: "A nasty cone with nice properties - insight by copositive optimization"
(basic concepts and ideas, along with a quick resume of interior-point methods and conic optimiation)
: "Data analysis, machine learning, ternary and other hard decision problems: how copositive optimization can help
" (including a discussion on the average hardness of related problems)
: "The role of copositivity in optimality conditions and relaxation bounds"
(going beyond quadratic optimization in a classical setup)
Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a (convex) cone subject to linear constraints. Such type of problems have the advantage that every local solution is automatically a global one, so that (exact) local solvers cannot be trapped in inefficient local solutions. Still, conic formulations be used to tackle NP-hard problems, by shifting all complexity into the boundary description of the cone.
The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems, or, on the continuous side, for (fractional) polynomial problems.
The presentations will focus on principles, algorithms and applications, along with some recent success stories.