Stochastic Variational Inequality Research Articles

Well-posedness of stochastic variational inequalities with discontinuous drifts

Well-posedness of stochastic variational inequalities with discontinuous drifts

Notes on random optimal control equilibrium problem via stochastic inverse variational inequalities

The main objective of the paper is to analyze how policymakers influence the random oligopolistic market equilibrium problem. To this purpose, random optimal control equilibrium conditions are introduced. Since the random optimal regulatory tax is characterized by a stochastic inverse variational inequality, existence and well-posedness results on such an inequality are proved. At last a numerical example is discussed.

One-dimensional McKean–Vlasov stochastic variational inequalities and coupled BSDEs with locally Hölder noise coefficients

In this article, we investigate three classes of equations: the McKean–Vlasov stochastic differential equation (MVSDE), the MVSDE with a subdifferential operator referred to as the McKean–Vlasov stochastic variational inequality (MVSVI), and the coupled forward–backward MVSVI. The latter class encompasses the FBSDE with reflection in a convex domain as a special case. We establish the well-posedness, in terms of the existence and uniqueness of a strong solution, for these three classes in their general forms. Importantly, we consider stochastic coefficients with locally Hölder continuity and employ different strategies to achieve that for each class.

Stochastic Bregman extragradient algorithm with line search for stochastic mixed variational inequalities

In this paper, we present a stochastic Bregman extragradient algorithm with line search for solving a class of stochastic mixed variational inequality, which not only does not require any information of the Lipschitz constant but allows to search a potentially larger step size per iteration. Compared with the existing algorithms, the proposed algorithm allows different step sizes in the prediction and correction steps, thus enhancing the algorithm's flexibility. Under the generalized monotonicity, we derive the almost sure convergence, the iteration complexity O ( 1 / ϵ ) and the oracle complexity O ( 1 / ϵ 2 ) for our algorithm. Furthermore, under the generalized strong monotonicity and the sample size increases at a geometric rate, the linear convergence rate of the proposed algorithm with respect to the Bregman distance between the iterations and solution is established. Numerical results demonstrate a favourable comparison of the proposed algorithm with existing ones.

A note on progressive hedging algorithm for multistage stochastic variational inequalities

Progressive hedging algorithm (PHA) is a long-standing algorithm originally designed for stochastic programming problems and has recently been extended to solving multistage stochastic variational inequalities (SVIs), both are important tools for processing mathematical programming problems with uncertainty. In this note, we first show that PHA for two-stage stochastic linear complementarity problems, a special case of multistage SVIs, is an application of the Douglas-Rachford operator splitting algorithm and then extend this result to multistage SVIs. To do this, we first reformulate the focused problems as zero-finding problems of the sum of two maximal monotone operators. This reformulation plays a key role in establishing the connections Since Douglas-Rachford operator splitting algorithm can be viewed as a proximal point algorithm, our result sharpens the understanding of PHA.

Modified forward-backward algorithm for stochastic mixed variational inequality problems

Modified forward-backward algorithm for stochastic mixed variational inequality problems

A randomized progressive hedging algorithm for stochastic variational inequality

A randomized progressive hedging algorithm for stochastic variational inequality

An inexact algorithm for stochastic variational inequalities

We present a new Progressive Hedging Algorithm to solve Stochastic Variational Inequalities in the formulation introduced by Rockafellar and Wets in 2017, allowing the generated subproblems to be approximately solved with an implementable tolerance condition. Our scheme is based on Hybrid Inexact Proximal Point methods and generalizes the exact algorithm developed by Rockafellar and Sun in 2019, providing stronger convergence results. We also show some numerical experiments in two-stage Nash games.

A stochastic fractional differential variational inequality with Lévy jump and its application

This paper investigates a stochastic fractional differential variational inequality with Lévy jump (SFDVI with Lévy jump), which comprises a stochastic fractional differential equation with Lévy jump and a stochastic variational inequality. By employing the successive approximation method as well as the projection technique, we establish unique existence of the solution for the SFDVI with Lévy jump under some mild conditions. Moreover, the main results are applied to obtain the unique existence of the solution for the spatial price equilibrium problem in stochastic environments.

Variable sample-size optimistic mirror descent algorithm for stochastic mixed variational inequalities

Variable sample-size optimistic mirror descent algorithm for stochastic mixed variational inequalities

Discrete approximation for two-stage stochastic variational inequalities

Discrete approximation for two-stage stochastic variational inequalities

Stochastic Approximation for Estimating the Price of Stability in Stochastic Nash Games

The goal in this paper is to approximate the Price of Stability (PoS) in stochastic Nash games using stochastic approximation (SA) schemes. PoS is amongst the most popular metrics in game theory and provides an avenue for estimating the efficiency of Nash games. In particular, knowing the value of PoS can help with designing efficient networked systems, including transportation networks and power market mechanisms. Motivated by the absence of efficient methods for computing the PoS, first we consider stochastic optimization problems with a nonsmooth and merely convex objective function and a merely monotone stochastic variational inequality (SVI) constraint. This problem appears in the numerator of the PoS ratio. We develop a randomized block-coordinate stochastic extra-(sub)gradient method where we employ a novel iterative penalization scheme to account for the mapping of the SVI in each of the two gradient updates of the algorithm. We obtain an iteration complexity of the order ϵ − 4 that appears to be best known result for this class of constrained stochastic optimization problems, where ϵ denotes an arbitrary bound on suitably defined infeasibility and suboptimality metrics. Second, we develop an SA-based scheme for approximating the PoS and derive lower and upper bounds on the approximation error. To validate the theoretical findings, we provide preliminary simulation results on a networked stochastic Nash Cournot competition.

