Elsevier

Operations Research Letters

Volume 46, Issue 5, September 2018, Pages 543-547
Operations Research Letters

Quantitative stability of multistage stochastic programs via calm modifications

https://doi.org/10.1016/j.orl.2018.08.007Get rights and content

Abstract

In this paper, we revisit the quantitative stability of multistage stochastic programs. Different from the single calm modification used in Küchler (2008), we introduce two types of calm modifications which leads to a much simpler proof and tighter upper bound for the difference of optimal values of multistage stochastic programs under different stochastic processes than those of Küchler (2008). In addition, we avoid those restrictive assumptions in Küchler (2008) and the filtration distance in Heitsch et al. (2006). Finally, we illustrate our results with two numerical examples.

Introduction

We consider the following T-stage (T2) stochastic linear program (see [[6], [12]]): infx1D1c1,x1+E[infx2D2(x1,ξ2)c2(ξ2),x2+E[infx3D3(x2,ξ3)c3(ξ3),x3++E[infxTDT(xT1,ξT)cT(ξT),xT]]],where , denotes the inner product in finite dimensional Hilbert space. ξ=(ξt)t=1TLT(Ω,F,P;RsT) is a RsT-valued stochastic process defined on the probability space (Ω,F,P), with finite Tth order absolute moments, and x=(xt)t=1T is the sequence of decision variables. We use bold letters, for example ξ or x, to denote random vectors in contrast to their realizations ξ or x. The corresponding filtration of ξ is {Ft}t=1T, defined by Ft=σ(ξt) for t=1,2,T, here ξt(ξ1,ξ2,,ξt) and especially ξT=ξ. The similar notation is adopted for other variables. Of course, we have that {,Ω}=F1F2FT=F. Specially, F1={,Ω} indicates that ξ1=ξ1 is deterministic. For t=1,2,,T, we use Ξt and Ξt to denote the support sets of ξt and ξt, respectively. The corresponding probability measures are denoted by Pt and Pt. Cost vectors c1Rn and ct:ΞtRn, t=2,,T, are affinely linear mappings with respect to ξt. D1Rn and the feasible solution multifunctions Dt:Xt1×ΞtRn are defined by Dt(xt1,ξt)={xtXtRn:Atxt+Bt(ξt)xt1=ht(ξt)},where Xt are nonempty polyhedral sets; the recourse matrix AtRm×n, the technology matrix Bt:ΞtRm×n and the right-hand side vector ht:ΞtRm are also affinely linear mappings with respect to ξt for t=2,3,,T. The affine linearity of c(ξ)(c1(ξ1),c2(ξ2),,cT(ξT)), means that c(ξ)c(ξˆ)Kξξˆ,c(ξ)Kmax{1,ξ}hold for some K1 and any ξ,ξˆΞT.

In the last decade, the stability analysis of multistage stochastic programs has been investigated in a number of works, see, for example, [[2], [4], [5], [10], [12]] and the references therein. The quantitative stability results have a significant impact on suitable methods for approximating the underlying continuous data process, which in return make it possible for us to solve original multistage stochastic programs by solving large scale deterministic optimization problems.

In early work [3], the authors studied the quantitative stability by assuming implicitly that filtrations of the original and approximate stochastic processes are consistent. To extend this result to a general situation where the filtration is also perturbed, Römisch and his coauthors employed the so-called filtration distance to describe the variation of filtrations in [4]. The main reason for introducing the filtration distance was that they adopted the feasible solutions in the α-level set to describe the optimal values under different stochastic processes. Then, Eichhorn and Römisch extended in [2] the risk-neutral result in [4] to the risk-averse case with polyhedral risk measures introduced in [1]. Considering the computational difficulty of filtration distances, Küchler used in [5] some strong recursive assumptions to avoid the filtration distance, and obtained the quantitative stability assertion for a class of measurable perturbations. For more information on this topic, we refer readers to [[7], [10], [11]] and the references therein.

In this paper, we aim at simplifying the proof and strengthening the quantitative stability conclusion in [5] by adopting two types of calm modifications. Our quantitative stability result also avoids the tricky filtration distance. For these purposes, we adopt the following notation in what follows. ξtmax1itξi, here ξi is the Euclidean norm in Rs for 1it and t=1,2,,T. Analogously, we define xt=max1itxi and xi is the Euclidean norm in Rn for 1it and t=1,2,,T. For sets S1,S2Rn, d(S1,S2)sups1S1d(s1,S2), here d(s1,S2)infs2S2s1s2 and dH(S1,S2)max{d(S1,S2),d(S2,S1)}. υ(ξ) denote the optimal value of problem (1) under the stochastic process ξ.

We need the following Lipschitzian results about Dt, t=2,,T, which can be found in [8, Example 9.35].

Proposition 1.1

For Dt(xt1,ξt),t=2,,T, defined in (2) , the following assertions hold: dH(Dt(xt1,ξt),Dt(xˆt1,ξt))Bmax{1,ξt}xˆt1xt1,dH(Dt(xt1,ξt),Dt(xt1,ξˆt))Bmax{1,xt1}ξˆtξtfor some constant B>0.

