This talk introduces the global optimization engine in FICO Xpress, allowing to solve general mixed-integer nonlinear problems to global optimality. We will discuss how existing features of the MILP and local NLP solvers are used and explain several different extensions for, e.g., spatial branching and convexification cuts, and their performance implications. Recent improvements and their impacts on the performance of the solver are also discussed.

This talk aims to showcase the latest developments in RAPOSa, a global solver for polynomial optimization problems, with a primary focus on its extension to address mixed-integer problems. The talk will delve into various facets of the key challenges encountered in this extension, along with some numerical experiments designed to assess the performance of different approaches, both relative to one another and also relative to other state-of-the-art global solvers for MINLP problems.

Regarding the extension to mixed integer problems, we will discuss several approaches, targeted to improve the quality of the final bounds returned by the algorithm. In particular, for the upper bounds we will discuss the integration of the calls to auxiliary local solvers (discrete and continuous) and the trade-offs that arise when trying to do it efficiently.

Finally, we will also briefly touch on other recent developments in RAPOSa, which will primarily revolve around the use of (nonlinear) conic constraints, such as SOCP and SDP constraints, inside RAPOSa's branch-and-bound scheme. In particular, we will discuss the effect of using these conic constraints to tighten, at different places of the algorithm, the linear RLT relaxations on which RAPOSa is built upon.

We consider solving mixed-integer nonlinear optimization problems to global
optimality. Our solver is based on the common branch-and-bound paradigm, but
includes a number of specialized techniques to deal with nonconvex constraints
and nonconvex objective functions. In this talk we will outline a few of these
components from a theoretical and computational point of view, with some
emphasis on relaxation schemes and cutting planes. Further we will preview
some features of the next Gurobi version that enhance MINLP solving and
modeling capabilities.

Branch-and-bound algorithms are the standard in mixed integer optimization, both with convex and non-convex continuous relaxations. Spatial branching, i.e., branching on continuous variables, is a necessary tool of global nonlinear solvers for tackling non-convexities arising in the objective function and/or the constraints. Spatial branching is harder than branching on integer variables and it has been comparably less studied. Similar to other techniques that were originally intended for MINLP but are now common in the MIP world, we show that it can be successfully used on a special class of convex mixed integer quadratic optimization problems with application in classification, where it is far more effective than the standard integer branching machinery of a commercial MIP solver.