Existence and convergence for stochastic differential variational inequalities

In this paper, we consider a class of stochastic differential variational inequalities (for short, SDVIs) consisting of an ordinary differential equation and a stochastic variational inequality. The existence of solutions to SDVIs is established under the assumption that the leading operator in the stochastic variational inequality is $P$-function and $P_{0}$-function, respectively. Then, by using the sample average approximation and time stepping methods, two approximated problems corresponding to SDVIs are introduced and convergence results are obtained.

Variable sample-size operator extrapolation algorithm for stochastic mixed variational inequalities

In this paper, we present a variable sample-size operator extrapolation algorithm for solving a class of stochastic mixed variational inequalities. One distinctive feature of our algorithm is that it updates a single search sequence by solving a prox-mapping subproblem and computing an evaluation of the expected mapping at each iteration and hence it may significantly reduce computation load. In particular, the iteration sequence generated by our algorithm always belongs to the feasible region. We show that, under some moderate conditions, the proposed algorithm can achieve O(1/T) ergodic convergence rate in terms of the expected restricted gap function, where T denotes the number of iterations. We derive some results related to the convergence rate of the Bregman distance between iterates and solutions, the iteration complexity, and the oracle complexity for the proposed algorithm when the sample size increases at a geometric rate. We also investigate the sublinear convergence rate in terms of the residual function under the generalized monotonicity condition. Numerical experiments on stochastic network game, stochastic sparse traffic assignment problems and sparse classification problem indicate that the proposed algorithm is promising compared with some existing algorithms.

On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities

Abstract A new class of stochastic differential equations (SDEs) is introduced in this article, which is driven by the generalized stochastic mixed variational inequality (GS-MVI). First, the property of the solution sets of the GS-MVI is proved by Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and Aumann’s measurable selection theorem. Next, we obtain the Carathéodory property of the solution set, with which the discussed SDEs can be transformed to stochastic differential inclusions (SDIs). The solution set of the proposed SDEs is proved to be nonempty through the existence of the solutions of the corresponding SDIs by the tools of fixed point theorem.

Euler scheme for solving a class of stochastic differential variational inequalities with some applications

In this paper, we introduce and study Euler scheme for solving a class of stochastic differential variational inequalities (SDVIs). We show the convergence of the sequence construed by this Euler scheme for solving SDVIs under some mild conditions. Moreover, we apply the obtained results to simulate the electrical circuits with diodes and the collapse of the bridge problems in stochastic environment.

Stochastic approximation based confidence regions for stochastic variational inequalities

The sample average approximation (SAA) and the stochastic approximation (SA) are two popular schemes for solving the stochastic variational inequalities problem (SVIP). In the past decades, theories on the consistency of the SAA solutions and SA solutions have been well studied. More recently, the asymptotic confidence regions of the true solution to SVIP have been constructed when the SAA scheme is implemented. It is of fundamental interest to develop confidence regions of the true solution to the SVIP when the SA scheme is employed. In this paper, we discuss the framework of constructing asymptotic confidence regions for the true solution of SVIP with a focus on stochastic dual average method. We first establish the asymptotic normality of the SA solutions both in ergodic sense and non-ergodic sense. Then the online methods of estimating the covariance matrices in the normal distributions are studied. Finally, practical procedures of building the asymptotic confidence regions of solutions to SVIP with numerical simulations are presented.

Sample average approximation method for a class of stochastic vector variational inequalities

In the present paper, the expected-value (EV) reformulation of a class of stochastic vector variational inequalities (SVVI) is investigated. By using the regularized gap function, the EV reformulation of SVVI is transformed into a constrained optimization problem. Then a sample average approximation (SAA) method is proposed for solving the constrained optimization problem. Under suitable assumptions, the limiting behaviors of the optimal values and optimal solutions of the approximation problem are investigated. Finally, the rates of convergence in the different senses of optimal solutions for sample average approximation problem are discussed under the error bound condition.

A Localized Progressive Hedging Algorithm for Solving Nonmonotone Stochastic Variational Inequalities

The progressive hedging algorithm (PHA) is an effective solution method for solving monotone stochastic variational inequalities (SVIs). However, this validity is based on the assumption of global maximal monotonicity. In this paper, we propose a localized PHA for solving nonmonotone SVIs and show that its validity is based on the weaker assumption of locally elicitable maximal monotonicity. Furthermore, we prove that such assumption holds when the mapping involved in the SVI is locally elicitable monotone or locally monotone. The local convergence of the proposed algorithm is established, and it is shown that the localized PHA has the rate of linear convergence under some mild assumptions. Some numerical experiments, including a two-stage orange market problem and randomly generated two-stage piecewise stochastic linear complementarity problems, indicate that the proposed algorithm is efficient. Funding: This work was supported by the National Natural Science Foundation of China [Grant 12171271]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.0017 .

A Prediction–Correction ADMM for Multistage Stochastic Variational Inequalities

A Prediction–Correction ADMM for Multistage Stochastic Variational Inequalities