## Spectral Gap Optimization of Divergence Type Diffusion Operators

In this paper, we address the problem of maximizing the spectral gap of a divergence type diffusion operator. Our main application of interest is characterizing the distribution of a swarm of agents that evolve on a bounded domain in Rn according to a Markov process. A subclass of the divergence type operators that we introduce … Read more

## FINITE ELEMENT MODEL UPDATING FOR STRUCTURAL APPLICATIONS

A novel method for performing model updating on finite element models is presented. The approach is particularly tailored to modal analyses of buildings, by which the lowest frequencies, obtained by using sensors and system identification approaches, need to be matched to the numerical ones predicted by the model. This is done by optimizing some unknown … Read more

## AN INEQUALITY-CONSTRAINED SQP METHOD FOR EIGENVALUE OPTIMIZATION

We consider a problem in eigenvalue optimization, in particular find- ing a local minimizer of the spectral abscissa – the value of a parameter that results in the smallest magnitude of the largest real part of the spectrum of a matrix system. This is an important problem for the stabilization of control sys- tems. Many … Read more

## The Spectral Bundle Method with Second-Order Information

The spectral bundle method was introduced by Helmberg and Rendl to solve a class of eigenvalue optimization problems that is equivalent to the class of semidefinite programs with the constant trace property. We investigate the feasibility and effectiveness of including full or partial second-order information in the spectral bundle method, building on work of Overton … Read more

## An acceleration procedure for optimal first-order methods

We introduce in this paper an optimal first-order method that allows an easy and cheap evaluation of the local Lipschitz constant of the objective’s gradient. This constant must ideally be chosen at every iteration as small as possible, while serving in an indispensable upper bound for the value of the objective function. In the previously … Read more

## A randomized Mirror-Prox method for solving structured large-scale matrix saddle-point problems

In this paper, we derive a randomized version of the Mirror-Prox method for solving some structured matrix saddle-point problems, such as the maximal eigenvalue minimization problem. Deterministic first-order schemes, such as Nesterov’s Smoothing Techniques or standard Mirror-Prox methods, require the exact computation of a matrix exponential at every iteration, limiting the size of the problems … Read more

## Structural optimization of the Ziegler’s pendulum: singularities and exact optimal solutions

Structural optimization of non-conservative systems with respect to stability criteria is a research area with important applications in fluid-structure interactions, friction-induced instabilities, and civil engineering. In contrast to optimization of conservative systems where rigorously proven optimal solutions in buckling problems have been found, for non-conservative optimization problems only numerically optimized designs were reported. The proof … Read more

## 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

## The Rotational Dimension of a Graph

Given a connected graph $G=(N,E)$ with node weights $s\in\R^N_+$ and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of $G$: Find $v_i\in\R^{|N|}$, $i\in N$ so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted barycenter … Read more

## Embedded in the Shadow of the Separator

We study the problem of maximizing the second smallest eigenvalue of the Laplace matrix of a graph over all nonnegative edge weightings with bounded total weight. The optimal value is the \emph{absolute algebraic connectivity} introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Using semidefinite programming techniques and … Read more