arXiv subjects

Uday V. Shanbhag

Publications and source records attributed to Uday V. Shanbhag.

At least 37 records · Page 2Linked to original sources

arXiv2021

No-regret distributed learning in subnetwork zero-sum games

In this paper, we consider a distributed learning problem in a subnetwork zero-sum game, where agents are competing in different subnetworks. These agents are connected through time-varying graphs where each agent has its own cost function and can receive information from its neighbors. We propose a distributed mirror descent algorithm for computing a Nash equilibrium and establish a sublinear regret bound on the sequence of iterates when the graphs are uniformly strongly connected and the cost functions are convex-concave. Moreover, we prove its convergence with suitably selected diminishing stepsizes for a strictly convex-concave cost function. We also consider a constant step-size variant of the algorithm and establish an asymptotic error bound between the cost function values of running average actions and a Nash equilibrium. In addition, we apply the algorithm to compute a mixed-strategy Nash equilibrium in subnetwork zero-sum finite-strategy games, which have merely convex-concave (to be specific, multilinear) cost functions, and obtain a final-iteration convergence result and an ergodic convergence result, respectively, under different assumptions.

math.OC↗

arXiv2021

Stochastic Relaxed Inertial Forward-Backward-Forward splitting for Monotone Inclusions in Hilbert spaces

We consider monotone inclusions defined on a Hilbert space where the operator is given by the sum of a maximal monotone operator $T$ and a single-valued monotone, Lipschitz continuous, and expectation-valued operator $V$. We draw motivation from the seminal work by Attouch and Cabot on relaxed inertial methods for monotone inclusions and present a stochastic extension of the relaxed inertial forward-backward-forward (RISFBF) method. Facilitated by an online variance reduction strategy via a mini-batch approach, we show that (RISFBF) produces a sequence that weakly converges to the solution set. Moreover, it is possible to estimate the rate at which the discrete velocity of the stochastic process vanishes. Under strong monotonicity, we demonstrate strong convergence, and give a detailed assessment of the iteration and oracle complexity of the scheme. When the mini-batch is raised at a geometric (polynomial) rate, the rate statement can be strengthened to a linear (suitable polynomial) rate while the oracle complexity of computing an $ε$-solution improves to $O(1/ε)$. Importantly, the latter claim allows for possibly biased oracles, a key theoretical advancement allowing for far broader applicability. By defining a restricted gap function based on the Fitzpatrick function, we prove that the expected gap of an averaged sequence diminishes at a sublinear rate of $O(1/k)$ while the oracle complexity of computing a suitably defined $ε$-solution is $O(1/ε^{1+a})$ where $a>1$. Numerical results on two-stage games and an overlapping group Lasso problem illustrate the advantages of our method compared to stochastic forward-backward-forward (SFBF) and SA schemes.

math.OC↗

arXiv2021

Zeroth-order randomized block methods for constrained minimization of expectation-valued Lipschitz continuous functions

We consider the minimization of an $L_0$-Lipschitz continuous and expectation-valued function, denoted by $f$ and defined as $f(x)\triangleq \mathbb{E}[\tilde{f}(x,ω)]$, over a Cartesian product of closed and convex sets with a view towards obtaining both asymptotics as well as rate and complexity guarantees for computing an approximate stationary point (in a Clarke sense). We adopt a smoothing-based approach reliant on minimizing $f_η$ where $f_η(x) \triangleq \mathbb{E}_{u}[f(x+ηu)]$, $u$ is a random variable defined on a unit sphere, and $η> 0$. In fact, it is observed that a stationary point of the $η$-smoothed problem is a $2η$-stationary point for the original problem in the Clarke sense. In such a setting, we derive a suitable residual function that provides a metric for stationarity for the smoothed problem. By leveraging a zeroth-order framework reliant on utilizing sampled function evaluations implemented in a block-structured regime, we make two sets of contributions for the sequence generated by the proposed scheme. (i) The residual function of the smoothed problem tends to zero almost surely along the generated sequence; (ii) To compute an $x$ that ensures that the expected norm of the residual of the $η$-smoothed problem is within $ε$ requires no greater than $\mathcal{O}(\tfrac{1}{ηε^2})$ projection steps and $\mathcal{O}\left(\tfrac{1}{η^2 ε^4}\right)$ function evaluations. These statements appear to be novel and there appear to be few results to contend with general nonsmooth, nonconvex, and stochastic regimes via zeroth-order approaches.

