In this paper, we consider the least squares semidefinite programming with a large number of equality and inequality constraints. One difficulty in finding an efficient method for solving this problem is due to the presence of the inequality constraints. In this paper, we propose to overcome this difficulty by reformulating the problem as a system of semismooth equations with two level metric p...

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated ...

Convergence analysis of a nested iterative scheme proposed by Bank,Welfert and Yserentant (BWY) ([Numer. Math., 666: 645-666, 1990]) for solving saddle point system is presented. It is shown that this scheme converges under weaker conditions: the contraction rate for solving the (1, 1) block matrix is bound by ( √ 5− 1)/2. Similar convergence result is also obtained for a class of inexact Uzawa...

We consider a general nonsymmetric second-order linear elliptic PDE in the framework of Lax-Milgram lemma. formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers mesh-refinement inexact iterative solution arising systems. More precisely, solver employs, as outer loop, so-called Zarantonello iteration to symmetrize system and, inner uniformly cont...

In this paper, we propose an inexact alternating direction method with square quadratic proximal  (SQP) regularization for  the structured variational inequalities. The predictor is obtained via solving SQP system  approximately  under significantly  relaxed accuracy criterion  and the new iterate is computed directly by an explicit formula derived from the original SQP method. Under appropriat...

‎In this paper‎, ‎we propose a feasible interior-point method for‎ ‎convex quadratic programming over symmetric cones‎. ‎The proposed algorithm relaxes the‎ ‎accuracy requirements in the solution of the Newton equation system‎, ‎by using an inexact Newton direction‎. ‎Furthermore‎, ‎we obtain an‎ ‎acceptable level of error in the inexact algorithm on convex‎ ‎quadratic symmetric cone programmin...

The proximal point algorithm (PPA) is classical and popular in the community of Optimization. In practice, inexact PPAs which solves the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact PPA with a new inexact criterion for solving convex minimization, and show that the iteration-complexity of this...

Abstract Exact and inexact Douglas–Rachford splitting methods are developed to solve the large-scale sparse absolute value equation (AVE) $Ax - |x| =b$, where $A\in \mathbb {R}^{n\times n}$ $b\in {R}^n$. The method adopts a relative error tolerance and, therefore, in inner iterative processes, LSQR is employed find qualified approximate solution of each subproblem, resulting lower cost for iter...

We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, inner system has to be solved which in theory done exactly. Whenever is getting large, exact solver is, however, no longer efficient or even feasible and iterative methods must used. focus this article on numerical showing inf...