A generally nonconvex optimization problem with equality constraints is studied. The problem is introduced as an “inf sup” of a generalized augmented Lagrangian function. A dual problem is defined as the “sup inf’ of the same generalized augmented Lagrangian. Sufftcient conditions are derived for constructing the augmented Lagrangian function such that the extremal values of the primal and dual...

A primal-dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of Kunisch and Rendl [17]. Based on a guess of the active set, a primal-dual pair (x,α) is computed that satisfies stationarity and the complementary condition. If x is not feasible, the variables connected to the infeasibilities are added to the active set...

In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primaldual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. T...

Primal-dual algorithms have played an integral role in recent developments in approximation algorithms, and yet there has been little work on these algorithms in the context of LP relaxations that have been strengthened by the addition of more sophisticated valid inequalities. We introduce primal-dual schema based on the LP relaxations devised by Carr, Fleischer, Leung & Phillips for the minimu...

In this paper, we propose a new primal-dual algorithm for minimizing f(x) + g(x) + h(Ax), where f , g, and h are convex functions, f is differentiable with a Lipschitz continuous gradient, and A is a bounded linear operator. It has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. For example, it reduces to the Chambolle-Pock algorithm when f = 0 and a...

In this paper we propose a perturbed proximal primal dual algorithm (PProx-PDA) for an important class of optimization problems whose objective is the sum of smooth (possibly nonconvex) and convex (possibly nonsmooth) functions subject to a linear equality constraint. This family of problems has applications in a number of statistical and engineering applications, for example in high-dimensiona...

This paper presents a canonical dual approach for solving a nonconvex global optimisation problem with a sum of double-well and log-sum-exp functions. Such a problem arises extensively in mechanics, robot designing, information theory and network communication systems. It includes fourth-order polynomial minimisation problems and minimax problems. Based on the canonical duality theory, this non...

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadr...

Based on a preconditioned version of the randomized block-coordinate forward-backward algorithm recently proposed in [23], several variants of block-coordinate primal-dual algorithms are designed in order to solve a wide array of monotone inclusion problems. These methods rely on a sweep of blocks of variables which are activated at each iteration according to a random rule, and they allow stoc...

Linear programming problems with trapezoidal fuzzy variables (FVLP) have recently attracted some interest. Some methods have been developed for solving these problems by introducing and solving certain auxiliary problems. Here, we apply a linear ranking function to order trapezoidal fuzzy numbers. Then, we establish the dual problem of the linear programming problem with trapezoidal fuzzy varia...