math.OC↗

arXiv2021

A relaxed-inertial forward-backward-forward algorithm for Stochastic Generalized Nash equilibrium seeking

In this paper we propose a new operator splitting algorithm for distributed Nash equilibrium seeking under stochastic uncertainty, featuring relaxation and inertial effects. Our work is inspired by recent deterministic operator splitting methods, designed for solving structured monotone inclusion problems. The algorithm is derived from a forward-backward-forward scheme for solving structured monotone inclusion problems featuring a Lipschitz continuous and monotone game operator. To the best of our knowledge, this is the first distributed (generalized) Nash equilibrium seeking algorithm featuring acceleration techniques in stochastic Nash games without assuming cocoercivity. Numerical examples illustrate the effect of inertia and relaxation on the performance of our proposed algorithm.

math.OC↗

arXiv2020

On the analysis of variance-reduced and randomized projection variants of single projection schemes for monotone stochastic variational inequality problems

Classical extragradient schemes and their stochastic counterpart represent a cornerstone for resolving monotone variational inequality problems. Yet, such schemes have a per-iteration complexity of two projections onto a convex set and require two evaluations of the map, the former of which could be relatively expensive if $X$ is a complicated set. We consider two related avenues where the per-iteration complexity is significantly reduced: (i) A stochastic projected reflected gradient method requiring a single evaluation of the map and a single projection; and (ii) A stochastic subgradient extragradient method that requires two evaluations of the map, a single projection onto $X$, and a significantly cheaper projection (onto a halfspace) computable in closed form. Under a variance-reduced framework reliant on a sample-average of the map based on an increasing batch-size, we prove almost sure (a.s.) convergence of the iterates to a random point in the solution set for both schemes. Additionally, both schemes display a non-asymptotic rate of $\mathcal{O}(1/K)$ where $K$ denotes the number of iterations; notably, both rates match those obtained in deterministic regimes. To address feasibility sets given by the intersection of a large number of convex constraints, we adapt both of the aforementioned schemes to a random projection framework. We then show that the random projection analogs of both schemes also display a.s. convergence under a weak-sharpness requirement; furthermore, without imposing the weak-sharpness requirement, both schemes are characterized by a provable rate of $\mathcal{O}(1/\sqrt{K})$ in terms of the gap function of the projection of the averaged sequence onto $X$ as well as the infeasibility of this sequence. Preliminary numerics support theoretical findings and the schemes outperform standard extragradient schemes in terms of the per-iteration complexity.

math.OC↗

arXiv2020

A Variable Sample-size Stochastic Quasi-Newton Method for Smooth and Nonsmooth Stochastic Convex Optimization

Classical theory for quasi-Newton schemes has focused on smooth deterministic unconstrained optimization while recent forays into stochastic convex optimization have largely resided in smooth, unconstrained, and strongly convex regimes. Naturally, there is a compelling need to address nonsmoothness, the lack of strong convexity, and the presence of constraints. Accordingly, this paper presents a quasi-Newton framework that can process merely convex and possibly nonsmooth (but smoothable) stochastic convex problems. We propose a framework that combines iterative smoothing and regularization with a variance-reduced scheme reliant on using increasing sample-sizes of gradients. We make the following contributions. (i) We develop a regularized and smoothed variable sample-size BFGS update (rsL-BFGS) that generates a sequence of Hessian approximations and can accommodate nonsmooth convex objectives by utilizing iterative regularization and smoothing. (ii) In strongly convex regimes with state-dependent noise, the proposed variable sample-size stochastic quasi-Newton scheme admits a non-asymptotic linear rate of convergence while the oracle complexity of computing an $ε$-solution is $\mathcal{O}(κ^{m+1}/ε)$ where $κ$ is the condition number and $m\geq 1$. In nonsmooth (but smoothable) regimes, using Moreau smoothing retains the linear convergence rate. To contend with the possible unavailability of Lipschitzian and strong convexity parameters, we also provide sublinear rates; (iii) In merely convex but smooth settings, the regularized VS-SQN scheme rVS-SQN displays a rate of $\mathcal{O}(1/k^{(1-\varepsilon)})$. When the smoothness requirements are weakened, the rate for the regularized and smoothed VS-SQN scheme worsens to $\mathcal{O}(k^{-1/3})$. Such statements allow for a state-dependent noise assumption under a quadratic growth property.

