A Globally Convergent Parallel SSLE Algorithm for Inequality Constrained Optimization

A newparallel variable distribution algorithmbased on interior point SSLE algorithm is proposed for solving inequality constrained optimization problems under the condition that the constraints are block-separable by the technology of sequential system of linear equation. Each iteration of this algorithm only needs to solve three systems of linear equations with the same coefficient matrix to obtain the descent direction. Furthermore, under certain conditions, the global convergence is achieved.

In addition, parallel variable distribution (PVD) algorithm [10] is a method that distributes the variables among parallel processors.The problem is parted into many respective subproblems and each subproblem is arranged to a different processor in it.Each processor has the primary responsibility for updating its block of variables while allowing the remaining secondary variables to change in a restricted fashion along some easily computable directions.In 2002, Sagastizábal and Solodov [11] proposed two new variants of PVD for the constrained case.Without assuming convexity of constraints, but assuming block-separable structure, they showed that PVD subproblems can be solved inexactly by solving their quadratic programming approximations.Han et al. [12] proposed an asynchronous PVT algorithm for solving large-scale linearly constrained convex minimization problems with the idea in 2009, which is based on the idea that a constrained optimization problem is equivalent to a differentiable unconstrained optimization problem by introducing the Fischer function.In 2011, Zheng et al. [13] gave a parallel SSLE algorithm, in which the PVD subproblems are solved inexactly by serial sequential linear equations, for solving large-scale constrained optimization with block-separable structure.Without assuming the convexity of constraints, the algorithm is proved to be globally convergent to a KKT point.
In this paper, we use Zhu [8] as our main reference on SSLE-type PVD method for problem (1).Suppose that the problem (1) has the following block-separable structure: Then, the problem is distributed into  parallel subproblems which have been computed by the  parallel processors.In the algorithm, at each iteration, the search direction is obtained by solving three systems of linear equations with the same coefficient matrix, which guarantees that every iteration is feasible.Thereby, the computational effort of the proposed algorithm is reduced further.Furthermore, its global convergence is obtained under some suitable conditions.
The remaining part of this paper is organized as follows.In Section 2, a parallel SSLE algorithm is presented.Global convergence is established under some basic assumptions in Section 3.And concluding remarks are given in the last section.

Description of Algorithm
Now we state our algorithm as follows.

Computation of the Main Search Direction. Establish a convex combination of 𝑑 0𝑘
and  1  : where 1.4 Computation of the High-Order Corrected Direction.Set Solve the following system of linear equations: where Step 3 (update).Obtain  +1  by updating the positive definite matrix    using some quasi-Newton formulas.Set and

Global Convergence of Algorithm
We make the following general assumptions and let them hold throughout the paper. (H3. Then from the assumption (H3.3), it shows that   = 0 (∀  ∈   (  )).
(1) It is obvious according to the definition of the KKT point of (1).
According to (H3.1), (H3.2), and (H3.4), we might assume that there exists a subsequence  as well, such that In order to obtain the global convergence of the algorithm, we assume the following condition.