1,399 research outputs found
Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities
Stochastic variational inequalities (SVI) model a large class of equilibrium
problems subject to data uncertainty, and are closely related to stochastic
optimization problems. The SVI solution is usually estimated by a solution to a
sample average approximation (SAA) problem. This paper considers the normal map
formulation of an SVI, and proposes a method to build asymptotically exact
confidence regions and confidence intervals for the solution of the normal map
formulation, based on the asymptotic distribution of SAA solutions. The
confidence regions are single ellipsoids with high probability. We also discuss
the computation of simultaneous and individual confidence intervals
Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators on Domains Given by Linear Minimization Oracles
The majority of First Order methods for large-scale convex-concave saddle
point problems and variational inequalities with monotone operators are
proximal algorithms which at every iteration need to minimize over problem's
domain X the sum of a linear form and a strongly convex function. To make such
an algorithm practical, X should be proximal-friendly -- admit a strongly
convex function with easy to minimize linear perturbations. As a byproduct, X
admits a computationally cheap Linear Minimization Oracle (LMO) capable to
minimize over X linear forms. There are, however, important situations where a
cheap LMO indeed is available, but X is not proximal-friendly, which motivates
search for algorithms based solely on LMO's. For smooth convex minimization,
there exists a classical LMO-based algorithm -- Conditional Gradient. In
contrast, known to us LMO-based techniques for other problems with convex
structure (nonsmooth convex minimization, convex-concave saddle point problems,
even as simple as bilinear ones, and variational inequalities with monotone
operators, even as simple as affine) are quite recent and utilize common
approach based on Fenchel-type representations of the associated
objectives/vector fields. The goal of this paper is to develop an alternative
(and seemingly much simpler) LMO-based decomposition techniques for bilinear
saddle point problems and for variational inequalities with affine monotone
operators
A Unifying Geometric Solution Framework and Complexity Analysis for Variational Inequalities
In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids, and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems. The analysis rests upon the assumption of strong-f-monotonicity, which is weaker than strict and strong monotonicity. Since linear programs satisfy this assumption, the general framework applies to linear programs
Progress on the Strong Eshelby's Conjecture and Extremal Structures for the Elastic Moment Tensor
We make progress towards proving the strong Eshelby's conjecture in three
dimensions. We prove that if for a single nonzero uniform loading the strain
inside inclusion is constant and further the eigenvalues of this strain are
either all the same or all distinct, then the inclusion must be of ellipsoidal
shape. As a consequence, we show that for two linearly independent loadings the
strains inside the inclusions are uniform, then the inclusion must be of
ellipsoidal shape. We then use this result to address a problem of determining
the shape of an inclusion when the elastic moment tensor (elastic
polarizability tensor) is extremal. We show that the shape of inclusions, for
which the lower Hashin-Shtrikman bound either on the bulk part or on the shear
part of the elastic moment tensor is attained, is an ellipse in two dimensions
and an ellipsoid in three dimensions
On the Efficient Solution of Variational Inequalities; Complexity and Computational Efficiency
In this paper we combine ideas from cutting plane and interior point methods in order to solve variational inequality problems efficiently. In particular, we introduce a general framework that incorporates nonlinear as well as linear "smarter" cuts. These cuts utilize second order information on the problem through the use of a gap function. We establish convergence as well as complexity results for this framework. Moreover, in order to devise more practical methods, we consider an affine scaling method as it applies to symmetric, monotone variationalinequality problems and demonstrate its convergence. Finally, in order to further improve the computational efficiency of the methods in this paper, we combine the cutting plane approach with the affine scaling approach
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