Numerical Methods of Linear and Nonlinear Optimization

The project has been concerned with developing new numerical techniques to solve large scale linear and nonlinear programming problems. Early work focused on sequential quadratic programming techniques for nonlinear programming. Subsequently, all work was focused on interior point methods for large scale linear and nonlinear programming. Initially, the focus of the research was on both dual-affine and primal-dual algorithm for linear programming. Substantial computational experience demonstrated the superiority of the primal-dual methods, and subsequent research focused on improving the efficiency of these methods, both by adding higher order methods via predictor- corrector techniques and by improving the linear algebra to take advantage of both sparsity and machine architecture. Most recently, research has focused on large scale quadratic programming. A primal-dual predictor-corrector method has been devised and shown to be very promising computationally for problems with diagonal or sparsely structured Hessian matrices. For problems with dense Hessians, a pure primal conjugate projected gradient algorithm shows promise on small problems. It remains to be tested on large-scale problems.