Covariance regularization in inverse space

This paper proposes to apply Gaussian graphical models to estimate the large-scale normal distribution in the context of data assimilation from a relatively small number of data from the satellite. Data assimilation is a field which fits simulation models to observation data developed mainly in meteorology and oceanography. The optimization problem tends to be huge … Read more

Analysis and Generalizations of the Linearized Bregman Method

This paper reviews the Bregman methods, analyzes the linearized Bregman method, and proposes fast generalization of the latter for solving the basis pursuit and related problems. The analysis shows that the linearized Bregman method has the exact penalty property, namely, it converges to an exact solution of the basis pursuit problem if and only if … Read more

Eigenvalue techniques for proving bounds for convex objective, nonconvex programs

We describe techniques combining the S-lemma and computation of projected quadratics which experimentally yield strong bounds on the value of convex quadratic programs with nonconvex constraints Citation unpublished report, Columbia University, March 2009 Article Download View Eigenvalue techniques for proving bounds for convex objective, nonconvex programs

Graph Realizations Associated with Minimizing the Maximum Eigenvalue of the Laplacian

In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes … Read more

Stochastic Nash Equilibrium Problems: Sample Average Approximation and Applications

This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The well known sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a … Read more

Explicit Sensor Network Localization using Semidefinite Representations and Facial Reductions

The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP, completion … Read more

Cutting Plane Methods and Subgradient Methods

Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of … Read more

Strengthening lattice-free cuts using non-negativity

In recent years there has been growing interest in generating valid inequalities for mixed-integer programs using sets with 2 or more constraints. In particular, Andersen (2007) and Borozan and Cornue’jols (2007) study sets defined by equations that contain exactly one integer variable per row. The integer variables are not restricted in sign. Cutting planes … Read more

Constrained Infinite Group Relaxations of MIPs

Recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized. In this paper, we consider a stronger relaxation of general mixed integer sets by allowing constraints, such as bounds, on the free integer variables in the continuous group relaxation. We generalize a number of results for the continuous … Read more

Iteration-complexity of first-order augmented Lagrangian methods for convex programming

This paper considers a special class of convex programming (CP) problems whose feasible regions consist of a simple compact convex set intersected with an affine manifold. We present first-order methods for this class of problems based on an inexact version of the classical augmented Lagrangian (AL) approach, where the subproblems are approximately solved by means … Read more