The paper is devoted to domain decomposition methods (DDM in short) for the 18 Stokes problem with the slip boundary conditions. The original domain is cut into 19 two sub-domains and the augmented Lagrangian formulation for separate resulting 20 Poisson problems in both domains is used for computations. To relate solutions of 21 these two sub-problems to the original solution, one has to intro...

A numerical method for the computation of the best constant in a Sobolev inequality involving the spacesH2(Ω) and C0(Ω) is presented. Green’s functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagr...

First-order methods (FOMs) have been widely used for solving large-scale problems. A majority of existing works focus on problems without constraint or with simple constraints. Several recent studied FOMs complicated functional In this paper, we design a novel augmented Lagrangian (AL)–based FOM nonconvex objective and convex functions. The new method follows the framework proximal point (PP) m...

The aim of this work is to propose a new numerical method for solving the mechanical frictional contact problem in the general case of multi-bodies in a three dimensional space. This method is called adapted augmented Lagrangian method (AALM) and can be used in a multi-physical context (like thermo-electro-mechanical fields problems). This paper presents this new method and its advantages over ...

The paper focuses on the convergence rate of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Under a set of assumptions of sufficient conditions, including the componentwise strict complementarity condition, the constraint nondegeneracy condition and the second order sufficient condition, we first study some properties of the augmented Lagrangian and then ...

This paper describes several methods for solving nonlinear complementarity problems. A general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem, obtaining primal, dual, and primal-dual formulations. We derive Bregman-function-based generalized proximal algorithms for each of these formulations, generating three classes of co...

We consider the class of semi-infinite programming problems which became in recent years a powerful tool for the mathematical modelling of many real-life problems. In this paper, we study an augmented Lagrangian approach to semi-infinite problems and present necessary and sufficient conditions for the existence of corresponding augmented Lagrange multipliers. Furthermore, we discuss two particu...

In the context of SQP methods or, more recently, of sequential semidefinite programming methods, it is common practice to construct a positive semidefinite approximation of the Hessian of the Lagrangian. The Hessian of the augmented Lagrangian is a suitable approximation as it maintains local superlinear convergence under appropriate assumptions. In this note we give a simple example that the o...

Dual decomposition has been recently proposed as a way of combining complementary models, with a boost in predictive power. However, in cases where lightweight decompositions are not readily available (e.g., due to the presence of rich features or logical constraints), the original subgradient algorithm is inefficient. We sidestep that difficulty by adopting an augmented Lagrangian method that ...