If we define F(x,ξ)=t=1Tct(ξt),xt, model (1) can be equivalently rewritten as (see, for example, [[4], [5], [8]]) min{E[F(x,ξ)]:xD(ξ)},where D(ξ) is a collection of decision processes x=(x1,x2,,xT) with x1D1 and measurable mappings xtDt(xt1,ξt) for t=2,,T.

Another way to reformulate the multistage stochastic linear program (1) is the dynamic programming method. Concretely, let Qt:Xt1×ΞtR denote the recourse function at the tth stage, which is defined recursively by Qt(xt1,ξt)=infxtDt(xt1,ξt)ct(ξt),xt+EQt+1(xt,ξt+1)|ξt=ξtfor t=T,T1,,1, here QT+10 and x01. Then problem (1) is equivalent to minx1D1c1,x1+EQ2(x1,ξ2).

Of particular interest in this paper, we consider the following measurable perturbation of ξ.

Definition 1.2 Approximation of Stochastic Process, [5]

A stochastic process ξ̃ on the probability space (Ω,F,P) is called an approximation of ξ, if there exist measurable mappings ft:ΞtΞt for t=1,2,,T, such that the following conditions are satisfied:

(a) ξ̃t=ft(ξt) for t=1,2,,T;

(b) fT(ΞT)ΞT;

(c) f1(ξ1)=ξ1 for every ξ1Ξ1;

(d) fT(ξT)LT(Ω,F,P;RsT).

Here, ft(ξt)(f1(ξ1),f2(ξ2),,ft(ξt)) for t=1,2,,T.

To guarantee that the feasible solution set in each stage under the perturbed stochastic process ξ̃ is nonempty, we need the following commonly used assumption.

Assumption 1

There exists a δ>0 such that for any perturbed stochastic process ξ̃ with ξ̃ξδ, x1D1 and xτDτ(xτ1,ξ̃τ), τ=2,,t1, Dt(xt1,ξ̃t) is nonempty Pt-a.s. for t=2,3,,T.

The relatively complete recourse assumption is widely used in the stability analysis of stochastic programs, see the review [9] for two-stage stochastic programming problems and [[2], [4]] for the multistage case.

To continue our discussion, in the same way as that in [5, Assumption 2.3], we introduce the following growth condition.

Assumption 2

There exists a positive number C1 such that, for every measurable mapping xt1:Ξt1Xt1, there exists an optimal solution xt(ξt) to problem (3) such that xt(ξt)Cmax{1,xt1(ξt1)}max{1,ξt},Pta.s.for t=2,3,,T. Specially, we have x1C for any x1D1.

Remark 1.3

Assumption 2 holds automatically when Xt, 1tT, are bounded. From Assumption 2, there exists a subset Ξ̄tΞt with Pt(ΞtΞ̄t)=0, such that for any ξtΞ̄t, we have xt(ξt)Cmax{1,xt1(ξt1)}max{1,ξt}=Cmaxxt1(ξt1)max{1,ξt}C2max{1,xt2(ξt2)}max{1,ξt1}max{1,ξt}C2xt2(ξt2)max{1,ξt}2.Then, we recursively obtain xt(ξt)Ctmax{1,ξt}t1,Pta.s.,t=2,,T.

Section snippets

Main results

We present our main results about quantitative stability of multistage stochastic linear programs when the original stochastic process is perturbed by an approximation defined in Definition 1.2. To this end, we introduce the following two types of calm modifications.

Definition 2.1 Calm Modifications

For an optimal solution x under stochastic process ξ satisfying the growth condition (4), we call

  • (i)

    x̄(ξˆ)=(x̄1,x̄2(ξˆ2),,x̄T(ξˆT)) the class I calm modification

Numerical examples

To illustrate our theoretical results, we consider here two examples.

Example 3.1

Example 2.6 in [4]

Consider the following multistage stochastic programming problem: minEt=1Tξtxt(xt,st)R+2,(xt,st)isFtmeasurable,stst1=xt,t=2,,T,s1=0,sT=a>0.,where R+2={yR2:y0}. We consider the 3-stage scenario trees for ξϵ and ξ̃ described in Fig. 1. That is, ξϵ1=(3,2+ϵ,3+ϵ) for ϵ>0, ξϵ2=(3,2,3) and ξϵ3=(3,2,1) with the same probability being 13. Its perturbation stochastic process is ξ̃1=(3,2,3) and ξ̃2=(3,2,1) endowed

Acknowledgments

This work was supported by the National Natural Science Foundation of China [grant numbers 11571270 and 71371152].

References (12)

  • ReaicheM.

    A note on sample complexity of multistage stochastic programs

    Oper. Res. Lett.

    (2016)
  • RömischW.

    Stability of stochastic programming problems

    Handbooks Oper. Res. Management Sci.

    (2003)
  • EichhornA. et al.

    Polyhedral risk measures in stochastic programming

    SIAM J. Optim.

    (2005)
  • EichhornA. et al.

    Stability of multistage stochastic programs incorporating polyhedral risk measures

    Optimization

    (2008)
  • FiedlerO. et al.

    Stability in multistage stochastic programming

    Ann. Oper. Res.

    (1995)
  • HeitschH. et al.

    Stability of multistage stochastic programs

    SIAM J. Optim.

    (2006)
There are more references available in the full text version of this article.

Cited by (3)

View full text