Mathematical Sciences: Algorithms for Large Scale Optimization

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Research will be carried out into algorithms for minimizing nonsmooth composite functions of many variables. Instances of such functions are integrable and bounded data fitting, and penalty functions for nonlinear programming. The general approach is to use a trust-region framework, and iterative methods, in the spirit of gradient projection methods, to solve the subproblem at each iteration. It is intended that the algorithms will be most suitable for nonlinear problems in which the constraints are partially separable (i.e. the contraint derivative matrix has a block-diagonal structure). Important examples of such problems include constrained discrete optimal control, and multicommodity network flow. Particular attention will be given to implementation of the algorithms on parallel computer architectures. Iterative solution methods for large-scale linear programming problems will also be investigated. These methods, which use as a framework some well-established techniques for general constrained optimization, have the advantage that factorization of the constraint matrix is never required. Instead, iterative methods (such as successive over-relaxation or a gradient projection/conjugate gradient technique) are used to solve the convex quadratic programming subproblems at each iteration. Implementation of these various algorithms on vector/parallel processors will be carried out, to determine both the effectiveness of the underlying approach, and the extent to which computer architectures can be utilized.