Gradient Based Projection Method for Constrained Optimization

We introduce a continuous-time gradient based optimization scheme for the convex programming problem. The dynamics of the optimization parameter are described by a continuous projection of the map's gradient. Its mechanics parallel that of an augmented steepest descent method with the exception that it behaves as an interior point method. The projection affects the flow field as if subject to an interior point barrier function. Under mild assumptions the optimization trajectories are shown to stay entirely within the feasible region and converge to the constrained optimum. The approach also simultaneously solves the Lagrangian dual problem even though the dynamics are not governed by it.