15th EUROPT Workshop on Advances in Continuous Optimization
Montréal, Canada, 12 — 14 July 2017
15th EUROPT Workshop on Advances in Continuous Optimization
Montréal, Canada, 12 — 14 July 2017
FactorizationFree Methods for LargeScale Optimization
Jul 13, 2017 01:30 PM – 03:10 PM
Location: TD Assurance Meloche Monnex
Chaired by Dominique Orban
4 Presentations

01:30 PM  01:55 PM
Efficient computed tomography reconstruction algorithms in cylindrical coordinates
Three dimensional Xray computed tomography (CT) is of great importance for medical diagnosis, nondestructive testing, etc. CT reconstruction can be viewed as a very large optimization problem, possibly with box constraints, in which the computation time is critical. Here, we present an approach in which the medium under study is represented in cylindrical coordinates so as to take advantage of invariances occurring in the data collection process and derive a highly parallel structure. In this communication, we discuss the issues of numerical efficiency, conditioning of the problem, development of specific preconditioners and ability to account for box constraints.

01:55 PM  02:20 PM
Factorizationfree methods for computed tomography
We study a tomographic reconstruction problem in cylindrical coordinates. A change of variables involving a Fourier matrix attempts to improve the conditioning of the Hessian but introduces linear inequality constraints. The scale and density of the problem call for factorizationfree methods. We argue that projections into the feasible set can be computed efficiently by solving a boundconstrained leastsquares problem with a fast linear operator. In this talk, we focus on a BarzilaiBorwein projected gradient method and a trustregion projected Newton method. We compare two solvers for the projection subproblem: a twometric projection algorithm and a trustregion projected Newton method. The performance of several combinations of these techniques is assessed using realistic synthetic data.

02:20 PM  02:45 PM
A FactorizationFree InteriorPoint Method for Constrained LeastSquares Problems with Exact Regularization
We present an interiorpoint method for linear leastsquares problems with linear constraints. The method employs the exact primaldual regularization procedure of Friedlander and Orban (2012) to treat degenerate constraints. The method is factorizationfree in the sense that neither the constraint Jacobian nor the leastsquares operator need be available as explicit matrices; only matrixoperator products are required. Numerical results demonstrate the effectiveness and robustness of the method.

02:45 PM  03:10 PM
A FactorizationFree Regularized Sequential Quadratic Optimization Method
When formulated appropriately, augmented Lagrangian methods require the solution of a symmetric quasidefinite linear system at each iteration. The latter are indefinite but their strong relationships with definite systems enable specialized linear algebra and make them prime candidates for matrixfree implementations. We illustrate how the adequatelyformulated augmented Lagrangian method for equalityconstrained problems provides the motivation for regularized sequential quadratic optimization. We present an efficient factorizationfree implementation for largescale problems, describe global and fast local convergence results, and report on numerical experiments. We conclude with comments on extensions to inequality constraints.