Equivalence of the Polyak-Łojasiewicz-Kurdyka Exponent Via Difference-of-Moreau-Envelope Smoothing

For unconstrained nonsmooth difference-of-convex (DC) optimization problems, the difference-of-Moreau-envelope (DME) smoothing serves as a significant smooth approximation for them. Maintaining DC structure, the resulted DME-based model has a one-to-one correspondence for the stationary points with the original DC problem. This has led to the development of DME-specific algorithms to indirectly solve the DC problems by solving their DME-based models. In this paper, we obtain the global convergence and the specific local convergence rate of various DME-specific algorithms to find the stationary points of the corresponding DC problems. These results are based on the Polyak-Łojasiewicz-Kurdyka (PLK) property and the specific PLK exponent assumed on the DME-based model or the potential function designed in the DME-specific algorithm. More importantly, we establish the equivalence of the PLK exponent between the DC problems and their DME-based models. Combined with our local convergence rate result, we are allowed to show the linear and sublinear convergence rates of these specific algorithms. Moreover, the equivalence result also provides a new tool to explore the PLK exponent of the DC problem from that of its DME-based model. An example is then provided to show that the PLK exponent of a nonconvex compressed sensing model that incorporates a logistic penalty term is 1/2, which was previously unknown.