Penalized logistic regression (PLR) is a widely used supervised learning model. In this paper, we consider its applications in largescale data problems and resort to a stochastic primal-dual approach for solving PLR. In particular, we employ a random sampling technique in the primal step and a multiplicative weights method in the dual step. This technique leads to an optimization method with su...

We consider a generic convex optimization problem associated with regularized empirical risk minimization of linear predictors. The problem structure allows us to reformulate it as a convex-concave saddle point problem. We propose a stochastic primal-dual coordinate method, which alternates between maximizing over one (or more) randomly chosen dual variable and minimizing over the primal variab...

A Stochastic Variance Reduced Primal Dual Fixed Point Method for Linearly Constrained Separable Optimization

We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution o...

In this paper we present several \infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to nd a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an-solution for an-perturbation of an initial strictly (primal and dual) feasible prob...

In this paper we present several \infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to nd a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an-solution for an-perturbation of an initial strictly (primal and dual) feasible prob...

This paper deals with a semidefinite program (SDP) having free variables, which often appears in practice. To apply the primal-dual interior-point method, we usually need to convert our SDP into the standard form having no free variables. One simple way of conversion is to represent each free variable as a difference of two nonnegative variables. But this conversion not only expands the size of...

In this paper the abstract of the thesis ”New Interior Point Algorithms in Linear Programming” is presented. The purpose of the thesis is to elaborate new interior point algorithms for solving linear optimization problems. The theoretical complexity of the new algorithms are calculated. We also prove that these algorithms are polynomial. The thesis is composed of seven chapters. In the first ch...

Solving large scale convex semidefinite programming (SDP) problems has long been a challenging task numerically. Fortunately, several powerful solvers including SDPNAL, SDPNAL+ and QSDPNAL have recently been developed to solve linear and convex quadratic SDP problems to high accuracy successfully. These solvers are based on the augmented Lagrangian method (ALM) applied to the dual problems with...