astronomically large with increase of T : The SDDP method suggests a computationally tractable approach to solving SAA, and hence the true problem. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

13 SDDP algorithm Introduced by Pereira and Pinto(1985,1991). It is based on building piecewise linear outer approximations of the convex cost-to-go functions. The distinguishing feature of the SDDP approach is random sampling from the set of scenarios in the forward step of the algorithm. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

14 SDDP algorithm Introduced by Pereira and Pinto(1985,1991). It is based on building piecewise linear outer approximations of the convex cost-to-go functions. The distinguishing feature of the SDDP approach is random sampling from the set of scenarios in the forward step of the algorithm. Almost sure convergence of the SDDP algorithm was proved in Philpott and Guan(2008) under mild regularity conditions. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

15 SDDP algorithm Introduced by Pereira and Pinto(1985,1991). It is based on building piecewise linear outer approximations of the convex cost-to-go functions. The distinguishing feature of the SDDP approach is random sampling from the set of scenarios in the forward step of the algorithm. Almost sure convergence of the SDDP algorithm was proved in Philpott and Guan(2008) under mild regularity conditions. An analysis of the SDDP algorithm applied to two stage stochastic programming indicates that its computational complexity grows fast with increase of the number of state variables. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

16 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: true problem Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

17 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SAA Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

18 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Forward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

19 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Forward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

20 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Forward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

21 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Forward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

22 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Backward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

23 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Backward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

24 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Backward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

25 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 1: Backward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

26 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 2: Forward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

27 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 2: Forward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

28 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 2: Forward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

29 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 2: Backward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

30 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 2: Backward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

31 SDDP algorithm: Illustration (a) Stage 1 (b) Stage 2 (c) Stage 3 Figure: SDDP iteration 2: Backward step... Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

32 Outline 1 Motivation 2 Risk neutral approach Risk neutral formulation Methodology SDDP algorithm 3 Worst-case-expectation approach Worst-case-expectation formulation The (WCE) algorithm Analysis of the worst-case-expectation policy 4 Conclusion Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

33 Worst-case-expectation formulation Suppose that ξ t = (ξ 1 t, ξ 2 t ): uncertain parameters : (ξ 1 2,..., ξ1 T ) Ξ1 = Ξ 1 2 Ξ1 T compact random parameters : ξ 2 2,..., ξ2 T with a specified probability distribution Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

34 Worst-case-expectation formulation Suppose that ξ t = (ξ 1 t, ξ 2 t ): uncertain parameters : (ξ 1 2,..., ξ1 T ) Ξ1 = Ξ 1 2 Ξ1 T compact random parameters : ξ2 2,..., ξ2 T with a specified probability distribution Nested formulation: Min c1 T x 1 + ρ 2 ξ1 min [ c2 T x ρ T ξ[t 1] min ] ct T x T A 1x 1=b 1 x 1 0 where ρ t ξ[t 1] [ ] = B 2x 1+A 2x 2=b 2 x 2 0 sup (ξ 1 2,...,ξ1 T ) Ξ1 E ξ 2 [t 1] [ ], t = 2,..., T. B T x T 1 +A T x T =b T x T 0 Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

35 Worst-case-expectation formulation Suppose that ξ t = (ξ 1 t, ξ 2 t ): uncertain parameters : (ξ 1 2,..., ξ1 T ) Ξ1 = Ξ 1 2 Ξ1 T compact random parameters : ξ2 2,..., ξ2 T with a specified probability distribution Nested formulation: Min c1 T x 1 + ρ 2 ξ1 min [ c2 T x ρ T ξ[t 1] min ] ct T x T A 1x 1=b 1 x 1 0 where ρ t ξ[t 1] [ ] = B 2x 1+A 2x 2=b 2 x 2 0 Dynamic Programming equations: Q t (x t 1, ξ 1 t, ξ 2 t ) = sup (ξ 1 2,...,ξ1 T ) Ξ1 E ξ 2 [t 1] [ ], t = 2,..., T. Min B tx t 1 +A tx t=b t, x t 0 B T x T 1 +A T x T =b T x T 0 { } ct T x t + Q t+1 (x t ) where: Q t+1 (x t ) = sup ξ 1 t+1 Ξ 1 t+1 E { Q t+1 (x t, ξ 1 t+1, ξ2 t+1 )} for 2 t T. such that: Q T +1 (.) 0 and B 1 = 0. Key assumption : stagewise independence Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

36 The (WCE) algorithm : Backward step Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

37 The (WCE) algorithm : Backward step At stage t = T for a given bt 1 and b Tj = (bt 1, b2 Tj ), we solve N problems Min c x T R n T T x T s.t. B T x T 1 + A T x T = b Tj, x T 0, j = 1,..., N. T We have : q T ( x T 1, b 1 T ) = N 1 N j=1 Q Tj( x T 1, b 1 T ). Q T ( x T 1 ) = sup q T ( x T 1, bt 1 )? bt 1 Ξ1 T Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

