Sample Average Approximation Method for Chance Constrained ...

More documents

Recommendations

Info

distribution in chance constraint by an empirical distribution corresponding to a randomsample. This approach is well known for stochastic programs with expected valuesobjectives [9]. SAA methods for chance constrained problems have been investigatedin [10] and [11].The remainder of the paper is organized as follows. In Section 2 we provide theoreticalbackground for the SAA approach, showing convergence of the optimal value of theapproximation to the optimal value of the true problem. In addition, following [8] wedescribe how to construct bounds for the optimal value of chance constrained problemsof the form (1). In Section 3, we present a chance constrained portfolio selection problem.We apply the SAA method to obtain upper bounds as well as candidate solutionsto the problems. In addition we present several numerical experiments that indicatehow one should tune the parameters of the SAA approach. In Section 4 we presenta simple blending problem modeled as a joint chance constrained problem. Section 5concludes the paper and suggests directions for future research.We use the following notation throughout the paper. The integer part of numbera ∈ R is denoted by ⌊a⌋. By Φ(z) we denote the cumulative distribution function (cdf)of standard normal random variable and by z α the corresponding critical value, i.e.,4
Φ(z α ) = 1 − α, for α ∈ (0, 1),B(k; p, N) := ∑ k( N)i=0 i p i (1 − p) N−i , k = 0, ..., N, (2)denotes the cdf of binomial distribution. For sets A, B ⊂ R n we denote byD(A, B) := sup x∈A dist(x, B) (3)the deviation of set A from set B.2 Theoretical BackgroundIn order to simplify the presentation we assume in this section that the constraintfunction G : R n ×Ξ → R is real valued. Of course, a number of constraints G i (x, ξ) ≤ 0,i = 1, . . .,m, can be equivalently replaced by one constraint withG(x, ξ) := max1≤i≤m G i(x, ξ) ≤ 0.Such operation of taking maximum preserves convexity of functions G i (·, ξ). We assumethat the set X is closed, the function f(x) is continuous and the function G(x, ξ) is a5
Page 1 and 2: Sample Average Approximation Method
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 14 and 15: us choose two positive integers M a
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 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?