math.OC↗

arXiv2020

Distributed Variable Sample-Size Gradient-response and Best-response Schemes for Stochastic Nash Equilibrium Problems over Graphs

This paper considers a stochastic Nash game in which each player minimizes an expectation valued composite objective. We make the following contributions. (I) Under suitable monotonicity assumptions on the concatenated gradient map, we derive optimal rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme when the sample-size increases at a geometric rate. If the sample-size increases at a polynomial rate of degree $v > 0$, the mean-squared errordecays at a corresponding polynomial rate while the iteration and oracle complexities to obtain an $ε$-NE are $\mathcal{O}(1/ε^{1/v})$ and $\mathcal{O}(1/ε^{1+1/v})$, respectively. (II) We then overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. In the resulting scheme, when the sample-size and the consensus steps grow at a geometric and linear rate, computing an $ε$-NE requires similar iteration and oracle complexities to (VS-PGR) with a communication complexity of $\mathcal{O}(\ln^2(1/ε))$; (III) Under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where each player solves a sample-average BR problem. Akin to (I), we also give the rate statements, oracle and iteration complexity bounds. (IV) Akin to (II), the distributed variant achieves similar iteration and oracle complexities to the centralized (VS-PBR) with a communication complexity of $\mathcal{O}(\ln^2(1/ε))$ when the communication rounds per iteration increase at a linear rate. Finally, we present some preliminary numerics to provide empirical support for the rate and complexity statements.

math.OC↗

arXiv2020

Asynchronous Variance-reduced Block Schemes for Composite Nonconvex Stochastic Optimization: Block-specific Steplengths and Adapted Batch-sizes

We consider the minimization of a sum of an expectation-valued coordinate-wise $L_i$-smooth nonconvex function and a nonsmooth block-separable convex regularizer. We propose an asynchronous variance-reduced algorithm, where in each iteration, a single block is randomly chosen to update its estimates by a proximal variable sample-size stochastic gradient scheme, while the remaining blocks are kept invariant. Notably, each block employs a steplength that is in accordance with its block-specific Lipschitz constant while block-specific batch-sizes are random variables updated at a rate that grows either at a geometric or polynomial rate with the (random) number of times that block is selected. We show that every limit point for almost every sample path is a stationary point and establish the ergodic non-asymptotic rate $\mathcal{O}(1/K) $. Iteration and oracle complexity to obtain an $ε$-stationary point are shown to be $\mathcal{O}(1/ε)$ and $\mathcal{O}(1/ε^2)$, respectively. Furthermore, under a $ μ$-proximal Polyak-Łojasiewicz (PL) condition with the batch size increasing at a geometric rate, we prove that the suboptimality diminishes at a {\em geometric} rate, the {\em optimal} deterministic rate while iteration and oracle complexity to obtain an $ε$-optimal solution are proven to be $\mathcal{O}( (L_{\rm max}/μ) \ln(1/ε))$ and $\mathcal{O}\left((L_{\rm ave}/μ) (1/ε)^{1+c} \right)$ with $c\geq 0$, respectively. In pursuit of less aggressive sampling rates, when the batch sizes increase at a polynomial rate of degree $v \geq 1$, suboptimality decays at a corresponding polynomial rate while the iteration and oracle complexity to obtain an $ε-$optimal solution are provably $\mathcal{O} ( v(1/ε)^{1/v})$ and $\mathcal{O} \left(e^v v^{2v+1}(1/ε)^{1+1/v}\right)$, respectively.