38 The (WCE) algorithm : Backward step At stage t = T for a given bt 1 and b Tj = (bt 1, b2 Tj ), we solve N problems Min c x T R n T T x T s.t. B T x T 1 + A T x T = b Tj, x T 0, j = 1,..., N. T We have : q T ( x T 1, b 1 T ) = N 1 N j=1 Q Tj( x T 1, b 1 T ). Q T ( x T 1 ) = sup q T ( x T 1, bt 1 )? bt 1 Ξ1 T Suppose that we can sample from sets { Ξ 1 } t 2 t T. Sample L points bt 1 l, l = 1,..., L, from Ξ1 T and compute subgradient (at x T 1 ) γ T l = N 1 N j=1 Add the corresponding cutting planes Q Tj ( x T 1, b 1 T l ). q T ( x T 1, b 1 T l ) + γt T l (x T 1 x T 1 ), l = 1,..., L, to the collection of cutting planes of Q T ( ). Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

39 The (WCE) algorithm: Sampling from the uncertainty set We consider as uncertainty set Ξ 1 t, t = 2,..., T, an ellipsoid, centered at ξ t : Ξ 1 t := {ξ : (ξ ξ t ) T A(ξ ξ t ) r t } where A is a positive definite matrix and r t > 0. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

40 The (WCE) algorithm: Sampling from the uncertainty set We consider as uncertainty set Ξ 1 t, t = 2,..., T, an ellipsoid, centered at ξ t : Ξ 1 t := {ξ : (ξ ξ t ) T A(ξ ξ t ) r t } where A is a positive definite matrix and r t > 0. Dominating points in Ξ 1 t : D = { ξ Ξ 1 t : does not exist ξ Ξ 1 t such that ξ ξ and ξ ξ } = arg max { a T ξ : (ξ ξ) T A(ξ ξ) r }. a 0, a =1 where r = ( ξ 1 2 u ) 2, for a given u > 0, and A = Σ 1. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

41 The (WCE) algorithm: Forward step (same as before) Q 2 ( ),..., Q T ( ) and a feasible solution x 1 an implementable policy: For a realization ξ t = (c t, A t, B t, b t ), t = 2,..., T, decisions x t, t = 1,..., T, are computed recursively going forward with x 1 being the chosen feasible solution of the first stage problem, and x t being an optimal solution of Min c x t T x t + Q t+1 (x t ) s.t. A t x t = b t B t x t 1, x t 0, t = 2,..., T t These optimal solutions can be used as trial decisions in the backward step of the algorithm. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

42 The (WCE) algorithm: Summary In the numerical approach outlined above, the key steps are: Problem: Evaluating expectation, Approach: Construct the SAA problem. Problem: Computing maximum for the uncertain parameters, Approach: Sampling from the uncertainty sets. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

43 Analysis of the worst-case-expectation policy Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

44 Analysis of the worst-case-expectation policy Figure: 120 stages policy values for risk neutral and (WCE) (u = 3%) Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

45 Analysis of the worst-case-expectation policy Figure: Individual stage costs for risk neutral and (WCE) (u = 3%) Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

46 Outline 1 Motivation 2 Risk neutral approach Risk neutral formulation Methodology SDDP algorithm 3 Worst-case-expectation approach Worst-case-expectation formulation The (WCE) algorithm Analysis of the worst-case-expectation policy 4 Conclusion Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

47 Conclusion A methodology that combines robust and stochastic programming approaches is suggested. A variant of the SDDP algorithm for solving this class of problems is suggested. Comparison with risk neutral approach: The worst-case-expectation approach constructs a policy that is less sensitive to unexpected demand increase with a reasonable loss on average when compared to the risk neutral method. The computational experiments for our problem show that there is practically no increase in CPU time when compared to the risk neutral approach. Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

48 References A. Eichhorn and W. Römisch, Polyhedral risk measures in stochastic programming, SIAM Journal on Optimization, 16 (2005), V. Guigues and W. Römisch, Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures, SIAM Journal on Optimization, 22 (2012), M.V.F. Pereira and L.M.V.G Pinto, Stochastic optimization of a multireservoir hydroelectric system a decomposition approach, Water Resources Research, 21 (1985), no. 6, M.V.F. Pereira and L.M.V.G. Pinto, Multi-stage stochastic optimization applied to energy planning, Mathematical Programming, 52 (1991), A.B. Philpott and Z. Guan, On the convergence of stochastic dual dynamic programming and related methods, Operations Research Letters, 36 (2008), A.B. Philpott and V.L. de Matos, Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion, E.J.O.R., 218 (2012), A. Ruszczyński and A. Shapiro, Conditional risk mappings, Mathematics of Operations Research, 31 (2006), A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, A. Shapiro, Analysis of stochastic dual dynamic programming method, E.J.O.R., 209 (2011), [Shapiro(2011A)] A. Shapiro, Topics in Stochastic Programming, CORE Lecture Series, Universite Catholique de Louvain, [Shapiro(2011B)] Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15

49 Thank you for your attention! Wajdi Tekaya (CSA) Worst-case-expectation approach July 10, / 15