In this article, necessary optimality conditions for mathematical programming problems under generalized equation constraints problems are studied in Asplund spaces. We consider a very general version of the problem and derive necessary optimality conditions under various hypothesis on the problem data and sacrificing the differentiability assumption.

In recent times, a variety of nonconvex feasibility problems have been solved empirically by employing Douglas–Rachford (DR) splitting methods. However, this theory is not adequate in explaining the observed success and is more concerned with the local convergence. In this paper, we study the convergence of the DR splitting method for finding a point of intersection of a closed ball and a sphere in the [Formula: see text]-dimensional Euclidean spaces. Also, we provide the region for the global convergence of the DR splitting method.

A displacement aggregation strategy is applied in modified limited memory Broyden Fletcher Goldfarb Shanno (M-LBFGS) algorithm to solve the large-scale unconstrained optimization problems. A displacement aggregation helps in store less memory while updating a new iterate to approximate the inverse Hessian matrix as it discards the generating linear dependence vectors. It has been observed that M-LBFGS scheme achieves the same theoretical convergence properties as the full memory scheme or the limited memory BFGS. Also, Numerical results show that displacement aggregation in adaptive M-LBFGS outperforms M-LBFGS.

No abstract

No abstract