Sample Average Approximation Method for Chance Constrained ...

More documents

Recommendations

Info

us choose two positive integers M and N, and letθ N := B ( ⌊γN⌋; α, N )and L be the largest integer such thatB(L − 1; θ N , M) ≤ β. (12)Next generate M independent samples ξ 1,m , . . .,ξ N,m , m = 1, . . ., M, each of size N, ofrandom vector ξ. For each sample solve the associated optimization problemNmin f(x), s.t.x∈X∑ (1l (0,∞) G(x, ξ j,m ) ) ≤ γN, (13)j=1and hence calculate its optimal value ˆϑ m N , m = 1, . . ., M. That is, solve M times thecorresponding SAA problem at the significance level γ. It may happen that problem(13) is either infeasible or unbounded from below, in which case we assign its optimalvalue as +∞ or −∞, respectively. We can view ˆϑ m N , m = 1, . . ., M, as an iid sample ofthe random variable ˆϑ N , where ˆϑ N is the optimal value of the respective SAA problemat significance level γ. Next we rearrange the calculated optimal values in the nonde-14
creasing order as followsˆϑ(1)N≤ · · · ≤ ˆϑ(M)N, i.e., ˆϑ(1)N(2)is the smallest, ˆϑNis the secondsmallest etc, among the values ˆϑ m (L)N , m = 1, . . .,M. We use the random quantity ˆϑNasa lower bound of the true optimal value ϑ ∗ . It is possible to show that with probabilityat least 1 − β, the random quantityˆϑ(L)Nis below the true optimal value ϑ∗ , i.e.,ˆϑ(L)Nis indeed a lower bound of the true optimal value with confidence at least 1 − β (see 3[8]). We will discuss later how to choose the constants M, N and γ.3 A Chance Constrained Portfolio ProblemConsider the following maximization problem subject to a single chance constraintmax E[ r T x ] , s.t. prob { r T x ≥ v } ≥ 1 − α. (14)x∈XHere x ∈ R n is vector of decision variables, r ∈ R n is a random (data) vector (withknown probability distribution), v ∈ R and α ∈ (0, 1) are constants, e is a vector whosecomponents are all equal to 1 andX := {x ∈ R n : e T x = 1, x ≥ 0}.3 In [8] this lower bound was derived for γ = 0. It is straightforward to extend the derivations tothe case of γ > 0.15
Page 1 and 2: Sample Average Approximation Method
Page 4 and 5: distribution in chance constraint b
Page 6 and 7: Carathéodory function, i.e., G(x,
Page 8 and 9: (i) for any sequence x k converging
Page 10 and 11: y a version of the uniform law of l
Page 12 and 13: f(¯x) ≥ ϑ ∗ . Also f(ˆx N )
Page 16 and 17: Note that, of course, E [ r T x ] =
Page 18 and 19: problem (5), with γ = 0, to be inf
Page 20 and 21: estimation of the parameters was ta
Page 22 and 23: original problem, we observe in Fig
Page 24 and 25: 3.3 Upper BoundsA method to compute
Page 26 and 27: priate choice of the parameters sin
Page 28 and 29: It is possible to write the SAA of
Page 30 and 31: 5 ConclusionsWe have discussed chan
Page 32 and 33: References1. Charnes, A., Cooper, W
Page 34 and 35: 15. Markowitz, H.: Portfolio select
Page 36: 1.011.0051.00.99530 80 130 1800.10.

Sample Average Approximation Method for Chance Constrained ...

Create successful ePaper yourself

Delete template?

Save as template?