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Abstract. We propose an extragradient method with stepsizes bounded away from zero for stochastic variational inequalities requiring only pseudo-monotonicity. We provide convergence and complexity analysis, allowing for an unbounded feasible set, unbounded operator, non-uniform variance of the oracle and, also, we do not require any regularization. Alongside the stochastic approximation procedure, we iteratively reduce the variance of the stochastic error. Our method attains the optimal oracle complexity O(1/ǫ 2 ) (up to a logarithmic term) and a faster rate O(1/K) in terms of the mean (quadratic) natural residual and the D-gap function, where K is the number of iterations required for a given tolerance ǫ > 0. Such convergence rate represents an acceleration with respect to the stochastic error. The generated sequence also enjoys a new feature: the sequence is bounded in L p if the stochastic error has finite p-moment. Explicit estimates for the convergence rate, the oracle complexity and the p-moments are given depending on problem parameters and distance of the initial iterate to the solution set. Moreover, sharper constants are possible if the variance is uniform over the solution set or the feasible set. Our results provide new classes of stochastic variational inequalities for which a convergence rate of O(1/K) holds in terms of the mean-squared distance to the solution set. Our analysis includes the distributed solution of pseudo-monotone Cartesian variational inequalities under partial coordination of parameters between users of a network.

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We consider the vector optimization problem of finding weakly efficient points for maps from a Hilbert space X to a Banach space Y , with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Y with a nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization. In this extension, the subproblems consist of finding weakly efficient points for suitable regularizations of the original map. We present both an exact and an inexact version, in which the subproblems are solved only approximately, within a constant relative tolerance. In both cases, we prove weak convergence of the generated sequence to a weakly efficient point, assuming convexity of the map with respect to C and C-completeness of the initial section. In cases where this last assumption fails, we still establish that the generating sequence is, in a suitable sense, a minimizing one. We also exhibit a particular instance of the algorithm for which, under a mild hypothesis on C, the weak limit of the generated sequence is an efficient, rather than a weakly efficient, point. Introduction.We discuss methods for vector-valued optimization in the following setting. We consider maps F : X → Y , where X is a real Hilbert space and Y is a real Banach space containing a closed, convex, and pointed cone C with nonempty interior, which defines a partial order C in Y , given by y C y if and only if y − y belongs to C, with its associate relation ≺ C , given by y ≺ C y if and only if y − y belongs to the interior int(C) of C. Actually, we admit the possibility that F takes the value +∞; this is made precise in section 2. Our goal is to analyze methods for finding a weakly efficient minimizer of F with respect to C , meaning a point a ∈ X such that there exists no x ∈ X satisfying F (x) ≺ C F (a). This paper is part of a wider research program consisting of the extension to vector-valued optimization of several iterative methods for scalar-valued methods. In these extensions, we attempt to define the iterates in the vector-valued case by considering the order C in Y , mimicking, whenever possible, the role of the usual order in R in the corresponding algorithm for scalar-valued optimization.Several methods already have been extended in this fashion in a finite dimensional setting. The case of the steepest descent method for multiobjective optimization (i.e., when C is the nonnegative orthant of R n ) was dealt with in [7]; the same method for general finite dimensional vector optimization (i.e., for partial orders given by rather general cones in R n ) is analyzed in [10]. An extension of the projected gradient method to the case of convexly constrained vector optimization, for the order given by a rather general cone in R n , can be found in [9].

We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to an ek-subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizes ek are exogenously given, satisfying ~=0 c~k ec, ~=0 c~ < ec, and ek is chosen so that ek ~ 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results.

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Several finite procedures for determining the step size of the steepest descent method for unconstrained optimization, without performing exact one-dimensional minimizations, have been considered in the literature. The convergence analysis of these methods requires that the objective function have bounded level sets and that its gradient satisfy a Lipschitz condition, in order to establish just stationarity of all cluster points. We consider two of such procedures and prove, for a convex objective, convergence of the whole sequence to a minimizer without any level set boundedness assumption and, for one of them, without any Lipschitz condition.