Inexact proximal memoryless quasi-Newton methods based on the Broyden family for minimizing composite functions

Inexact proximal DC Newton-type method for nonconvex composite functions

“…• We propose an inexact proximal DC Newton-type method, which is an extension of the inexact proximal Newton-type method [21] to DC optimization problems, and show its global convergence. The framework of the method (Algorithm 1) is similar to the inexact SQA method proposed by Liu and Takeda [17], but the details are different.…”

“…Although usual proximal mappings can be easily obtained for some special cases [1,7], the computing cost of scaled proximal mapping is very expensive. Hence, inexact proximal Newton-type methods, which inexactly calculate scaled proximal mappings, have been proposed (see, for example, [5,13,16,21,30]). The proximal Newton-type method is also known as the Successive Quadratic Approximation (SQA) method.…”

“…al. [21]. Then, the scaled proximal mappings can be obtained by solving a twodimensional system of semi-smooth equations (31).…”

“…In Section 2, we propose an inexact proximal DC Newton-type method. We briefly introduce the inexact proximal Newton-type method [21] in Section 2.1, and we extend the inexact proximal Newton-type method to DC optimization problems in Section 2.2.…”

“…In Section 3, we show the global convergence properties of the proposed method. In Section 4, we introduce the memoryless quasi-Newton formula [21] (Section 4.1) and give an efficient method for computing scaled proximal mappings (Section 4.2). In Section 5, we present some numerical experiments to show the efficiency of the proposed method in comparison with existing methods.…”