If I have a minimization problem in which both the objective function and constraint are nonconvex. I use gradient projection method to solve the problem iteratively.
If we relax the constraint and still use the same method with the same intial point, can we achieve a smaller converged objective value for sure? or equivalently, can we still arrive at the same stationary point for sure? But the simulation give me the negative answer...
Thank you very much!!!
By relaxation the feasible set stays equal or becomes larger. Thus, the global minimum might decrease (depending on the objective function).
However, since you have a non convex problem, gradient projection method computes a stationary point, a local minimum at best. So it may give you the same, a larger, or a smaller value. You can't really predict what would happen.