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...

The purpose of this paper is to present a new approach for solving linear programming, which has some interesting theoretical properties. In each step of the iteration, we trace a direction completely different from primal simplex method, dual simplex method, primal-dual method and interior point method. The new method is impervious to primal degeneracy and can reach a pair of exact primal and ...

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these Primal–Dual splitting methods....

In this paper, primal-dual methods for general nonconvex nonlinear optimization problems are considered. The proposed methods are exterior point type methods that permit primal variables to violate inequality constraints during the iterations. The methods are based on the exact penalty type transformation of inequality constraints and use a smooth approximation of the problem to form primal-dua...

The primal-dual method (or primal-dual schema) is another means of solving linear programs. The basic idea of this method is to start from a feasible solution y to the dual program, then attempt to find a feasible solution x to the primal program that satisfies the complementary slackness conditions. If such an x cannot be found, it turns out that we can find a better y in terms of its objectiv...

In this work, we study the properties of central paths, defined with respect to a large class of penalty and barrier functions, for convex semidefinite programs. The type of programs studied here is characterized by the minimization of a smooth and convex objective function subject to a linear matrix inequality constraint. So, it is a particular case of convex programming with conic constraints...

In this paper a symmetric primal-dual transformation for positive semideenite programming is proposed. For standard SDP problems, after this symmetric transformation the primal variables and the dual slacks become identical. In the context of linear programming, existence of such a primal-dual transformation is a well known fact. Based on this symmetric primal-dual transformation we derive Newt...

Duality results on countably infinite linear programs are scarce. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix. Subspaces that permit a dual with this transpose do not admit an interior point. This difficulty has stumped researche...

We introduce and study the primal-dual exterior point (PDEP) method for convex optimization problems. The PDEP is based on the Nonlinear Rescaling (NR) multipliers method with dynamic scaling parameters update. The NR method at each step alternates finding the unconstrained minimizer of the Lagrangian for the equivalent problem with both Lagrange multipliers and scaling parameters vectors updat...

Received: date / Revised version: date Abstract. A class Ψ of strongly concave and smooth functions ψ : R → R with specific properties is used to transform the terms of the classical Lagrangian associated with the constraints. The transformation is scaled by a positive vector of scaling parameters, one for each constraint. Each step of the Lagrangian Transformation (LT) method alternates uncons...