Bound constrained optimization

This thesis is divided into three parts. In the first part we study a preconditioner proposed by Nash and Sofer when applied to a certain class of problems. The main examples within this class that motivate this work are elliptic variational inequalities with boundary constraints. The class of problems can in general be stated as nonlinear minimization problems although the elliptic variational problems we consider are quadratic problems. We use the projected Newton method to solve these problems and we apply the preconditioner to the linear system that must be solved within the algorithm. The elliptic variational problems are continuous and must therefore be discretized in order to solve them numerically. The discretization of the continuous problems lead to finite dimensional problems that have box constraints and this allows us to use the projected Newton algorithm. The natural approach would however solve these problems with an algorithm based on the continuous problems. Projected Newton is not such an algorithm. We do not take the natural approach because there are significant difficulties (which we will discuss) involved with this approach. The projected Newton algorithm does however converge to the solutions of the discretized problems which suffices. The theory of the preconditioning is based on the locality properties that the discretized operators inherit from the continuous operators and on the smoothing properties of their inverses. We use these properties to analyze the preconditioning of the projected Newton algorithm when used in the context described above. We present numerical results that demonstrate our theoretical results.; In the second part of this thesis we study convergence properties of the BFGS algorithm in the presence of noise. Our motivation was an attempt to explain the observed convergence behavior of the implicit filtering algorithm with a quasi-Newton update. The implicit filtering algorithm is an optimization algorithm that was designed to solve problems where the objective function is the sum of a smooth and easy to minimize function and a small perturbation. We examine both the local and global convergence properties of the BFGS algorithm in the presence of noise. Our theoretical results are based on the existing local and global theory for the convergence of the BFGS quasi-Newton update. We extend these results to functions with perturbations whose amplitudes decay to zero the closer the perturbations are to the solution.; In the final part of the thesis we study a real industrial application of implicit filtering. The application is the optimal design of a high-speed automotive valve train. We discuss the model of the valve train in depth and explain exactly what the optimization problem is. We discuss implementation issues for the model and the implicit filtering algorithm and present a typical result obtained by the implicit filtering algorithm.