Walter Murray

New approaches to linear and nonlinear programming

This report describes technical progress during the past twelve months on DOE Contract DE-FG-87ER25030 and requests support for the third year. The project involves study of the theoretical properties and computational performance of techniques that solve linear and nonlinear programs by means of nonlinear transformations. The group at the Systems Optimization Laboratory (SOL) were the first to recognize the connection between Karmarkars projective method and the logarithmic barrier method. It is now generally recognized that essentially all interior-point methods for linear programming inspired by Karmarkars method are closely related to application of Newtons method to a sequence of barrier functions. Each barrier function is defined from the objective function and a barrier term that is infinite along the boundary of the feasible region. As the weight on the barrier term is reduced to zero, the solution of the subproblem becomes closer to the solution of the original problem.