Normalized penalty gradient flow: A continuous-time approach for solving constrained nonconvex nonsmooth optimization problems

To avoid calculating penalty parameters, this paper introduces a continuous-time approach that combines the normalized gradient flow with the penalty method to solve the nonconvex nonsmooth optimization problem with a convex constraint set. Subsequently, this approach is extended to solve nonconvex nonsmooth optimization with a box constraint set and affine equality constraints. Compared with the current methods, the proposed approach does not require calculating the penalty parameters, allows the objective function to be nonconvex or nonsmooth, and allows the initial point to be arbitrarily selected. It is proved that the solution trajectory with any initial point converges to the critical point set of the optimization problem. Finally, the effectiveness of the proposed approach is authenticated through several numerical experiments.