This chapter presents an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions. Along the sequence generated, the duality gap converges to zero at least linearly with a global convergence ratio (1 Yf/n); each iteration reduces the duality gap by at least Yf/n. Here n denotes the size of the probl...

We study the extension of the Chambolle–Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. Local convergence is shown under technical conditions including metric regularity of the corresponding primal-dual optimality conditions. We also show convergence for a Nesterov-type accelerated variant provided one part of the functional i...

In this paper we present a primal-dual interior-point algorithm to solve a class of multi-objective network flow problems. More precisely, our algorithm is an extension of the single-objective primal infeasible dual feasible inexact interior point method for multi-objective linear network flow problems. Our algorithm is contrasted with standard interior point methods and experimental results on...

The primal-dual optimization algorithm developed in Chambolle and Pock (CP) (2011 J. Math. Imag. Vis. 40 1-26) is applied to various convex optimization problems of interest in computed tomography (CT) image reconstruction. This algorithm allows for rapid prototyping of optimization problems for the purpose of designing iterative image reconstruction algorithms for CT. The primal-dual algorithm...

In this paper we present a primal-dual interior-point algorithm to solve a class of multi-objective network flow problems. More precisely, our algorithm is an extension of the single-objective primal-dual infeasible and inexact interior point method for multi-objective linear network flow problems. A comparison with standard interior point methods is provided and experimental results on bi-obje...

This paper presents a brief review and some new developments on the canonical duality theory with applications to a class of variational problems in nonconvex mechanics and global optimization. These nonconvex problems are directly related to a large class of semi-linear partial differential equations in mathematical physics including phase transitions, post-buckling of large deformed beam mode...

Traditionally, a multiuser problem is a constrained optimization problem characterized by a set of users, an objective given by a sum of user-specific utility functions, and a collection of linear constraints that couple the user decisions. The users do not share the information about their utilities, but do communicate values of their decision variables. The multiuser problem is to maximize th...

We propose a new class of algorithms for linear cost network flow problems with and without gains. These algorithms are based on iterative improvement of a dual cost and operate in a manner that is reminiscent of coordinate ascent and Gauss-Seidel relaxation methods. We compare our coded implementations of these methods with mature state-of-theart primal simplex and primal-dual codes, and find ...

These are the notes of my talk presented in the colloquium on discrete curvature at the CIRM, in Luminy (France) on November 21st, 2013, in which we study the space of triangulations from a purely geometric point of view and revisit the results presented in [21] and [20] (joint works with Patrick Mullen, Fernando De Goes and Mathieu Desbrun). Motivated by practical numerical issues in a number ...

The primal-dual algorithm for linear programming is very effective for solving network flow problems. For the method to work, an initial feasible solution to the dual is required. In this article, we show that, for the shortest path problem in a positively weighted graph equipped with a consistent heuristic function, the primal-dual algorithm will become the well-known A* algorithm if a special...