Saturday, 14:20 - 16:00 - Room 105
Stream: Linear and nonlinear optimization
We deal with the numerical solution of ill-posed nonlinear systems and nonlinear least-squares problems and consider procedures in the class of trust-region methods. We propose a trust-region approach that ensures regularizing properties and is so suitable to solve the above mentioned problems. We provide a trust region radius choice ensuring regularizing properties and giving rise to a procedure that has the potential to handle also the noisy case as a solution of the unperturbed problem is approached. Results of some numerical tests will be presented, too.
When we solve nonlinear programming problems (NLPs), algorithms for NLPs sometimes crash numerically and fail to find a local optimal solution, even if an appropriate constraint qualification would be satisfied at a solution.
To overcome and analyze such difficulties, we have implemented an NLP solver based on the SQP method, which uses multiple precision arithmetic. This solver is equipped with automatic differentiation, then we can try to solve various NLPs easily.
In this research, we analyze the detail of numerical behavior of the SQP method for some ill-posed NLPs by using our solver.
The Levenberg-Morisson-Marquardt algorithm (LMM) is one of the most popular algorithms for the solution of nonlinear least squares problems. Motivated by the problem structure in data assimilation, we consider in this work the extension of the LMM algorithm to the scenarios where the linearized least squares subproblems are solved inexactly and/or the gradient model is noisy and accurate only within a certain probability. Under appropriate assumptions, we show that the modiï¬ed algorithm converges globally and almost surely to a ï¬rst order stationary point.
Aircraft conflicts resolution, to avoid losses of separation between aircraft trajectories during flights, is crucial in Air Traffic Management. We propose a purely continuous optimization model relying on an exact l1-penalty function, to deal with the aircraft separation constraints. The decision levers are both aircraft speed and heading-angle changes. The removal of the infinite-dimensional feature of the separation constraints introduced in (Cafieri &Durand,2014) is exploited, and a linearization of angle-related nonlinear terms is proposed. Numerical results validate the proposed approach
This talk presents the concept of the effective hull of a set and the method of its approximation. The effective hull is essentially an intersection of the of Edgeworth-Pareto hulls for all possible combinations of maximization and minimization of criteria. We show that the effective hull of a set contains the set and is contained by its convex hull. We also developed a method for numerical approximation of an effective hull with the given precision. In the talk we outline an important application of the proposed approach for describing working set of robotic manipulators.