An inertial-type stochastic self-adaptive algorithm for stochastic pseudomonotone variational inequality problem

This paper considers a class of two-stage stochastic linear variational inequality problems whose first stage problems are stochastic linear box-constrained variational inequality problems and second stage problems are stochastic linear complementary problems having a unique solution. We first give conditions for the existence of solutions to both the original problem and its perturbed problems. Next we derive quantitative stability assertions of this two-stage stochastic problem under total variation metrics via the corresponding residual function. Moreover, we study the discrete approximation problem. The convergence and the exponential rate of convergence of optimal solution sets are obtained under moderate assumptions respectively. Finally, through solving a non-cooperative game in which each player’s problem is a parameterized two-stage stochastic program, we numerically illustrate our theoretical results.

In this paper, we propose a new class of variational inequality problems, say, uncertain variational inequality problems based on uncertainty theory in finite Euclidean spaces $R^{n}$ . It can be viewed as another extension of classical variational inequality problems besides stochastic variational inequality problems. Note that both stochastic variational inequality problems and uncertain variational inequality problems involve uncertainty in the real world, thus they have no conceptual solutions. Hence, in order to solve uncertain variational inequality problems, we introduce the expected value of uncertain variables (vector). Then we convert it into a classical deterministic variational inequality problem, which can be solved by many algorithms that are developed on the basis of gap functions. Thus the core of this paper is to discuss under what conditions we can convert the expected value model of uncertain variational inequality problems into deterministic variational inequality problems. Finally, as an application, we present an example in a noncooperation game from economics.

We consider a stochastic variational inequality (SVI) problem with a continuous and monotone mapping over a compact and convex set. Traditionally, stochastic approximation (SA) schemes for SVIs have relied on strong monotonicity and Lipschitzian properties of the underlying map. We present a regularized smoothed SA (RSSA) scheme where in the stepsize, smoothing, and regularization parameters are diminishing sequences. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to a solution in an almost-sure sense. Additionally, we provide rate estimates that relate iterates to their counterparts derived from the Tikhonov trajectory associated with a deterministic problem.

We consider a stochastic variational inequality (SVI) problem with a continuous and monotone mapping over a compact and convex set. Traditionally, stochastic approximation (SA) schemes for SVIs have relied on strong monotonicity and Lipschitzian properties of the underlying map. We present a regularized smoothed SA (RSSA) scheme where in the stepsize, smoothing, and regularization parameters are diminishing sequences. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to a solution in an almost-sure sense. Additionally, we provide rate estimates that relate iterates to their counterparts derived from the Tikhonov trajectory associated with a deterministic problem.

Traditionally, most stochastic approximation (SA) schemes for stochastic variational inequality (SVI) problems have required the underlying mapping to be either strongly monotone or monotone and Lipschitz continuous. In contrast, we consider SVIs with merely monotone and non-Lipschitzian maps. We develop a regularized smoothed SA (RSSA) scheme wherein the stepsize, smoothing, and regularization parameters are reduced after every iteration at a prescribed rate. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to the least norm solution in an almost sure sense, extending the results in Koshal et al. (IEEE Trans Autom Control 58(3):594–609, 2013) to the non-Lipschitzian regime. Additionally, we provide rate estimates that relate iterates to their counterparts derived from a smoothed Tikhonov trajectory associated with a deterministic problem. To derive non-asymptotic rate statements, we develop a variant of the RSSA scheme, denoted by aRSSA $$_r$$ , in which we employ a weighted iterate-averaging, parameterized by a scalar r where $$r = 1$$ provides us with the standard averaging scheme. The main contributions are threefold: (i) when $$r<1$$ and the parameter sequences are chosen appropriately, we show that the averaged sequence converges to the least norm solution almost surely and a suitably defined gap function diminishes at an approximate rate $$\mathcal{O}({1}\slash {\root 6 \of {k}})$$ after k steps; (ii) when $$r<1$$ , and smoothing and regularization are suppressed, the gap function admits the rate $$\mathcal{O}({1}\slash {\sqrt{k}})$$ , thus improving the rate $$\mathcal{O}(\ln (k)/\sqrt{k})$$ under standard averaging; and (iii) we develop a window-based variant of this scheme that also displays the optimal rate for $$r < 1$$ . Notably, we prove the superiority of the scheme with $$r < 1$$ with its counterpart with $$r=1$$ in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large. We present the performance of the developed schemes on a stochastic Nash–Cournot game with merely monotone and non-Lipschitzian maps.

In this paper, we consider stochastic mixed vector variational inequality problems. Firstly, we present an equivalent form for the stochastic mixed vector variational inequality problems. Secondly, we present a deterministic bi-criteria model for giving the reasonable resolution of the stochastic mixed vector variational inequality problems and further propose the approximation problem for solving the given deterministic model by employing the smoothing technique and the sample average approximation method. Thirdly, we obtain the convergence analysis for the proposed approximation problem while the sample space is compact. Finally, we propose a compact approximation method when the sample space is not a compact set and provide the corresponding convergence results.

