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...

Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to bot...

An approach to determine primal and dual stepsizes in the infeasible{ interior{point primal{dual method for convex quadratic problems is presented. The approach reduces the primal and dual infeasibilities in each step and allows diierent stepsizes. The method is derived by investigating the eecient set of a multiobjective optimization problem. Computational results are also given.

We consider a Primal-Dual Augmented Lagrangian (PDAL) method for optimization problems with equality constraints. Each step of the PDAL requires solving the Primal-Dual linear system of equations. We show that under the standard second-order optimality condition the PDAL method generates a sequence, which locally converges to the primal-dual solution with quadratic rate.

Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate a sequence of iterates that are feasible with respe...

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...

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...

The primal-dual method increases the dual variables gradually until some dual constraint becomes tight. Then, the primal variable corresponding to the tight dual constraint is ‘bought’ (or selected), and the process continues till we get a feasible primal solution. Next, we compare the value of the primal solution to the value of the dual solution to get an appropriate approximation factor (or ...

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 ...

We consider the total variation minimization model with consistent finite element discretization. It has been shown in the literature that this model can be reformulated as a saddle-point problem and be efficiently solved by the primal-dual method. The convergence for this application of the primal-dual method has also been analyzed. In this paper, we focus on a more general primaldual scheme w...