Copositive optimization – Recent developments and applications

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 cone subject to linear constraints. 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. Here some of the recent success stories are told, along with principles, algorithms and applications.

► Copositive optimization is an important problem class within conic optimization which receives much attention nowadays.
► It allows reformulation of diverse NP-hard problems, including nonconvex, mixed-binary, and fractional quadratic problems.
► Combinatorial applications: quadratic assignment, bounding the clique and the crossing number, graph coloring and partitioning.
► Applications in the continuous domain: stability in switched systems, optimality conditions, and tight convex underestimation.
► Further applications: rigid body mechanics, networks in queueing, traffic, reliability, uncertainty in Q2 mixed-integer LPs.