Convergence Properties of Minimization Algorithms for Convex Constraints Using a Structured Trust Region

We present in this paper a class of trust region algorithms in which the structure of the problem is explicitly used in the very denition of the trust region itself. This development is intended to re BLOCKINect the possibility that some parts of the problem may be more \trusted" than others, a commonly occurring situation in large-scale nonlinear applications. After describing the structured trust region mechanism, we prove global convergence for all algorithms in our class. We also prove that, when convex constraints are present, the correct set of such constraints active at the problem's solution is identied by these algorithms after a nite number of iterations. Abstract We present in this paper a class of trust region algorithms in which the structure of the problem is explicitly used in the very denition of the trust region itself. This development is intended to re BLOCKINect the possibility that some parts of the problem may be more \trusted" than others, a commonly occurring situation in large-scale nonlin-ear applications. After describing the structured trust region mechanism, we prove global convergence for all algorithms in our class. We also prove that, when convex constraints are present, the correct set of such constraints active at the problem's solution is identied by these algorithms after a nite number of iterations. 1 Introduction Trust region algorithms have enjoyed a long and successful history as tools for the solution of nonlinear, nonconvex, optimization problems. They have been studied and applied to This long lasting interest is probably justied by the attractive combination of a solid convergence theory, a noted algorithmic robustness, the existence of numerically ecient implementations and an intuitively appealing justication. The main idea behind trust region algorithms is that, if a nonlinear function (objective and/or constraints) is expensive to compute or dicult to handle explicitly, one can replace it by a suitable model. This model may be \trusted" within a certain trust region around the current point, whose size (the trust region radius) is then expanded if the model and function suciently agree, or decreased if they dier too much. The minimization then proceeds by replacing the dicult nonlinear function(s) with the corresponding easier model(s). It is remarkable that, up to now, all algorithms that we are aware of use a single trust region radius to measure the degree of trustworthiness of the models employed, even if several dierent functions are involved. This choice is …