Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

We present algebraic multilevel preconditioners for linear systems arising from the discretization of systems of coupled elliptic partial differential equations (PDEs). These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary … Read more

Simultaneous Pursuit of Out-of-Sample Performance and Sparsity in Index Tracking Portfolios

Index tracking is a passive investment strategy in which an investor purchases a set of assets to mimic a market index. The tracking error, the difference between the performances of the index and the portfolio, may be minimized by buying all the assets contained in the index. However, this strategy results in a considerable amount … Read more

A preconditioning framework for sequences of diagonally modified linear systems arising in optimization

We propose a framework for building preconditioners for sequences of linear systems of the form $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive semidefinite and $\Delta_k$ is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region … Read more

The Gram dimension of a graph

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner products on the edges of the graph. The class of graphs satisfying $\gd(G) … Read more

The Lagrange method and SAO with bounds on the dual variables

We consider the general nonlinear programming problem with equality and inequality constraints when the variables x are confined to a compact set. We regard the Lagrange multipliers as dual variables lambda, those of the inequalities being nonnegative. For each lambda, we let phi(lambda) be the least value of the Lagrange function, which occurs at x=x(lambda), … Read more


In this paper, we consider a class of optimization problems having the following characteristics: there exists a fixed number k which does not depend on the size n of the problem such that if we randomly change the value of k variables, it has the ability to find a new solution that is better than … Read more

Some remarks on stability of generalized equations

The paper concerns the computation of the graphical derivative and the regular (Frechet) coderivative of the solution map to a class of generalized equations, where the multi-valued term amounts to the regular normal cone to a (possibly nonconvex) set given by C2 inequalities. Instead of the Linear Independence qualification condition, standardly used in this context, … Read more

About error bounds in metric spaces

The paper presents a general primal space classification scheme of necessary and sufficient criteria for the error bound property incorporating the existing conditions. Several primal space derivative-like objects – slopes – are used to characterize the error bound property of extended-real-valued functions on metric sapces. Citation Published in D. Klatte et al. (eds.), Operations Research … Read more

Metric regularity of the sum of multifunctions and applications

In this work, we use the theory of error bounds to study of metric regularity of the sum of two multifunctions, as well as some important properties of variational systems. We use an approach based on the metric regularity of epigraphical multifunctions. Our results subsume some recent results by Durea and Strugariu Citation XLIM (UMR-CNRS … Read more

The Decision Rule Approach to Optimization under Uncertainty: Methodology and Applications

Dynamic decision-making under uncertainty has a long and distinguished history in operations research. Due to the curse of dimensionality, solution schemes that naively partition or discretize the support of the random problem parameters are limited to small and medium-sized problems, or they require restrictive modeling assumptions (e.g., absence of recourse actions). In the last few … Read more