Large-scale nonlinear optimization

Researchers: Anders Forsgren and Tove Odland, partially in cooperation with Philip E. Gill and Elizabeth Wong (UCSD).

The goal of this project is the development of computationally efficient methods for solving large sparse smooth nonlinear optimization problems. Recent work has been focused on linear algebraic procedures for solving quadratic programs. Such problems are of fundamental importance on their own but the major importance is that they form the cornerstone in many methods for solving general nonlinear optimization problems.

One subproject is devoted to active-set methods for convex quadratic programming, where we have devised a symmetric formulation of primal and dual active-set methods. A shifting strategy of the bounds has been devised that allows a general problem to be solved as a sequence of quadratic programs, without the need for a separate feasibility phase.

A second subproject is devoted to solving systems of linear equations for which the matrix is symmetric. We have derived an unnormalized Krylov subspace framework for solving such equations. In this framework, we find a solution if the equations are compatible and a certificate of incompatibility otherwise. The method of conjugate gradients is obtained as a special case. In addition, we have investigated the relationship between quasi-Newton methods and the method of conjugate gradients for minimizing a strictly convex quadratic function.