Extended Formulations in Mixed-integer Convex Programming

We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations. We then connect the popular modeling framework of disciplined convex programming (DCP) to the existence of extended formulations independent of conic representability. We present evidence that our approach can yield significant gains in practice, with the solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201

High performance interior point methods for three-dimensional finite element limit analysis

The ability to obtain rigorous upper and lower bounds on collapse loads of various structures makes finite element limit analysis an attractive design tool. The increasingly high cost of computing those bounds, however, has limited its application on problems in three dimensions. This work reports on a high-performance homogeneous self-dual primal-dual interior point method developed for three-dimensional finite element limit analysis. This implementation achieves convergence times over 4.5× faster than the leading commercial solver across a set of three-dimensional finite element limit analysis test problems, making investigation of three dimensional limit loads viable. A comparison between a range of iterative linear solvers and direct methods used to determine the search direction is also provided, demonstrating the superiority of direct methods for this application. The components of the interior point solver considered include the elimination of and options for handling remaining free variables, multifrontal and supernodal Cholesky comparison for computing the search direction, differences between approximate minimum degree [1] and nested dissection [13] orderings, dealing with dense columns and fixed variables, and accelerating the linear system solver through parallelization. Each of these areas resulted in an improvement on at least one of the problems in the test set, with many achieving gains across the whole set. The serial implementation achieved runtime performance 1.7× faster than the commercial solver Mosek [5]. Compared with the parallel version of Mosek, the use of parallel BLAS routines in the supernodal solver saw a 1.9× speedup, and with a modified version of the GPU-enabled CHOLMOD [11] and a single NVIDIA Tesla K20c this speedup increased to 4.65×

Deciphering Okmok Volcano's restless years (2002-2005)

Thesis (Ph.D.) University of Alaska Fairbanks, 2015Okmok Volcano is an active island-arc shield volcano located in the central Aleutian islands of Alaska. It is defined by a 10-km-diameter caldera that formed in two cataclysmic eruptions, the most recent being ~2050 years ago. Subsequent eruptions created several cinder cones within the caldera. The youngest of these, Cone A, was the active vent from 1815 through its 1997 eruption. On July 12 2008 Okmok erupted from new vents located northwest of Cone D. Between 2001 and 2004, geodetic measurements showed caldera inflation. These studies suggested that new magma might be entering the system. In 2002, a newly installed seismic network recorded quasi-periodic ("banded") seismic tremor signals occurring at the rate of two or more episodes per hour. This tremor was a near-continuous signal from the day the seismic network was installed. Although the volcano was not erupting, it was clearly in a state of unrest. This unrest garnered considerable attention because the volcano had erupted just six years prior. The seismic tremor potentially held insight as to whether the unrest was a remnant of the 1997 eruption, or whether it signaled a possible rejuvenation of activity and the potential for eruption. To determine the root cause and implications of this remarkable seismic tremor sequence, I created a catalog of over ~17,000 tremor events recorded between 2003 and mid-2005. Tremor patterns evolved on the scale of days, but remained the dominant seismic signal. In order to facilitate the analysis of several years of data I created a MATLAB toolbox, known as "The Waveform Suite". This toolbox made it feasible for me to work with several years of digital data and forego my introductory analyses that were based on paper "helicorder" records. I first attempted to locate the tremor using the relative amplitudes of the seismograms to determine where the tremor was being created. Candidate tremor locations were constrained to a few locations along a corridor between Cone A and the caldera center. I then determined theoretical ratios between a reference station and stations nearby the candidate sources. Results suggested that the signal originated in the shallow portion of the corridor connecting the surface of Cone A to the top of the central magma chamber. This study also suggested that the source migrated along this corridor. I integrated the tremor patterns with other studies and proposed that heat and pressure from continued injections of magma were responsible for maintaining an open venting system at Cone A. The tremor resulted from the boiling of a shallow hydrothermal system in the vicinity of Cone A and volatiles potentially coming from the magma itself. The tremor catalog demonstrates that the seismic signal waned during the study period suggesting that fewer fresh volatiles entered the system, which may have allowed the pathways connecting the magma and volatiles to the surface to close up. By the time new magma entered the system in 2006, this network of pathways was closed, forcing the volatiles to seek a new exit. In hindsight, the 2003-2005 period of varied and waning seismic tremor, and the inferred end of massive open venting, may have been a pivotal era at Okmok that eventually led to the 2008 eruption

Adjoint-based predictor-corrector sequential convex programming for parametric nonlinear optimization

This paper proposes an algorithmic framework for solving parametric optimization problems which we call adjoint-based predictor-corrector sequential convex programming. After presenting the algorithm, we prove a contraction estimate that guarantees the tracking performance of the algorithm. Two variants of this algorithm are investigated. The first one can be used to solve nonlinear programming problems while the second variant is aimed to treat online parametric nonlinear programming problems. The local convergence of these variants is proved. An application to a large-scale benchmark problem that originates from nonlinear model predictive control of a hydro power plant is implemented to examine the performance of the algorithms.Comment: This manuscript consists of 25 pages and 7 figure

Advanced flight control system study

The architecture, requirements, and system elements of an ultrareliable, advanced flight control system are described. The basic criteria are functional reliability of 10 to the minus 10 power/hour of flight and only 6 month scheduled maintenance. A distributed system architecture is described, including a multiplexed communication system, reliable bus controller, the use of skewed sensor arrays, and actuator interfaces. Test bed and flight evaluation program are proposed