A smoothing extended sequential quadratic method for difference-of-convex optimization over a convex composite inequality constraint

Abstract

We consider the problem of minimizing a difference-of-convex objective over a convex composite inequality constraint and a compact convex set constraint. To solve this problem, we extend the ESQM in [1] via incorporating a variable smoothing scheme. In essence, in each iteration of our algorithm, we apply one proximal gradient step to a smoothed penalty function, constructed based on a smooth approximation of the convex composite constraint function; and we design explicit rules to update the smoothing and penalty parameters. Under suitable constraint qualifications, we establish an iteration complexity of $O(\hat{I}{\mu }^{-3})$ for obtaining an $(\hat{I}\mu ,\hat{I}\mu )$-KKT point. Moreover, in the convex setting, we show that the whole sequence generated by our algorithm is convergent and derive its local convergence rate under a standard Hölderian growth condition.