Abstract
The possibilities of exploiting the special structure of d.c. programs, which consist of optimising the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this paper we propose an algorithm which allows the evaluation of both the concave and the convex part by their proximal points. Additionally, we allow a smooth part, which is evaluated via its gradient. In the spirit of primal-dual splitting algorithms, the concave part might be the composition of a concave function with a linear operator, which are, however, evaluated separately. For this algorithm we show that every cluster point is a solution of the optimisation problem. Furthermore, we show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem. Convergence of the iterates is shown if this objective function satisfies the Kurdyka–Łojasiewicz property. In the last part, we apply the algorithm to an image processing model.
Highlights
The possibilities of exploiting the special structure of d.c. programs, which consist of optimising the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997
We show the connection to the Toland dual problem and prove a descent property for the objective function values of a primal-dual formulation of the problem
Convergence of the iterates is shown if this objective function satisfies the Kurdyka–Łojasiewicz property
Summary
Optimisation problems where the objective function can be written as a difference of two convex functions arise naturally in several applications, such as image processing [1], machine learning [2], optimal transport [3] and sparse signal recovering [4]. We go one step further by proposing an algorithm, where both the convex and concave parts are evaluated via proximal steps. We consider a linear operator in the concave part of the objective function, which is evaluated in a forward manner in the spirit of primal-dual splitting methods. 3, we propose a double-proximal d.c. algorithm, which generates both a primal and a dual sequence of iterates and show several properties which make it comparable to DCA. We prove a descent property for the objective function values of a primal-dual formulation and that every cluster point of the sequence of. 4, we show global convergence of our algorithm and convergence rates for the iterates in certain cases, provided that the objective function of the primal-dual reformulation satisfies the Kurdyka–Łojasiewicz property; in other words, it is a KŁ function. We close our paper with some numerical examples addressing an image deblurring and denoising problem in the context of different d.c. regularisations
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