Chance constrained optimization - applications, properties and numerical issues

Transcription

1 Chance constrained optimization - applications, properties and numerical issues Dr. Abebe Geletu Ilmenau University of Technology Department of Simulation and Optimal Processes (SOP) May 31, 2012

2 This Chancy, Chancy, Chancy World Leonard Rastrigin Uncertainty is the only certainty... John Allen Paulos, Temple University, Philadelphia

3 Topics Chance constrained optimization problems History and major contributions Some application areas Properties and difficulties of chance constraints Approximation strategies for chance constraints A new analytic approximation Conclusion

4 Chance constrained optimization problems (CCOPT ) min x E [f (x, ξ)] (1) s.t. either Pr{g i (x, ξ) 0} α i, i = 1,..., m. (2) or Pr{g i (x, ξ) 0, i = 1,..., m} α. (3) f, g : R n R p R are at least differentiable w.r.t. x X R n ; x- a vector of deterministic variables; ξ Ω R p - a vector of random variables with joint probability density function φ(ξ).

5 Chance constrained optimization problems... Pr{ }, E[ ] are probability, expectation operators; Pr{g(x, ξ) 0} α - chance or probability constraint. 1 2 α 1 probability (reliability) level. Single chance constraints Joint chance constraints Pr{g i (x, ξ) 0} α i, i = 1,..., m. Pr{g i (x, ξ) 0, i = 1,..., m} α. The random vector ξ: can be either a Gaussian or non-gaussian. (See Geletu et al for recent review article.)

6 History and major contributions The beginning: Charnes, Cooper & Symonds 1958, Major contribution: Prekopa 1972,1973, 1995, 2001, Major references: Ben-Tal, El Ghaoui & Nemirovski: Robust optimization, Birge & Lauveaux: Introduction to stochastic programming, Kall & Wallace : Stochastic Programming, Kibzun & Kan : Stochastic programming problems. with probability and quantile functions, Marti: Stochastic Optimization Methods, Prékopa: Stochastic programming, Ruszczyński & Shapiro: Stochastic programming, 2003.

7 Some application areas Classical application areas Water reservoir management Optimal power flow (OPF) Financial risk management (using risk metrics like VaR and cvar) Reliability based (engineering) design optimization (RBDO) Modern applications Control and optimization based on prediction (eg. using weather forecast data, etc.) Reliable navigation and obstacle avoidance in unmanned autonomous ground/areal vehicles Optimal and reliable wind/soalr power generation Reliability and fault-tolerance in mechatronic systems etc.

8 Some application areas...

9 Some application areas...

10 Some application areas... Some related terminologies Chance Constraints Robustness Reliability /Zuverlässigkeit Risk metrics Risk management Fault tolerance Chance Constraints Worst case scenario Chance Constraints min Pr{Failure} max Pr{Reliability}

11 Properties and difficulties of chance constraints Meaning of chance constraints: Z := g(x, ξ) is a random variable. Hence, Pr{g(x, ξ) 0} + Pr{g(x, ξ) > 0} = 1. irrespective of the distribution of Z = g(x, ξ). reliability constraint For α near 1, Pr{g(x, ξ) 0} α; eg. α = 0.95, α = risk constraint Pr{g(x, ξ) > 0} 1 α; eg. 1 α = 0.05, α = x is feasible iff Pr{g(x, ξ) 0} α holds with reliability α. Let p(x) := Pr {g(x, ξ) 0}. Then the feasible set of CCOPT is P := {x X p(x) α}.

12 Properties and difficulties of chance constraints Important issues (A) Continuity (see Kibzun & Kan 1996, Birge & Lauveaux 1997, Prekopa 1995). (B) Differentiability of p( ) Uryasév 1994, Marti (C) Convexity of the feasible set P Prekopa 1995, Henrion & Strugarek. (D) Stability and regularity Henrion and Römisch (E) How to determine if a given x is feasible for CCOPT or not? (F) For a given x, how to determine the value p(x) = Pr {g(x, ξ) 0}? In general, (E) & (F) are not trivial issues.

13 Properties and difficulties of chance constraints Major difficulty: the value p(x) = Pr{g(x, ξ) 0} = {ξ Ω g(x,ξ) 0} is difficult to compute. Hence, CCOPT is known to be a hard-problem. But there are some special cases: Example 1: g(x, ξ) = a x + b ξ, ξ R and ξ N(µ, σ 2 ). Then φ(ξ)dξ Pr{g(x, ξ) 0} = Pr{a x + b ξ} = 1 Pr{a x + b ξ} { ( ξ µ a x + b ) } µ ( ) = 1 Pr = 1 Φ a x + b. σ σ

14 Properties and difficulties of chance constraints Example 1: contd... Hence, if g(x, ξ) = a x + b ξ, ξ R and ξ N(µ, σ 2 ), then ( ) Pr{g(x, ξ) 0} α Φ 1 (1 α) a x + b 0. The expression on the left is an exact analytic representation of the chance constraint. Example 2:(separable form) g(x, ξ) = g(x) + c ξ + b with ξ N(µ, σ 2 ). Example 3:(a linear transformation of ξ) g(x, ξ) = (Ax + b) ξ + b with ξ N(µ, σ 2 ).

