Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This version has updated reference

Experimental and data analysis techniques for deducing collision-induced forces from photographic histories of engine rotor fragment impact/interaction with a containment ring

An analysis method termed TEJ-JET is described whereby measured transient elastic and inelastic deformations of an engine-rotor fragment-impacted structural ring are analyzed to deduce the transient external forces experienced by that ring as a result of fragment impact and interaction with the ring. Although the theoretical feasibility of the TEJ-JET concept was established, its practical feasibility when utilizing experimental measurements of limited precision and accuracy remains to be established. The experimental equipment and the techniques (high-speed motion photography) employed to measure the transient deformations of fragment-impacted rings are described. Sources of error and data uncertainties are identified. Techniques employed to reduce data reading uncertainties and to correct the data for optical-distortion effects are discussed. These procedures, including spatial smoothing of the deformed ring shape by Fourier series and timewise smoothing by Gram polynomials, are applied illustratively to recent measurements involving the impact of a single T58 turbine rotor blade against an aluminum containment ring. Plausible predictions of the fragment-ring impact/interaction forces are obtained by one branch of this TEJ-JET method; however, a second branch of this method, which provides an independent estimate of these forces, remains to be evaluated