Motivated by multi-user optimization problems and non-cooperative Nash games in uncertain regimes, we consider stochastic Cartesian variational inequality problems where the set is given as the Cartesian product of a collection of component sets. First, we consider the case where the number of the component sets is large and develop a randomized block stochastic mirror-prox algorithm, where at each iteration only a randomly selected block coordinate of the solution vector is updated through implementing two consecutive projection steps. We show that when the mapping is strictly pseudo-monotone, the algorithm generates a sequence of iterates that converges to the solution of the problem almost surely. When the maps are strongly pseudo-monotone, we prove that the mean-squared error diminishes at the optimal rate. Second, we consider large-scale stochastic optimization problems with convex objectives and develop a new averaging scheme for the randomized block stochastic mirror-prox algorithm. We show that by using a different set of weights than those employed in the classical stochastic mirror-prox methods, the objective values of the averaged sequence converges to the optimal value in the mean sense at an optimal rate. Third, we consider stochastic Cartesian variational inequality problems and develop a stochastic mirror-prox algorithm that employs the new weighted averaging scheme. We show that the expected value of a suitably defined gap function converges to zero at an optimal rate.

Deterministic finite-dimensional variational inequality problems represent a tool for capturing a broad range of optimization and equilibrium problems in application set- tings ranging from traffic equilibrium models and energy markets to communication networks, among others. Increasingly, these settings are complicated by risk and uncer- tainty, and although deterministic models represent an important first step, a compre- hensive examination of the stochastic generalization of such problems is in order. One possible model for capturing uncertainty in this realm is a finite-dimensional stochastic variational inequality problem, which is defined by a deterministic set and a map whose components contain expectations. Unfortunately, traditional analytical and compu- tational techniques largely fail to extend naturally in resolving this problem when the expectations are defined over a general probability space. This tutorial intends to provide a review of a subset of the analytical and algorithmic advances that have emerged in the resolution of this problem. We motivate the study of the stochastic variational inequality problem by considering two applications, the first drawn from strategic bidding in power markets and the second arising in the design of cognitive radio systems. This discussion paves the way for the following sets of contributions of this tutorial and concludes by demonstrating the utility of the presented tools on one of the application settings. (i) First, existence statements for the stochastic varia- tional inequality problem are complicated by the need to evaluate a multidimensional integral. By combining Lebesgue convergence theorems with existence statements for finite-dimensional analogs, a set of integration-free existence statements are provided for the stochastic variational inequality problem. Furthermore, these statements are extended to the regime where the integrands of the expectation are multivalued maps arising, for instance, from a stochastic nonsmooth Nash game. (ii) Second, when the expectations are unavailable as closed-form expressions, deterministic schemes can- not easily resolve stochastic variational problems, prompting the need for leveraging Monte Carlo sampling schemes. One such avenue lies in the use of stochastic approx- imation schemes. The earliest among these relied on the strong monotonicity of the map. Hybrid variants of such schemes that combine Tikhonov regularization and proximal-point methods are also suggested, both of which weaken the need for strong monotonicity of the map and are characterized by desirable almost-sure convergence properties. A shortcoming of the earlier schemes is the limited guidance provided for the choice of step-length sequences. This concern is partially resolved in the presen- tation of a self-tuned step-length stochastic approximation scheme that prescribed a step-length rule that adapts to problem parameters such as Lipschitz and monotonic- ity constants.

Expected residual minimization method for stochastic variational inequality problems with nonlinear perturbations

Method of weighted expected residual for solving stochastic variational inequality problems

In this paper, we consider a class of stochastic variational inequality problems (SVIPs). Different from the classical variational inequality problems, the SVIP contains a mathematical expectation, which may not be evaluated in an explicit form in general. We combine a hybrid Newton method for deterministic cases with an unconstrained optimization reformulation based on the well-known D-gap function and sample average approximation (SAA) techniques to present an SAA-based hybrid Newton method for solving the SVIP. We show that the level sets of the approximation D-gap function are bounded. Furthermore, we prove that the sequence generated by the hybrid Newton method converges to a solution of the SVIP under appropriate conditions, and some numerical experiments are presented to prove the effectiveness and competitiveness of the hybrid Newton method. Finally, we apply this method to solve two specific traffic equilibrium problems.

This paper focus on the quantitative stability of a class of two-stage stochastic linear variational inequality problems whose second stage problems are stochastic linear complementarity problems with fixed recourse matrix. Firstly, we discuss the existence of solutions to this two-stage stochastic problems and its perturbed problems. Then, by using the corresponding residual function, we derive the quantitative stability of this two-stage stochastic problem under Fortet-Mourier metric. Finally, we study the sample average approximation problem, and obtain the convergence of optimal solution sets under moderate assumptions.

In this paper, a class of stochastic vector variational inequality (SVVI) problems are considered. By employing the idea of a D-gap function, the SVVI problem is reformulated as a deterministic model, which is an unconstrained expected residual minimization (UERM) problem, while it is reformulated as a constrained expected residual minimization problem in the work of Zhao et al. Then, the properties of the objective function are investigated and a sample average approximation approach is proposed for solving the UERM problem. Convergence of the proposed approach for global optimal solutions and stationary points is analyzed. Moreover, we consider another deterministic formulation, i.e., the expected value (EV) formulation for an SVVI problem, and the global error bound of a D-gap function based on the EV formulation is given.

We present a new method for solving the box-constrained stochastic linear variational inequality problem with three special types of uncertainty sets. Most previous methods, such as the expected value and expected residual minimization, need the probability distribution information of the stochastic variables. In contrast, we give the robust reformulation and reformulate the problem as a quadratically constrained quadratic program or convex program with a conic quadratic inequality quadratic program, which is tractable in optimization theory.

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.