Stochastic Variational Inequalities: Residual Minimization Smoothing Sample Average Approximations

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.

The ideas of approximation and continuity have been extensively investigated both for optimization problems and variational inequalities. In this paper, we study approximation issues for a class of VIs, called Stochastic Variational Inequalities (SVIs), that arise, for example, in stochastic programming and portfolio choice problems. SVI problems are special cases of a more general class of problems that we will study first, called Stochastic Ky Fan Inequalities (SKFIs). We also analyze the role of monotonicity in the analysis of both SVIs and SKFIs. Our interest in these problems is motivated by recent research in the theory of portfolio choice for investors who are not classical expected utility maximizers.

<p style='text-indent:20px;'>This paper focuses on the quantitative stability analysis of the expected residual minimization (ERM) formulation for a class of stochastic linear variational inequalities. Firstly, the existence of solutions of the ERM formulation and its perturbed problem is discussed. Then, the quantitative stability of the ERM formulation is derived under suitable probability metrics. Finally, the sample average approximation (SAA) problem of the ERM formulation is studied, and the rates of convergence of optimal solution sets are obtained under different assumptions.</p>

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

This paper conducts sensitivity analysis of random constraint and variational systems related to stochastic optimization and variational inequalities. We establish efficient conditions for well-posedness, in the sense of robust Lipschitzian stability and/or metric regularity, of such systems by employing and developing coderivative characterizations of well-posedness properties for random multifunctions and efficiently evaluating coderivatives of special classes of random integral set-valued mappings that naturally emerge in stochastic programming and stochastic variational inequalities.

Sample average approximation method is one of the well-behaved methods in the stochastic optimization.This paper presents a sample average approximation method based on a D-gap function for stochastic variational inequality problems.An unconstrained optimization reformulation is proposed for the expected-value formulation of stochastic variational inequality problems based on the D-gap function. An implementable sample average approximation method for the reformulation is established andit is proven that the optimal values and the optimal solutions of the approximation problems converge to their true counterpart with probability one as the sample size increases under some moderate assumptions.Finally, the preliminary numerical results for some test examples are reported, which show that the proposed method is promising.

A popular approach for addressing uncertainty in variational inequality problems is by solving the expected residual minimization (ERM) problem. This avenue necessitates distributional information associated with the uncertainty and requires minimizing nonconvex expectation-valued functions. We consider a distinctly different approach in the context of uncertain linear complementarity problems with a view towards obtaining robust solutions. Specifically, we define a robust solution to a complementarity problem as one that minimizes the worst-case of the gap function. In what we believe is amongst the first efforts to comprehensively address such problems in a distribution-free environment, we show that under specified assumptions on the uncertainty sets, the robust solutions to uncertain monotone linear complementarity problem can be tractably obtained through the solution of a single convex program. We also define uncertainty sets that ensure that robust solutions to non-monotone generalizations can also be obtained by solving convex programs. More generally, robust counterparts of uncertain non-monotone LCPs are proven to be low-dimensional nonconvex quadratically constrained quadratic programs. We show that these problems may be globally resolved by customizing an existing branching scheme. We further extend the tractability results to include uncertain affine variational inequality problems defined over uncertain polyhedral sets as well as to hierarchical regimes captured by mathematical programs with uncertain complementarity constraints. Preliminary numerics on uncertain linear complementarity and traffic equilibrium problems suggest that the presented avenues hold promise.

Method with batching for stochastic finite-sum variational inequalities in non-Euclidean setting

Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control Models

In this paper, we consider a class of one-dimensional reflected Backward Stochastic Differential Equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. Using a stochastic variational inequality, we characterize its solution.

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].

Method of weighted expected residual for solving stochastic variational inequality problems

The paper deals with the random time-dependent oligopolistic market equilibrium problem. For such a problem the firms’ point of view has been analyzed in Barbagallo and Guarino Lo Bianco (Optim. Lett. 14: 2479–2493, 2020) while here the policymaker’s point of view is studied. The random dynamic optimal control equilibrium conditions are expressed by means of an inverse stochastic time-dependent variational inequality which is proved to be equivalent to a stochastic time-dependent variational inequality. Some existence and well-posedness results for optimal regulatory taxes are obtained. Moreover a numerical scheme to compute the solution to the stochastic time-dependent variational inequality is presented. Finally an example is discussed.

This paper aims to optimise healthcare resource pre-positioning, patient scheduling, and patient transferring under uncertain demands and stochastic resource consumption. We propose a two-stage stochastic programming model that formulates the patient scheduling problem as a Markov decision process. To address complexities and uncertainties, we use Artificial Intelligence (AI) techniques to improve both model formulation and algorithmic performance. To tackle the limited data challenge, we introduce a Wasserstein distance-based ambiguity set and propose a two-stage distributionally robust optimisation (DRO) model, which derives a deterministic equivalent using the Lagrangian dual of non-anticipativity constraints. The solution process is accelerated with a scenario decomposition approach and the K-means clustering method. Both theoretical and numerical results demonstrate the consistency of the two-stage DRO model with the sample average approximation (SAA) method. The potential of AI to improve the model's performance is evident through the significant reduction in computation time achieved with the K-means clustering approach without compromising solution quality. Compared to the SAA method, the DRO model exhibits a considerable reduction in the waiting penalty cost for out-of-sample cases, ranging from 43.06 % to 81.23 % . Numerical results show that the proposed algorithm outperforms SAA methods and several benchmark policies in terms of computational efficiency and solutions.

In this paper, we consider stochastic vector variational inequality problems (SVVIPs). Because of the existence of stochastic variable, the SVVIP may have no solutions generally. For solving this problem, we employ the regularized gap function of SVVIP to the loss function and then give a low-risk conditional value-at-risk (CVaR) model. However, this low-risk CVaR model is difficult to solve by the general constraint optimization algorithm. This is because the objective function is nonsmoothing function, and the objective function contains expectation, which is not easy to be computed. By using the sample average approximation technique and smoothing function, we present the corresponding approximation problems of the low-risk CVaR model to deal with these two difficulties related to the low-risk CVaR model. In addition, for the given approximation problems, we prove the convergence results of global optimal solutions and the convergence results of stationary points, respectively. Finally, a numerical experiment is given.