Lagrangian relaxation is commonly used to generate bounds for mixed-integer linear programming problems. However, when the number of dualized constraints is very large (exponential in the dimension of the primal problem), explicit dualization is no longer possible. In order to reduce the dual dimension, different heuristics were proposed. They involve a separation procedure to dynamically selec...

The goal is to analyze the semi-smooth Newton method applied to the solution of contact problems with friction in two space dimensions. The primal-dual algorithm for problems with the Tresca friction law is reformulated by eliminating primal variables. The resulting dual algorithm uses the conjugate gradient method for inexact solving of inner linear systems. The globally convergent algorithm b...

The dual-primal isogeometric tearing and interconnecting (IETI-DP) method is the adaption of the dual-primal finite element tearing and interconnecting (FETI-DP) method to isogeometric analysis of scalar elliptic boundary value problems like, e.g., diffusion problems with heterogeneous diffusion coefficients. The purpose of this paper is to extent the already existing results on condition numbe...

We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical ̄nite element method based on the lowest order Raviart Thomas element, which yields approximations to both the primal and dual variables (the magnetic and electric ̄elds). We prove...

We consider a class of augmented Lagrangian methods for solving convex programming problems with inequality constraints. This class involves a family of penalty functions and specific values of parameters p, q, ỹ ∈ R and c > 0. The penalty family includes the classical modified barrier and the exponential function. The associated proximal method for solving the dual problem is also considered. ...

We consider a family of primal/primal-dual/dual search directions for the monotone LCP over the space of n n symmetric block-diagonal matrices. We consider two infeasible predictor-corrector path-following methods using these search directions, with the predictor and corrector steps used either in series (similar to the Mizuno-Todd-Ye method) or in parallel (similar to Mizuno et al./McShane's m...

A local acceleration method for primal-dual potential-reduction algorithms is introduced. The method developed here uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function, and can be regarded as a primal-dual variant of the Iri-Imai algorithm based on the multiplicative analogue of Karmarkar's potential function. When iterates are close to an optimal solu...

We show quasi-optimal a priori convergence results in the Land H−1/2-norm for the approximation of surface based Lagrange multipliers such as those employed in the mortar finite element method. We improve on the estimates obtained in the standard saddle point theory, where error estimates for both the primal and dual variables are obtained simultaneously and thus only suboptimal a priori estima...

Regularized empirical risk minimization problems are fundamental tasks in machine learning and data analysis. Many successful approaches for solving these problems are based on a dual formulation, which often admits more efficient algorithms. Often, though, the primal solution is needed. In the case of regularized empirical risk minimization, there is a convenient formula for reconstructing an ...

Motivated by various applications to computer vision, we consider the convex cost tension problem, which is the dual of the convex cost flow problem. In this paper, we first propose a primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated minimum cut problems. We show that the time complexity of the prim...