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 {\epsilon}-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.Comment: Computational Optimization and Applications, 201

On the resolution of misspecified convex optimization and monotone variational inequality problems

We consider a misspecified optimization problem that requires minimizing a function f(x;q*) over a closed and convex set X where q* is an unknown vector of parameters that may be learnt by a parallel learning process. In this context, We examine the development of coupled schemes that generate iterates {x_k,q_k} as k goes to infinity, then {x_k} converges x*, a minimizer of f(x;q*) over X and {q_k} converges to q*. In the first part of the paper, we consider the solution of problems where f is either smooth or nonsmooth under various convexity assumptions on function f. In addition, rate statements are also provided to quantify the degradation in rate resulted from learning process. In the second part of the paper, we consider the solution of misspecified monotone variational inequality problems to contend with more general equilibrium problems as well as the possibility of misspecification in the constraints. We first present a constant steplength misspecified extragradient scheme and prove its asymptotic convergence. This scheme is reliant on problem parameters (such as Lipschitz constants)and leads us to present a misspecified variant of iterative Tikhonov regularization. Numerics support the asymptotic and rate statements.Comment: 35 pages, 5 figure

Tractable ADMM Schemes for Computing KKT Points and Local Minimizers for ℓ0\ell_0-Minimization Problems

We consider an $\ell_0$-minimization problem where $f(x) + \gamma \|x\|_0$ 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$_{\rm cf}^{\mu, \alpha, \rho}$) 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$_{\rm cf}^{\mu, \alpha, \rho}$), 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.Comment: 47 pages, 3 table

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

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

On robust solutions to uncertain linear complementarity problems and their variants

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.Comment: 37 pages, 3 figures, 8 table

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

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 $\epsilon$-NE are $\mathcal{O}(1/\epsilon^{1/v})$ and $\mathcal{O}(1/\epsilon^{1+1/v})$, 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 $\epsilon$-NE requires similar iteration and oracle complexities to (VS-PGR) with a communication complexity of $\mathcal{O}(\ln^2(1/\epsilon))$; (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 $\mathcal{O}(\ln^2(1/\epsilon))$ 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

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

We consider the structured stochastic convex program requiring the minimization of $\mathbb{E}[\tilde f(x,\xi)]+\mathbb{E}[\tilde g(y,\xi)]$ 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 $\mathcal{O}(1/\epsilon)$. Preliminary applications on LASSO and distributed regression suggest that the scheme performs well compared to its competitors.Comment: 37 pages, 2 figures, 3 table

A Shared-Constraint Approach to Multi-leader Multi-follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria for it has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over the shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.Comment: The earlier manuscript was rejected. We felt it had too many themes crowding it and decided to make a separate paper from each theme. This submission draws some parts from the earlier manuscript and adds new results. Another parts is under review with the IEEE TAC (on arxiv) and another was published in Proc IEEE CDC, 2013. This submission is under review with Set-valued and Variational Analysi