math.OC↗

arXiv2019

Tractable ADMM Schemes for Computing KKT Points and Local Minimizers for $\ell_0$-Minimization Problems

We consider an $\ell_0$-minimization problem where $f(x) + γ\|x\|_0$ is minimized over a polyhedral set and the $\ell_0$-norm regularizer implicitly emphasizes sparsity of the solution. Such a setting captures a range of problems in image processing and statistical learning. Given the nonconvex and discontinuous nature of this norm, convex regularizers are often employed as substitutes. Therefore, far less is known about directly solving the $\ell_0$-minimization problem. Inspired by [19], we consider resolving an equivalent formulation of the $\ell_0$-minimization problem as a mathematical program with complementarity constraints (MPCC) and make the following contributions towards the characterization and computation of its KKT points: (i) First, we show that feasible points of this formulation satisfy the relatively weak Guignard constraint qualification. Furthermore, under the suitable convexity assumption on $f(x)$, an equivalence is derived between first-order KKT points and local minimizers of the MPCC formulation. (ii) Next, we apply two alternating direction method of multiplier (ADMM) algorithms to exploit special structure of the MPCC formulation: (ADMM$_{\rm cf}^{μ, α, ρ}$) and (ADMM$_{\rm cf}$). These two ADMM schemes both have tractable subproblems. Specifically, in spite of the overall nonconvexity, we show that the first of the ADMM updates can be effectively reduced to a closed-form expression by recognizing a hidden convexity property while the second necessitates solving a convex program. In (ADMM$_{\rm cf}^{μ, α, ρ}$), we prove subsequential convergence to a perturbed KKT point under mild assumptions. Our preliminary numerical experiments suggest that the tractable ADMM schemes are more scalable than their standard counterpart and ADMM$_{\rm cf}$ compares well with its competitors to solve the $\ell_0$-minimization problem.

math.OC↗

arXiv2019

SI-ADMM: A Stochastic Inexact ADMM Framework for Stochastic Convex Programs

We consider the structured stochastic convex program requiring the minimization of $\mathbb{E}[\tilde f(x,ξ)]+\mathbb{E}[\tilde g(y,ξ)]$ subject to the constraint $Ax + By = b$. Motivated by the need for decentralized schemes and structure, we propose a stochastic inexact ADMM (SI-ADMM) framework where subproblems are solved inexactly via stochastic approximation schemes. Based on this framework, we prove the following: (i) under suitable assumptions on the associated batch-size of samples utilized at each iteration, the SI-ADMM scheme produces a sequence that converges to the unique solution almost surely; (ii) If the number of gradient steps (or equivalently, the number of sampled gradients) utilized for solving the subproblems in each iteration increases at a geometric rate, the mean-squared error diminishes to zero at a prescribed geometric rate; (iii) The overall iteration complexity in terms of gradient steps (or equivalently samples) is found to be consistent with the canonical level of $\mathcal{O}(1/ε)$. Preliminary applications on LASSO and distributed regression suggest that the scheme performs well compared to its competitors.

math.OC↗

arXiv2019

Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants

We consider the stochastic variational inequality problem in which the map is expectation-valued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes are limited to monotone maps. However, non-monotone stochastic variational inequality problems are not uncommon and are seen to arise from product pricing, fractional optimization problems, and subclasses of economic equilibrium problems. Motivated by the need to address a broader class of maps, we make the following contributions: (i) We present an extragradient-based stochastic approximation scheme and prove that the iterates converge to a solution of the original problem under either pseudomonotonicity requirements or a suitably defined acute angle condition. Such statements are shown to be generalizable to the stochastic mirror-prox framework; (ii) Under strong pseudomonotonicity, we show that the mean-squared error in the solution iterates produced by the extragradient SA scheme converges at the optimal rate of O(1/k) statements that were hitherto unavailable K in this regime. Notably, we optimize the initial steplength by obtaining an ε-infimum of a discontinuous nonconvex function. Similar statements are derived for mirror-prox generalizations and can accommodate monotone SVIs under a weak-sharpness requirement. Finally, both the asymptotics and the empirical rates of the schemes are studied on a set of variational problems where it is seen that the theoretically specified initial steplength leads to significant performance benefits.

