On optimisation programmes with hidden convexity

The main result of this paper is a method for finding hidden convexity in some smooth nonconvex programmes. Specifically, the method is identifying an equivalent convex formulation of the underlying nonconvex programme. The main idea of our method is to view a constrained optimisation programme as a control system, where the dual variable plays a role of a control signal. Therefore the existence of a global stabilising feedback controller implies the existence of an equivalent convex optimisation programme. In detail, the case of programmes with linear constraints is considered, for which sufficient conditions for finding hidden convexity are derived. If these sufficient conditions are satisfied an equivalent convex formulation can be obtained by using control-theoretic tools without a considerable computational cost. The whole procedure can be seen as a generalisation of the augmented Lagrangian method. This observation allows to obtain a control-theoretic interpretation of the augmented Lagrangian method, and an extension to incorporate linear inequality constraints.