An algorithm for solving large-scale nonlinear ' programs with linear constraints is presented. The method combines efficient sparse-matrix techniques as in the revised simplex method with stable quasi-Newton methods for handling the nonlinearities. A general-purpose production code (MINOS) is described, along with computational experience on a wide variety of problems.

This paper discusses certain connections between nonlinear programming algorithms and the formulation of optimization problems for systems governed by state constraints. I work through the calculation of the sensitivities associated with the di erent formulations and present some useful relationships between them. These relationships have practical consequences; if one uses a reduced basis nonlinear programming algorithm, then the implementations for the di erent formulations need only di er in a single step.

We discuss a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems appear in many applications and are of a different nature than systems of equations. Our approach uses an optimization-based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids. Our goal is not to argue for a particular optimization-based multigrid algorithm, but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms. Furthermore, using several model problems we give evidence (both theoretical and numerical) that the optimization setting is well suited to multigrid algorithms. Some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation. In addition, we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization-based multigrid algorithm.

This paper discusses the calculation of sensitivities, or derivatives, for optimization problems involving systems governed by differential equations and other state relations. The subject is examined from the point of view of nonlinear programming, beginning with the analytical structure of the first and second derivatives associated with such problems and the relation of these derivatives to implicit differentiation and equality constrained optimization. We also outline an error analysis of the analytical formulae and compare the results with similar results for finite-difference estimates of derivatives. We then attend to an investigation of the nature of the adjoint method and the adjoint equations and their relation to directions of steepest descent. We illustrate the points discussed with an optimization problem in which the variables are the coefficients in a differential operator. This research was supportedby the National Aeronautics and Space Administrationunder NASA Contra...