A boosted DC algorithm for non-differentiable DC components with non-monotone line search

Abstract

We introduce a new approach to apply the boosted difference of convex functions algorithm (BDCA) for solving non-convex and non-differentiable problems involving difference of two convex functions (DC functions). Supposing the first DC component differentiable and the second one possibly non-differentiable, the main idea of BDCA is to use the point computed by the DC algorithm (DCA) to define a descent direction and perform a monotone line search to improve the decreasing the objetive function accelerating the convergence of the DCA. However, if the first DC component is non-differentiable, then the direction computed by BDCA can be an ascent direction and a monotone line search cannot be performed. Our approach uses a non-monotone line search in the BDCA (nmBDCA) to enable a possible growth in the objective function values controlled by a parameter. Under suitable assumptions, we show that any cluster point of the sequence generated by the nmBDCA is a critical point of the problem under consideration and provide some iteration-complexity bounds. Furthermore, if the first DC component is differentiable, we present different iteration-complexity bounds and prove the full convergence of the sequence under the Kurdyka-Lojasiewicz property of the objective function. Some numerical experiments show that the nmBDCA outperforms the DCA such as its monotone version.

An adaptive exact penalty method for nonsmooth optimal control problems with nonsmooth nonconvex state and control constraints

“…Our goal is to develop a class of local search methods for the problem (P) that combines the boosted DCA with nonmonotone line search [2,3,26] and the exact penalty technique based on the use of an L 1 exact penalty function and the steering exact penalty methodology [5,6,20,74]. In the case of the problem (P) this penalty function has the form Φ c (x, u) = J(x, u) + cϕ(x, u), where…”

“…[45,46,50,82]). Recently, an improved version of the DCA, called the boosted DCA [2,3,26], was proposed. In contrast to the original DCA, boosted DCA utilises line search (nonmonotone line search in the nonsmooth case; see Ref.…”

“…In contrast to the original DCA, boosted DCA utilises line search (nonmonotone line search in the nonsmooth case; see Ref. [26]) to improve its overall performance. As was demonstrated by numerical experiments in the aforementioned references, the boosted DCA significantly outperforms the standard DCA on various test problems.…”

“…Our goal is to develop a class of adaptive exact penalty methods, called the boosted steering exact penalty DCA (B-STEP-DCA), for constrained nonsmooth optimal control problems, whose DC structure is explicitly known, based on the combination of steering exact penalty techniques [5,6,20,74] and the boosted DCA with nonmonotone line search [2,3,26] under the assumption that all computations are performed with finite precision. The use of the steering exact penalty techniques allows the method to automatically and adaptively adjust the penalty parameter to ensure convergence of the constructed sequence and sufficient decrease of the infeasibility measure on each iteration.…”

“…In turn, the use of nonmonotone line search with adaptive choice of the trial step size and line search tolerance, proposed in Ref. [26] for unconstrained problems, is aimed at improving the overall efficiency of the method and reducing the number of iterations before termination. Moreover, the nonmonotone line search also has an innertial effect that in some cases might help the method to "jump over" some critical points and find a better solution (cf.…”