Convergence Analysis of an Inexact MBA Method for Constrained DC Problems

This paper concerns a class of constrained difference-of-convex (DC) optimization problems in which, the constraint functions are continuously differentiable and their gradients are strictly continuous. For such nonconvex and nonsmooth optimization problems, we develop an inexact moving balls approximation (MBA) method by a workable inexactness criterion for the solution of subproblems. This criterion is proposed by leveraging a global error bound for the strongly convex program associated with parametric optimization problems. We establish the full convergence of the iterate sequence under the Kurdyka-{\L}ojasiewicz (KL) property of the constructed potential function, achieve the local convergence rate of the iterate and objective value sequences under the KL property of the potential function with exponent $q\in[1/2,1)$, and provide the iteration complexity of $O(1/\epsilon^2)$ to seek an $\epsilon$-KKT point. A verifiable condition is also presented to check whether the potential function has the KL property of exponent $q\in[1/2,1)$. To our knowledge, this is the first implementable inexact MBA method with a complete convergence certificate. Numerical comparison with DCA-MOSEK, a DC algorithm with subproblems solved by MOSEK, is conducted on $\ell_1\!-\!\ell_2$ regularized quadratically constrained optimization problems, which demonstrates the advantage of the inexact MBA in the quality of solutions and running time.