Optimization

Joint work with Sven Leyffer (Argonne National Laboratory)
and Emilie Wanufelle (University of Namur).

Motivated by problems related to power systems analysis which give rise
to nonconvex mixed integer nonlinear programming (MINLP) problems,
we propose a global optimization method based on ideas and techniques
that can be easily extended to handle a large class of nonconvex MINLPs.

Our method decomposes the nonlinear functions appearing in the problem
to solve into one- and two-dimensional components for which piecewise

Joint work with Evrim Acar, and Daniel M. Dunlavy
(Sandia National Laboratories).

We discuss optimal low-rank approximation of matrices with non-negative entries, without the need of a regularization parameter. It will be shown that the standard SVD-approximation can be recovered via convex-optimization, which is why adding mild convex constraints often gives an optimal solution. Moreover, the issue of computability will be addressed by solving our new convex problem via the so-called Douglas-Rachford algorithm. We will see that if there is a unique optimal solution than also the non-convex Douglas-Rachford will locally converge to it.

Mathematical models are employed ubiquitously for description, prediction and decision making. In addressing end-goal objectives, great care needs to be devoted to attainment of appropriate balance of inexactness throughout the various stages of the end goal process (e.g. modeling and optimization). Disregard to such considerations, either entails redundant computation or impairment of the overall fidelity of the optimization process.

We survey some developments in machine learning and data analysis,
focusing on those in which optimization is an important
component. Some of these have possible relevance for industrial and
energy applications, for example, constraints and covariances could be
learned from process data rather than specified a priori. Some
possibilities along these lines will be proposed.

Deep brain stimulation is a therapy where an electrode is placed in the
brain and periodic electrical pulses are delivered for treatment of
diseases such as Parkinson's Disease and Epilepsy. This therapy has
been quite successful in Parkinson's and moderately successful in
epilepsy. Despite the wide range of stimulation parameters that can be
used, only a small range is used by the clinician setting them. A
closed loop optimization algorithm optimizing stimulation parameters

Parsimony, including sparsity and low rank, has been shown to successfully model
data in numerous machine learning and signal processing tasks. Traditionally, such
modeling approaches rely on an iterative algorithm that minimizes an objective
function with parsimony-promoting terms. The inherently sequential structure and
data-dependent complexity and latency of iterative optimization constitute a major

An increasing amount of the oil and gas produced in North America and the world is from unconventional reservoirs. Understanding the fractures within the reservoir plays an important role in developing this resource. The objective of this project is to infer from P-wave seismic amplitude variations with offset and azimuth (AVAz) the elastic parameters of the earth and then from these elastic parameters characterize the fractures (Figure 1).

Algorithms for design optimization are increasingly able to handle complex problem formulations. We will consider the design of a fuel tank consisting of four different disciplinary sub-system components- structures, aerodynamics, cost, and systems. This is a multi-disciplinary, design problem with multiple competing objectives. We will examine several formulations and how to best match the problem formulation with the choice of optimizer.