Confidence regions of stochastic variational inequalities: error bound approach

ABSTRACT

In this paper, we aim to build confidence regions of the true solution to the stochastic variational inequalities problem (SVIP) when the sample average approximation (SAA) scheme is implemented. A new approach based on error bound conditions admitted by the SVIP is proposed. This so-called error bound approach provides an upper bound of the distance between SAA solutions and the true solution set through the distance between the SAA function and the true counterpart at the SAA solutions. Certain statistical tools such as central limit theorem and Owen's empirical likelihood theorem are then employed to construct the asymptotic confidence regions of the solutions to SVIP. In particular, if the SVIP admits a global error bound condition, the non-asymptotic (uniform) confidence regions of the solutions are also approachable. Different from the conventional normal map approach, our error bound approach does not require any information regarding the derivative of the solution mapping with respect to perturbations of involved functions in SVIP. For constructing component-wise confidence regions, the validity of the error bound approach is guaranteed for those cases where the functions own separable structures.

Keywords:

• Stochastic programming
• stochastic variational inequalities
• (non-)asymptotic confidence regions
• empirical likelihood method

2010 Mathematics Subject Classifications:

• 90C33
• 90C15

Acknowledgements

We would like to thank the editor for organizing an effective review and two anonymous referees for insightful comments and constructive suggestions which help us significantly to consolidate the paper. The research is supported by the NSFC grants #11971090, #11971220, Fundamental Research Funds for the Central Universities under grant DUT19LK24 and Guangdong Basic and Applied Basic Research Foundation 2019A1515011152.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The uniqueness and Lipschitz of $z(\cdot )$ is with respect to the perturbation of the function $f(\cdot )$ in SVIP (3(3) $(y-x{)}^{T}f(x)\ge 0,\mathrm{\forall }y\in \mathcal{C},$(3) ).

2 For a given point, the difference of the SAA function and the true one can be taken as a random vector.

3 $df\left(x_0\right)(\cdot )$ denotes the B-derivative, see [16, Page 547] for details.

4 $C^1(\mathcal{C},{IR}^n)$ denotes the Banach space of continuously differentiable mappings $g:\mathcal{C}\to {IR}^n$.

5 The ball is defined in the Banach space of continuously differentiable mappings $f:\mathcal{C}\to {\mathbb{R}}^n,$ equipped with the norm defined as in [16, (9) page 4].

6 A solution $x^{\ast }\in {IR}^n$ to complementarity problem (16(16) $0\le x\perp h\left(x\right)\ge 0$(16) ) is said to be non-degenerate if $x_i^{\ast }\ne \left(h\right(x^{\ast }){)}_i,i=1,\dots ,n$.

7 The corresponding residual function $r_N(\cdot )$ means that $r_N(\cdot )$ and $r(\cdot )$ are induced by the same type error bound conditions, such as, normal map error bound condition. But $r_N(\cdot )$ is corresponding to SAA-SVIP (8(8) $(SAA-SVIP)(y-x{)}^T{f}_N(x)\ge 0,\mathrm{\forall }y\in \mathcal{C}.$(8) ) and $r(\cdot )$ is corresponding to SVIP (3(3) $(y-x{)}^{T}f(x)\ge 0,\mathrm{\forall }y\in \mathcal{C},$(3) ). Please see Examples 3.1–3.3 for details.

Additional information

Funding

This work was supported by National Natural Science Foundation of China 11971090 and 1971220.