math.OC↗

arXiv2019

A Proximal-Point Algorithm with Variable Sample-sizes (PPAWSS) for Monotone Stochastic Variational Inequality Problems

We consider a stochastic variational inequality (SVI) problem with a continuous and monotone mapping over a closed and convex set. In strongly monotone regimes, we present a variable sample-size averaging scheme (VS-Ave) that achieves a linear rate with an optimal oracle complexity. In addition, the iteration complexity is shown to display a muted dependence on the condition number compared with standard variance-reduced projection schemes. To contend with merely monotone maps, we develop amongst the first proximal-point algorithms with variable sample-sizes (PPAWSS), where increasingly accurate solutions of strongly monotone SVIs are obtained via (VS-Ave) at every step. This allows for achieving a sublinear convergence rate that matches that obtained for deterministic monotone VIs. Preliminary numerical evidence suggests that the schemes compares well with competing schemes.

math.OC↗

arXiv2019

Asynchronous Schemes for Stochastic and Misspecified Potential Games and Nonconvex Optimization

The distributed computation of equilibria and optima has seen growing interest in a broad collection of networked problems. We consider the computation of equilibria of convex stochastic Nash games characterized by a possibly nonconvex potential function. Our focus is on two classes of stochastic Nash games: (P1): A potential stochastic Nash game, in which each player solves a parameterized stochastic convex program; and (P2): A misspecified generalization, where the player-specific stochastic program is complicated by a parametric misspecification. In both settings, exact proximal BR solutions are generally unavailable in finite time since they necessitate solving parameterized stochastic programs. Consequently, we design two asynchronous inexact proximal BR schemes to solve the problems, where in each iteration a single player is randomly chosen to compute an inexact proximal BR solution with rivals' possibly outdated information. Yet, in the misspecified regime (P2), each player possesses an extra estimate of the misspecified parameter and updates its estimate by a projected stochastic gradient (SG) algorithm. By Since any stationary point of the potential function is a Nash equilibrium of the associated game, we believe this paper is amongst the first ones for stochastic nonconvex (but block convex) optimization problems equipped with almost-sure convergence guarantees. These statements can be extended to allow for accommodating weighted potential games and generalized potential games. Finally, we present preliminary numerics based on applying the proposed schemes to congestion control and Nash-Cournot games.

math.OC↗

arXiv2018

Linearly Convergent Variable Sample-Size Schemes for Stochastic Nash Games: Best-Response Schemes and Distributed Gradient-Response Schemes

This paper considers an $N$-player stochastic Nash game in which the $i$th player minimizes a composite objective $f_i(x) + r_i(x_i)$, where $f_i$ is expectation-valued and $r_i$ has an efficient prox-evaluation. In this context, we make the following contributions. (i) Under a strong monotonicity assumption on the concatenated gradient map, we derive ({\bf optimal}) rate statements and oracle complexity bounds for the proposed variable sample-size proximal stochastic gradient-response (VS-PGR) scheme; (ii) We overlay (VS-PGR) with a consensus phase with a view towards developing distributed protocols for aggregative stochastic Nash games. Notably, when the sample-size and the number of consensus steps at each iteration grow at a suitable rate, a linear rate of convergence can be achieved; (iii) Finally, under a suitable contractive property associated with the proximal best-response (BR) map, we design a variable sample-size proximal BR (VS-PBR) scheme, where the proximal BR is computed by solving a sample-average problem. If the batch-size for computing the sample-average is raised at a suitable rate, we show that the resulting iterates converge at a linear rate and derive the oracle complexity.

math.OC↗

arXiv2018

On Synchronous, Asynchronous, and Randomized Best-Response schemes for computing equilibria in Stochastic Nash games

