A general double-proximal gradient algorithm for d.c. programming

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 p...

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Vydáno v:Mathematical programming Ročník 178; číslo 1-2; s. 301 - 326
Hlavní autoři: Banert, Sebastian, Boț, Radu Ioan
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2019
Springer Nature B.V
Témata:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Convergence analysis
d.c. programming
Direct current
Full Length Paper
Image processing
Kurdyka-Lojasiewicz property
Linear operators
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operators (mathematics)
Optimization
Proximal-gradient algorithm
Theoretical
Toland dual
Toland dual
65K05
Kurdyka–Łojasiewicz property
49M29
90C26
d.c. programming
Proximal-gradient algorithm
Convergence analysis
ISSN:0025-5610, 1436-4646, 1436-4646
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Popis
Shrnutí: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.
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ISSN:0025-5610
1436-4646
1436-4646
DOI:10.1007/s10107-018-1292-2