09h45 - 10h10
R-Linear Convergence of Limited Memory Steepest Descent
The limited memory steepest descent method (LMSD) proposed by Fletcher is an extension of the Barzilai-Borwein "two-point step size" strategy for steepest descent methods for solving unconstrained optimization problems. It is known that the Barzilai-Borwein strategy yields a method with an R-linear rate of convergence when it is employed to minimize a strongly convex quadratic. Our work extends this analysis for LMSD, also for strongly convex quadratics. In particular, we show that the method is R-linearly convergent for any choice of the history length parameter. The results of numerical experiments are provided to illustrate behaviors of the method that are revealed through the theoretical analysis.
10h10 - 10h35
Solving large scale nonlinear optimization problems
We consider applications of nonlinear optimization problems with thousands of variables including positron emission tomography reconstruction and training of support vector machines, and discuss how fast projected gradient method can be used to address them.
10h35 - 11h00
An Evolving Subspace Method for Low Rank Minimization
I present a method for solving low rank minimization problems that combines subspace minimization techniques, inexact subspace conditions to terminate exploration of the subspace, and inexact singular value decompositions. Taking together, these features allow the algorithm to scale well, and in fact be competitive with nonconvex approaches that are often used. Convergence results are discussed and preliminary numerical experiments are provided.