Two‐stage stochastic linear complementarity problems (TSLCP) model a large class of equilibrium problems subject to data uncertainty, and are closely related to two‐stage stochastic optimization problems. The sample average approximation (SAA) method is one of the basic approaches for solving TSLCP and the consistency of the SAA solutions has been well studied. This paper focuses on building confidence regions of the solution to TSLCP when SAA is implemented. We first establish the error‐bound condition of TSLCP and then build the asymptotic and nonasymptotic confidence regions of the solutions to TSLCP by error‐bound approach, which is to combine the error‐bound condition with central limit theory, empirical likelihood theory, and large deviation theory.
The popular AB/push-pull method for distributed optimization problem may unify much of the existing decentralized first-order methods based on gradient tracking technique. More recently, the stochastic gradient variant of AB/Push-Pull method (S-AB) has been proposed, which achieves the linear rate of converging to a neighborhood of the global minimizer when the step-size is constant. This paper is devoted to the asymptotic properties of S-AB with diminishing step-size. Specifically, under the condition that each local objective is smooth and the global objective is strongly-convex, we first present the boundedness of the iterates of S-AB and then show that the iterates converge to the global minimizer with the rate O 1/ √ k . Furthermore, the asymptotic normality of Polyak-Ruppert averaged S-AB is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.
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