We propose a low-rank stochastic gradient descent (LR-SGD) method for solving a class of semidefinite programming (SDP) problems. LR-SGD has clear computational advantages over the standard SGD peers as its iterative projection step (a SDP problem) can be solved in an efficient manner. Specifically, LR-SGD constructs a low-rank stochastic gradient and computes an optimal solution to the project...

In this article, a modified gradient-projection algorithm (GPA) is introduced, which combines Xu’s idea of an alternative averaged mapping approach to the GPA and the general iterative method for nonexpansive mappings in Hilbert space introduced by Marino and Xu. Under suitable conditions, it is proved that the strong convergence of the sequences generated by implicit and explicit schemes to a ...

We consider concave minimization problems over nonconvex sets. Optimization problems with this structure arise in sparse principal component analysis. We analyze both a gradient projection algorithm and an approximate Newton algorithm where the Hessian approximation is a multiple of the identity. Convergence results are established. In numerical experiments arising in sparse principal component...

We consider solving the singular linear system arisen from the Poisson equation with the Neumann boundary condition. To handle the singularity, there are two usual approaches: one is to fix a Dirichlet boundary condition at one point, and the other seeks a unique solution in the orthogonal complement of the kernel. One may incorrectly presume that the two solutions are the similar to each other...

Adversarial training is the most empirically successful approach in improving robustness of deep neural networks for image classification. For text classification, however, existing synonym substitution based adversarial attacks are effective but not very efficient to be incorporated into practical training. Gradient-based attacks, which images, hard implemented due lexical, grammatical and sem...

A numerical method for the solution of the two-dimensional Euler equations for incompressible ow on locally reened grids is presented. The method is a second order Godunov-projection method adapted from Bell, Colella, and Glaz. Second order accuracy of the numerical method in time and space is established through numerical experiments. The main contributions of this work concern the formulation...

Inversion of ill-posed problem from measurement data have been proposed use: i) Conjugate gradient projection method with regularization; ii) Conditional gradient method with regularization; iii) SVD with constraints with regularization method and iv) The method for solving two-dimensional integral equation of convolution type for vector functions using DFT method for Tikhonov functional. The p...