Convergence Conditions for Variational Inequality Algorithms

Within the extensive variational inequality literature, researchers have developed many algorithms. Depending upon the problem setting, these algorithms ensure the convergence of (i) the entire sequence of iterates, (ii) a subsequence of the iterates, or (iii) averages of the iterates. To establish these convergence results, the literature repeatedly invokes several basic convergence theorems. In this paper, we review these theorems and a few convergence results they imply, and introduce a new result, called the orthogonality theorem, for establishing the convergence of several algorithms for solving a certain class of variational inequalities. Several of the convergence results impose a condition of strong-f-monotonicity on the problem function. We also provide a general overview of the properties of strong-f-monotonicity, including some new results (for example, the relationship between strong-f-monotonicity and convexity)

Data-Driven Estimation in Equilibrium Using Inverse Optimization

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and \emph{nonparametric} estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.Comment: 36 pages, 5 figures Additional theorems for generalization guarantees and statistical analysis adde

Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems

We prove that for compactly perturbed elliptic problems, where the corresponding bilinear form satisfies a Garding inequality, adaptive mesh-refinement is capable of overcoming the preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. As an important consequence of our analysis, one does not have to deal with the a-priori assumption that the underlying meshes are sufficiently fine. Hence, the overall conclusion of our results is that adaptivity has stabilizing effects and can overcome possibly pessimistic restrictions on the meshes. In particular, our analysis covers adaptive mesh-refinement for the finite element discretization of the Helmholtz equation from where our interest originated