Quantitative stability of multistage stochastic programs via calm modifications
Introduction
We consider the following -stage stochastic linear program (see [[6], [12]]): where denotes the inner product in finite dimensional Hilbert space. is a -valued stochastic process defined on the probability space , with finite th order absolute moments, and is the sequence of decision variables. We use bold letters, for example or , to denote random vectors in contrast to their realizations or . The corresponding filtration of is , defined by for , here and especially . The similar notation is adopted for other variables. Of course, we have that . Specially, indicates that is deterministic. For , we use and to denote the support sets of and , respectively. The corresponding probability measures are denoted by and . Cost vectors and , , are affinely linear mappings with respect to . and the feasible solution multifunctions are defined by where are nonempty polyhedral sets; the recourse matrix , the technology matrix and the right-hand side vector are also affinely linear mappings with respect to for . The affine linearity of , means that hold for some and any .
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 -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. , here is the Euclidean norm in for and . Analogously, we define and is the Euclidean norm in for and . For sets , , here and . denote the optimal value of problem (1) under the stochastic process .
We need the following Lipschitzian results about , , which can be found in [8, Example 9.35].
Proposition 1.1 For , defined in (2) , the following assertions hold: for some constant .
If we define , model (1) can be equivalently rewritten as (see, for example, [[4], [5], [8]]) where is a collection of decision processes with and measurable mappings for .
Another way to reformulate the multistage stochastic linear program (1) is the dynamic programming method. Concretely, let denote the recourse function at the th stage, which is defined recursively by for , here and . Then problem (1) is equivalent to
Of particular interest in this paper, we consider the following measurable perturbation of .
Definition 1.2 Approximation of Stochastic Process, [5] A stochastic process on the probability space is called an approximation of , if there exist measurable mappings for , such that the following conditions are satisfied: (a) for ; (b) ; (c) for every ; (d) . Here, for .
To guarantee that the feasible solution set in each stage under the perturbed stochastic process is nonempty, we need the following commonly used assumption.
Assumption 1 There exists a such that for any perturbed stochastic process with , and , , is nonempty -a.s. for .
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 such that, for every measurable mapping , there exists an optimal solution to problem (3) such that for . Specially, we have for any .
Remark 1.3 Assumption 2 holds automatically when , , are bounded. From Assumption 2, there exists a subset with , such that for any , we have Then, we recursively obtain
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 under stochastic process satisfying the growth condition (4), we call
the class I calm modification
Numerical examples
To illustrate our theoretical results, we consider here two examples.
Example 3.1 Consider the following multistage stochastic programming problem: where . We consider the -stage scenario trees for and described in Fig. 1. That is, for , and with the same probability being . Its perturbation stochastic process is and endowed Example 2.6 in [4]
Acknowledgments
This work was supported by the National Natural Science Foundation of China [grant numbers 11571270 and 71371152].
References (12)
A note on sample complexity of multistage stochastic programs
Oper. Res. Lett.
(2016)Stability of stochastic programming problems
Handbooks Oper. Res. Management Sci.
(2003)- et al.
Polyhedral risk measures in stochastic programming
SIAM J. Optim.
(2005) - et al.
Stability of multistage stochastic programs incorporating polyhedral risk measures
Optimization
(2008) - et al.
Stability in multistage stochastic programming
Ann. Oper. Res.
(1995) - et al.
Stability of multistage stochastic programs
SIAM J. Optim.
(2006)
Cited by (3)
On complexity of multistage stochastic programs under heavy tailed distributions
2021, Operations Research LettersCitation Excerpt :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.
QUANTITATIVE STABILITY OF THE ERM FORMULATION FOR A CLASS OF STOCHASTIC LINEAR VARIATIONAL INEQUALITIES
2022, Journal of Industrial and Management OptimizationSTABILITY OF A CLASS OF RISK-AVERSE MULTISTAGE STOCHASTIC PROGRAMS AND THEIR DISTRIBUTIONALLY ROBUST COUNTERPARTS
2021, Journal of Industrial and Management Optimization