An inexact SQP Newton method for convex SC1 minimization problems

Master'sMASTER OF SCIENC

Worst-Case Linear Discriminant Analysis as Scalable Semidefinite Feasibility Problems

In this paper, we propose an efficient semidefinite programming (SDP) approach to worst-case linear discriminant analysis (WLDA). Compared with the traditional LDA, WLDA considers the dimensionality reduction problem from the worst-case viewpoint, which is in general more robust for classification. However, the original problem of WLDA is non-convex and difficult to optimize. In this paper, we reformulate the optimization problem of WLDA into a sequence of semidefinite feasibility problems. To efficiently solve the semidefinite feasibility problems, we design a new scalable optimization method with quasi-Newton methods and eigen-decomposition being the core components. The proposed method is orders of magnitude faster than standard interior-point based SDP solvers. Experiments on a variety of classification problems demonstrate that our approach achieves better performance than standard LDA. Our method is also much faster and more scalable than standard interior-point SDP solvers based WLDA. The computational complexity for an SDP with $m$ constraints and matrices of size $d$ by $d$ is roughly reduced from $\mathcal{O}(m^3+md^3+m^2d^2)$ to $\mathcal{O}(d^3)$ ($m>d$ in our case).Comment: 14 page

Algorithms for Large Scale Nuclear Norm Minimization and Convex Quadratic Semidefinite Programming Problems

Ph.DDOCTOR OF PHILOSOPH

L1 data fitting for robust reconstruction in magnetic particle imaging: quantitative evaluation on Open MPI dataset

Magnetic particle imaging is an emerging quantitative imaging modality, exploiting the unique nonlinear magnetization phenomenon of superparamagnetic iron oxide nanoparticles for recovering the concentration. Traditionally the reconstruction is formulated into a penalized least-squares problem with nonnegativity constraint, and then solved using a variant of Kaczmarz method which is often stopped early after a small number of iterations. Besides the phantom signal, measurements additionally include a background signal and a noise signal. In order to obtain good reconstructions, a preprocessing step of frequency selection to remove the deleterious influences of the noise is often adopted. In this work, we propose a complementary pure variational approach to noise treatment, by viewing highly noisy measurements as outliers, and employing the l1 data fitting, one popular approach from robust statistics. When compared with the standard approach, it is easy to implement with a comparable computational complexity. Experiments with a public domain dataset, i.e., Open MPI dataset, show that it can give accurate reconstructions, and is less prone to noisy measurements, which is illustrated by quantitative (PSNR / SSIM) and qualitative comparisons with the Kaczmarz method. We also investigate the performance of the Kaczmarz method for small iteration numbers quantitatively

A Semismooth Newton-CG Augmented Lagrangian Method for Large Scale Linear and Convex Quadratic SDPS

Ph.DDOCTOR OF PHILOSOPH

Inexact Interior-Point Methods for Large Scale Linear and Convex Quadratic Semidefinite Programming

Ph.DDOCTOR OF PHILOSOPH