An efficient, exact, and generic quadratic programming solver for geometric optimization

We present a solver for quadratic programming problems, which is tuned for applications in computational geometry. The solver implements a generalization of the simplex method to quadratic programs. Unlike existing solvers, it is e cient if the problem is dense and has few variables or few constraints. The range of applications covers well-known problems like smallest enclosing ball, or polytope distance, but also linear programming problems like smallest enclosing annulus. We provide an exact implementation with only little overhead compared to pure oating-point code. Moreover, unlike all methods for these problems that were suggested (and implemented) before in computational geometry, the runtime in practice is not exponential in the dimension of the problem, which for example allows to compute smallest enclosing balls in dimensions up to 300 (beyond that, the exact arithmetic becomes the limiting factor). The solver follows the generic programming paradigm, and it will become part of the European computational geometry algorithms library Cgal.