Exact and inexact Newton linesearch interior-point algorithms for nonlinear programming problems
Argaez Ramos, Miguel
Tapia, Richard A.
Doctor of Philosophy
In the first part of this research we consider a linesearch globalization of the local primal-dual interior-point Newton's method for nonlinear programming recently introduced by El-Bakry, Tapia, Tsuchiya, and Zhang. Our linesearch uses a new merit function and a new notion of centrality. We establish a global convergence theory and present rather promising numerical experimentation. In the second part, we consider an inexact Newton's method implementation of the linesearch globalization algorithm given in the first part. This inexact method is designed to solve large scale nonlinear programming problems. The iterative solution technique uses an orthogonal projection-Krylov subspace scheme to solve the highly indefinite and nonsymmetric linear systems associated with nonlinear programming. Our iterative projection method maintains linearized feasibility with respect to both the equality constraints and complementarity condition. This guarantees that in each iteration the linear solver generates a descent direction, so that the iterative solver is not required to find a Newton step but rather cheaply provides a way to march toward an optimal solution of the problem. This makes the use of a preconditioner inconsequential except near the solution of the problem, where the Newton step is effective. Moreover, we limit the problem to finding a good preconditioner only for the Hessian of the Lagrangian function associated with the nonlinear programming problem plus a positive diagonal matrix. We report numerical experimentation for several large scale problems to illustrate the viability of the method.
Mathematics; Industrial engineering; Operations research