15 Properties and difficulties of chance constraints... An analytic representation can lead to a nonlinear optimization problem. Example 4 (Theorem 6.2., P. 83, B. Liu 2002) Let ξ = (ξ 1,..., ξ n, ξ n+1 ) and g(x, ξ) = x 1 ξ x n ξ n ξ n+1. If ξ 1,..., ξ n, ξ n+1 are independently normally distributed random variables, then Pr{g(x, ξ) 0} α n E[ξ i ]x i +Φ 1 (α) n V [ξ i ]xi 2 + V [ξ n+1 ] E[ξ n+1 )], i=1 i=1 where Φ is the standard normal distribution.

16 Properties and difficulties of chance constraints... The above result can be extended to { } p Pr w 0 (x) + ξ k w k (x) 0 α. (4) k=1 Garnier et al (Linearization) If ξ = (ξ 1,..., ξ p ) is a normal multivariate random vector with mean 0 and small variance, then which leads to (??) g(x, ξ) g(x, 0) + p i=1 g ξ i (x, 0)ξ, In general, there is no closed-form analytic representation. Nemirovski 2012 studies tractability of (??) for non-gaussian ξ.

17 Approximation strategies for chance constraint optimization It remains to use approximation methods. Approximation strategies (I) Back-mapping (II) robust optimization (III) sample average approximation

18 I. Back-mapping (projection) method Idea of back-projection (Wendt, Li and Wozny 2002): Find a monotonic relation between Z = g(x, ξ) and some random variable ξ j ; i.e. verify theoretically (see Geletu et al. 2010) or experimentally that there is a real-valued function ϕ such that, for any x X, Z = ϕ x (ξ j ); ϕ x ( ) is strictly increasing (ξ j Z) or decreasing (ξ j Z). ξ j = ϕ 1 x (Z). ξ Z Pr {g(x, ξ) 0} = Pr { ξ j ϕ 1 x (0) }. ξ Z Pr {g(x, ξ) 0} = Pr { ξ j ϕ 1 x (0) }.

19 Back projection... Abbildung: Back Projection of chance constraints Requires a global implicit function theorem (see Geletu et al. 2011).

20 I. Back-projection... Now, for ξ j Z, (CCOPT) is equivalent to (CCOPT ) s.t. min E [f (x, ξ)] x p(x) = Pr { ξ j ϕ 1 x (0) } α, u U. p(x) = p(x) = ϕ 1 x (0) φ(ξ)dξ (5) x ϕ 1 x (0)φ(ξ)dξ (6)

21 I. Back-projection... Advantages: usable if monotonic relations are easy to find; provides direct representation of chance constraints. Disadvantages: monotonic relations may not exist; monotonic relations can be difficult to verify.

22 II. Sample average approximation (SAA) Shapiro 2003, Pagnoncelli et al Define { 0, if g(x, ξ) > 0 I (0,+ ] (g(x, ξ)) = 1, if g(x, ξ) 0. Determine samples {ξ 1,..., ξ N } Ω (e.g. low discrepancy sequences like Fourer, Sobol or Niederreiter sequences). Replace the chance constrains with p N (x) = 1 N N k=1 ) I (,0] (g(x, ξ k ) α. (p N (x) =Relative-frequency count for the satisfaction of g(x, ξ) 0. )

23 II. Sample average approximation... Advantages: (SAA) avoids computation of multidimensional integrals; Convexity structures are preserved. Disadvantages: SAA leads to a non-smooth optimization problem. Feasibility of the obtained solution to the (CCOPT) is guaranteed only when N.

24 III. Robust optimization technique Robust optimization considers the (worst-case) problem (RO) min E [f (x, ξ)] x s.t. g(x, ξ) 0, ξ Ω, x X, where g(x, ξ) 0 is required to be satisfied for as many realizations of ξ from Ω as possible. Randomized solution based on scenario generation: Generate independent identically distributed random samples ξ 1,..., ξ N from Ω (Monte-Carlo method).

25 III. Robust optimization technique... Solve the optimization problem (NLP) RO 1 min x N N f (x, ξ k ) k=1 x X. Theorem (Califore & Campi 2005) s.t. g(x, ξ k ) 0, k = 1,..., N; Suppose α (0, 1) and f (, ξ) is convex w.r.t. x R n. If the number of random samples ξ 1,..., ξ N N 2n ( ) ( ) 1 2 (1 α) ln + ln 1 α 1 α ( ) 1 + 2n, α then the optimal solution obtained from (NLP) RO is an optimal solution of (RO) with reliability α.

