joint work with Chris Coey, Juan Pablo Vielma
We discuss the modeling capabilities of our new conic interior point solver, Hypatia.jl. We focus on problems that can be naturally formulated using nonsymmetric cones, such as the weighted sum of squares cone. The standard approach to solve these problems involves utilizing semidefinite programming reformulations, which unnecessarily increases the dimensionality of the optimization problem. We show the advantages of being able to optimize directly over known nonsymmetric cones and newly defined cones.
joint work with Paul Breiding
Homotopy continuation methods allow to compute all isolated solutions of polynomial systems as well as representatives for positive dimensional solution sets using techniques from algebraic geometry. After a short introduction to the fundamental constructions of the homotopy continuation approach, I will continue with describing an efficient approach to the solution of polynomial systems with parametric coefficients by using the monodromy action induced by the fundamental group of the regular locus of the parameter space. This is demonstrated using the Julia package HomotopyContinuation.jl.
joint work with Carleton Coffrin
The generalized moment problem (GMP) is an infinite dimensional linear optimization problem on measures. Relaxations based on LP, SOCP, SDP and other cones are known to approximate its solution. Since implementation of the different algorithms are spread over all platforms of mathematical computing and not necessarily able to take into account new algorithmic ideas or additional use of structure, it is difficult to compare known and newly emerging approaches in practice and to decide which hierarchy works best for a particular problem. With GMP.jl we provide a unified framework.