Element-by-Element Preconditioners for Large Partially Separable Optimization Problems

We study the solution of large-scale nonlinear optimization problems by methods which aim to exploit their inherent structure. In particular, we consider the property of partial separability, first studied by Griewank and Toint [Nonlinear Optimization, 1981, pp. 301–312]. A typical minimization method for nonlinear optimization problems approximately solves a sequence of simplified linearized subproblems. In this paper, we explore how partial separability may be exploited by iterative methods for solving these subproblems. We particularly address the issue of computing effective preconditioners for such iterative methods. We concentrate on element-by-element preconditioners which reflect the structure of the problem. We find that the performance of these methods can be considerably improved by amalgamating elements before applying the preconditioners. We report the results of numerical experiments which demonstrate the effectiveness of this approach.