Concave programming (CP) is a problem of maximizing convex function over a convex set of constraints. If the set is polyhedral, the solutions of this global optimization problem lie on its vertices. We consider a largely forgotten branch of research in CP, originated from the work of Hoang Tuy and based on the cone splitting techniques. The same type of methods could be applied to characterize vertices of the whole polyhedron and to compute its projection in the lower-dimensional subspace. We describe our Julia implementation of the projection method for polytopes.
joint work with Pontus Giselsson
We present a method for solving the general mixed constrained quadratic programming problem using an active set method on the dual problem. There are two main contributions; we present a new way of handling the factorization in the the dual problem, and show how iterative refinement can be used to handle the problems arising when the dual is not positive definite. An open source implementation in Julia is available, which can be used to solve problems using arbitrary-precision floating-point numbers.
COSMO is an operator splitting algorithm for convex optimisation problems with quadratic objective function and conic constraints. At each step the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The solver is able to exploit chordal sparsity in the problem data and to detect infeasible problems. The low per-iteration computational cost makes the method particularly efficient for large problems. Our Julia implementation COSMO.jl is open-source, extensible and performs well on a variety of problem classes.