This work considers a stochastic Nash game in which each player solves a parameterized stochastic optimization problem. In deterministic regimes, best-response schemes have been shown to be convergent under a suitable spectral property associated with the proximal best-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization at each iteration. Instead, we propose an inexact generalization in which an inexact solution is computed via an increasing number of projected stochastic gradient steps. Based on this framework, we present three inexact best-response schemes: (i) First, we propose a synchronous scheme where all players simultaneously update their strategies; (ii) Subsequently, we extend this to a randomized setting where a subset of players is randomly chosen to their update strategies while the others keep their strategies invariant; (iii) Finally, we propose an asynchronous scheme, where each player determines its own update frequency and may use outdated rival-specific data in updating its strategy. Under a suitable contractive property of the proximal best-response map, we derive a.s. convergence of the iterates for (i) and (ii) and mean-convergence for (i) -- (iii). In addition, we show that for (i) -- (iii), the iterates converge to the unique equilibrium in mean at a prescribed linear rate. Finally, we establish the overall iteration complexity in terms of projected stochastic gradient steps for computing an $ε-$Nash equilibrium and in all settings, the iteration complexity is ${\cal O}(1/ε^{2(1+c) + δ})$ where $c = 0$ in the context of (i) and represents the positive cost of randomization (in (ii)) and asynchronicity and delay (in (iii)). The schemes are further extended to linear and quadratic recourse-based stochastic Nash games.

math.OC↗

arXiv2018

On stochastic mirror-prox algorithms for stochastic Cartesian variational inequalities: randomized block coordinate and optimal averaging schemes

Motivated by multi-user optimization problems and non-cooperative Nash games in uncertain regimes, we consider stochastic Cartesian variational inequalities (SCVI) 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. For solving this type of problems, the classical stochastic approximation methods and their prox generalizations are computationally inefficient as each iteration becomes very costly. To address this challenge, we develop a randomized block stochastic mirror-prox (B-SMP) algorithm, where at each iteration only a randomly selected block coordinate of the solution is updated through implementing two consecutive projection steps. Under standard assumptions on the problem and settings of the algorithm, 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. To derive rate statements, we assume that the maps are strongly pseudo-monotone and obtain {a non-asymptotic mean squared error $\mathcal{O}\left(\frac{d}{k}\right)$, where $k$ is the iteration number and $d$ is the number of component sets. Second, we consider large-scale stochastic optimization problems with convex objectives. For this class of problems, we develop a new averaging scheme for the B-SMP algorithm. Unlike the classical averaging stochastic mirror-prox (SMP) method where a decreasing set of weights for the averaging sequence is used, here we consider a different set of weights that are characterized in terms of the stepsizes and a {parameter}. We show that using such weights, the objective values of the averaged sequence converges to the optimal value in the mean sense with the rate $\mathcal{O}\left(\frac{\sqrt{d}}{\sqrt{k}}\right)$.

math.OC↗

arXiv2017

Distributed computation of equilibria in misspecified convex stochastic Nash games

The distributed computation of Nash equilibria is assuming growing relevance in engineering where such problems emerge in the context of distributed control. Accordingly, we present schemes for computing equilibria of two classes of static stochastic convex games complicated by a parametric misspecification, a natural concern in the control of large-scale networked engineered system. In both schemes, players learn the equilibrium strategy while resolving the misspecification: (1) Monotone stochastic Nash games: We present a set of coupled stochastic approximation schemes distributed across agents in which the first scheme updates each agent's strategy via a projected (stochastic) gradient step while the second scheme updates every agent's belief regarding its misspecified parameter using an independently specified learning problem. We proceed to show that the produced sequences converge in an almost-sure sense to the true equilibrium strategy and the true parameter, respectively. Surprisingly, convergence in the equilibrium strategy achieves the optimal rate of convergence in a mean-squared sense with a quantifiable degradation in the rate constant; (2) Stochastic Nash-Cournot games with unobservable aggregate output: We refine (1) to a Cournot setting where we assume that the tuple of strategies is unobservable while payoff functions and strategy sets are public knowledge through a common knowledge assumption. By utilizing observations of noise-corrupted prices, iterative fixed-point schemes are developed, allowing for simultaneously learning the equilibrium strategies and the misspecified parameter in an almost-sure sense.

math.OC↗

arXiv2017

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work [44] and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occurring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively.

math.OC↗