26 III. Robust optimization technique... Advantages: there is no need to compute integrals; the problem (NLP) RO simple to implement and solve; it also preserves convexity structures. Disadvantages: solution of (NLP) RO may not be feasible to the (CCOPT); for a higher reliability level α, very large number of scenarios ξ 1,..., ξ N are required. Recent suggestion: scenarios reduction methods: Henrion, Küchler & Römisch 2009; Campi & Garetti 2011.

27 A new analytic approximation strategy Let This implies h(x, ξ) = { 0, if g(x, ξ) 0 1, if g(x, ξ) > 0. Pr{g(x, ξ) > 0} = E [h(x, ξ)] Pr{g(x, ξ) 0} α E [h(x, ξ)] 1 α. A general idea of analytic approximation (Geletu et al. 2012): Find a continuous (possibly smooth) parametric function ψ(τ, ) so that where τ > 0. E[h(x, ξ)] ψ(τ, x),

28 A new analytic approximation strategy... Hence M(τ) = {x ψ(τ, x) 1 α} P := {x E[h(x, ξ)] 1 α}.

29 A new analytic approximation strategy... Requirements on ψ(τ, x): (a) for each fixed τ, ψ(τ, x) should be easy to compute. (b) For each fixed x inf ψ(τ, x) = E [h(x, ξ)] τ>0 (b) sup τ M(τ) = P. Question: How to construct such a function ψ(τ, x)? Basically, ψ(τ, x) = E [Ψ(τ, g(x, ξ))], for Ψ(τ, s) : R + R R.

30 A new analytic approximation strategy... Some suggestions: Ψ 1 (τ, s) = exp(τs), τ > 0 (Pinter 1984) Ψ 2 (τ, s) = exp(τ 1 s), τ > 0 (Nemirovski and Shapiro 2006, Nemirovski 2011). Ψ 3 (τ, s) = τ α [s τ, 0] +, τ > 0 (Rockafellar and Uryasev 2000), where [s τ, 0] + = max{s τ, 0}. A new analytic approximation function (Geletu et al. 2012) Ψ 4 (τ, s) = with 0 < m 2 m 1 and τ > m 1τ 1 + m 2 τe 1 τ s

31 A new analytic approximation strategy... Abbildung: Comparison of Ψ 2 and Ψ 4

32 A new analytic approximation strategy... Theorem (A. Geletu 2012) 1 Let τ > 0. Then Ψ 4 (τ, s) > 0 for any value of s; Ψ 4 (τ, s) 1 for s 0. 2 For 0 < τ 1 2, then the function Ψ 4 (, s) non-decreasing w.r.t. τ; Hence, lim Ψ 4(τ, s) = τ 0 + { 1, if s 0 0, if s < 0 Hence, for any s R : lim τ 0 + Ψ 4 (τ, s) I [0,+ ) (s) = 0. lim ψ(τ, x) = lim E [Ψ 4(τ, g(x, ξ))] = lim E [h(x, ξ)] τ 0 + τ 0 + τ 0 +

33 A new analytic approximation strategy... Instead of CCOPT solve the problem (NLP) τ min E [f (x, ξ)] x (7) s.t. (8) x M(τ) = {x X ψ(τ, x)} 1 α, for each τ with the same objective function E [f (x, ξ)] as in (CCOPT).

34 A new analytic approximation strategy... Theorem (Geletu et al. 2012) If {τ k } any sequence, such that τ k 0 + and X is a compact set, then for each τ k, the feasible set M(τ k ) is compact. lim k + M(τ k ) = P. (in fact, lim k + H(M(τ k ), P) = 0 ) If x X is a limit point of a sequence {x k } of optimal solutions of (NLP) τk, then x is a solution of CCOPT. Computational advantage: Solution of NLP τ for a sufficiently small τ provides a good approximation for CCOPT.

35 A new analytic approximation strategy... Advantages: CCOPT becomes more tractable NLP τ can be solved using state-of-the-art optimization solvers solution of NLP τ is always feasible to the CCOPT Disadvantages: Objective and constraints may require the evaluation of high dimensional integrals convexity structures of CCOPT may not be preserved in the new approximation currently, the approximation works only for single chance constraints

36 Conclusions Analytic approximation facilitates tractability of nonlinear chance constrained optimization problems. For fast processes, linearization and analytic approximation might be imperative. Large-scale problems require faster evaluation of multidimensional probability integrals. This can be achieved through: efficient numerical integration techniques (like QMC, sparse-grid, etc., integration ) the design of new numerical integration technique parallel computation, etc.

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