On complexity of multistage stochastic programs under heavy tailed distributions

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Abstract

In this paper, the complexity of sample average approximation (SAA) of multistage stochastic programs under heavy tailed distributions is investigated. Specifically, we estimate confidence levels when the accuracy parameter and sample size are given under independently and identically distributed (iid) and non-iid conditional samples, respectively. Different from the existing works, we emphasize the impact of heavy tailed distributions, non-iid conditional sampling and stages dependence of the random process in multistage stochastic programs.

Introduction

Multistage stochastic programming is an important branch in stochastic optimization and has plenty of applications [12], [15], [20].To solve (multistage) stochastic programs, there are usually two commonly-used methods: decision rules (see e.g. [5], [6]) and scenario-based approaches (see e.g. [3], [7], [11], [17]). By scenarios, the intractable multistage stochastic programs are approximated by large scale scenario-based problems, which thus can be numerically solved. A follow-up question is: how does the scenario-based problem approximate the true counterpart? The research along that line can be roughly divided into two parts: stability analysis (see e.g. [8], [9]) and convergence analysis (see e.g. [16], [18], [19]). In this paper, we focus on the quantitative convergence analysis of multistage stochastic programs.

Compared with the rich literature on single-stage or two-stage stochastic optimization problems (see e.g. [11], [20]), works on the SAA convergence analysis of multistage cases are relatively few. In [18], Shapiro considered the conditional sampling procedure for multistage stochastic programs, and showed the asymptotic consistency between conditional sampling SAA problem and the original problem. For the complexity analysis of multistage stochastic programs, Shapiro discussed, in [19], the rate of convergence which can be used to derive how large the sample size should be to ensure the certain accuracy and confidence level. More recently, Reaiche [16] developed a lower bound for the sample complexity of a class of multistage stochastic programs.

However, there are still some interesting points that have not been stressed in existing works, such as heavy tailed distributions (whose tails fail to decay exponentially), stages dependence of random process, non-iid conditional sampling. For example, heavy tailed distributions have wide applications in financial risk management, insurance policy selection and analysis of resident income. The relevant data are significantly skewed with high leptokurtosis [14].

In view of this, this paper makes the following main contributions. Firstly, the stages dependence of the random process is considered. Secondly, we derive the complexity for both iid and non-iid conditional sampling multistage stochastic programs under heavy tailed distributions. Furthermore, we develop not only the convergence of objective functions but also the first stage here-and-now optimal solution sets, which is ignored in the existing works. It is noteworthy that this paper employs some results in [10], where single-stage or/and two-stage problems under heavy tailed distributions are considered, to help the derivation. However, we here stress the methodologies to multistage case under conditional sampling and heavy tailed distributions, which is different from [10].

The rest of the paper is organized as follows. In Section 2, dynamic multistage stochastic programming model and its SAA problem under conditional sampling are presented. In Section 3, iid conditional sampling case is investigated under heavy tailed distributions. Non-iid case is considered in Section 4. Some final remarks are given in Section 5.

Section snippets

Models

A T-stage stochastic programming problem (see e.g. [12], [15], [20]) can be written as: minx1X1f1(x1)+E[minx2X2(x1,ξ2)f2(x2,ξ2)+E[+E[minxTXT(xT1,ξT)fT(xT,ξT)]]],where xtRnt,t=1,,T are decision variables (Rk denotes the k-dimensional real space hereafter for k=1,2,); ξ2,,ξT are a sequence of random vectors (or random process) defined on sample space Ωt with ξt:Ωtsupp(ξt)Rst for t=2,,T, where supp(ξt) denotes the support set of ξt; X1Rn1 is deterministic and nonempty, Xt:Rnt1×RstRnt

SAA of (2) under iid conditional sampling

In this section, we assume that ξ21,,ξ2N1 are iid and for each i=1,,N1, ξ3i1,,ξ3iN2 are iid, too. To proceed, we need some auxiliary results.

Let ξ be a random vector supported on supp(ξ)Rs and XRn. Denote F:X×supp(ξ)R and κ:supp(ξ)R+ (R+ means the nonnegative real numbers). If F(x,ξ) is finite and there exist γ>0 and δ>0 such that ( denotes the Euclidean norm hereafter) |F(x,ξ)F(x,ξ)|κ(ξ)xxγfor all xX with xxδ and ξsupp(ξ), F is said to be H-calm at xX with modulus κ(ξ

Non-iid conditional sampling case

In this section, we relax the condition of iid random sample. That is, ξ21,,ξ2N1 are non-iid and ξ3i1,,ξ3iN2 are also non-iid for i=1,,N1. We need some preparations.

Let Y be a random variable. Y1,,YN are N general (probably non-iid) samples of Y.

Assumption 3

There exist α:RR+ being non-decreasing, β:NR+ with β(N) as N and C0>0, independent of N, such that lim supNEβ(N)α1Ni=1NYiC0andlim supNEβ(N)α1Ni=1NYiC0.

If Assumption 3 holds, we will say Y1,,YN satisfy Assumption 3 and so on in a

Conclusion

In this paper, we take T=3 for example to show rates of convergence under iid and non-iid conditional samplings relative to heavy tailed distributions. In fact, by a similar procedure, we can extend these results to the general T-stage case. For example, under certain assumptions (similar to Assumption 1, Assumption 2), (14) can be extended to Probsupx1X1|fˆN1,,NT1(x1)f(x1)|ϵC2N1p2ϵp++CTN1NT1NT1p2ϵp,where N1,,NT1 are sample sizes of stages 2 ,,T, and C2,,CT>0.

There are still some

Acknowledgments

This paper was supported by the Postdoctoral Science Foundation of China [Grant Number 2020M673117], Fundamental Research Funds for the Central Universities, China [Grant Number 2020CDJQY-A039] and the National Natural Science Foundation of China [Grant Number 11971078]. The authors would like to thank editors and the anonymous referee for the suggestions and comments which have helped to improve the presentation and quality of this paper greatly.

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