10h30 - 10h55
Efficient calculation of spectral bounds for Hessian matrices on hyperrectangles for deterministic global optimization
This presentation summarizes progress made over the past few years in the calculation of spectral bounds of interval Hessian matrices or, more generally, Hessian matrix sets for functions defined on hyperrectangles. Spectral bounds of this type play an important role in deterministic global optimization, specifically in alphaBB, an approach Professor Christodoulos A. Floudas proposed as a young researcher and continuously refined over his entire career. The presented approach deliberately avoids the computation of interval Hessian matrices. As a consequence, the method has a favorable computational complexity compared to classical approaches that do use interval Hessians. More importantly, the resulting spectral bounds are sometimes (but not in general) tighter than the tightest possible bounds that could be computed with the corresponding interval Hessian matrix.
10h55 - 11h20
Higher Order methods for solving systems of equations and for in global optimization
Professor Christodoulos A. Floudas twenty years ago proposed a method for solving
systems of equations via a global optimization approach.
Underdetermined systems of equations and global optimization problems with a
submanifold of solutions often pose problems for deterministic algorithms because
the branch-and-bound approach in this sitation usually suffers heavily from the
cluster effect, excessive subdivision of boxes close to the solution manifold.
In this lecture we will develop new methods to deal with solution manifolds,
including verified computing on compact matrix Lie groups to deal with continuous
symmetries, and a verified continuation method computing inclusion/exclusion areas
around the solution manifold.
11h20 - 11h45
Parallel branching algorithms for the deterministic global optimisation of general NLP problems.
Deterministic global optimisation (DGO) of general NLP problems is a well-known NP-hard problem. Despite remarkable algorithmic advances in the past decades, DGO algorithms for NLP problems are not designed to take advantage of modern multiprocessor hardware, unlike their mixed-integer counterparts. In this work, we propose the first set of parallel branching algorithms for continuous NLP problems, synchronous and asynchronous versions of which are implemented in our DGO software written in C++14 and MPI-3, originally based on the MINOTAUR framework. This implementation is used to investigate the challenges which must be overcome in order to achieve linear parallel scaling.