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An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions.

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Title: An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions.
Authors: Ferreira, Orizon P.1 (AUTHOR) orizon@ufg.br, Mordukhovich, Boris S.2 (AUTHOR) aa1086@wayne.edu, Santos, Wilkreffy M. S.3 (AUTHOR) wilkreffy.santos@gmail.com, de O. Souza, João Carlos4 (AUTHOR) joaocos.mat@ufpi.edu.br
Source: Journal of Optimization Theory & Applications. Feb2026, Vol. 208 Issue 2, p1-27. 27p.
Subjects: Optimization algorithms, Subdifferentials, Differentiable functions, Mathematical optimization, Iterative methods (Mathematics), Heuristic algorithms, Nonconvex programming
Abstract: In this paper, we introduce an inexact approach to the Boosted Difference of Convex Functions Algorithm (BDCA) for solving nonconvex and nondifferentiable problems involving the difference of two convex functions (DC functions). Specifically, when the first DC component is differentiable and the second may be nondifferentiable, BDCA utilizes the solution from the subproblem of the DC Algorithm (DCA) to define a descent direction for the objective function. A monotone linesearch is then performed to find a new point that improves the objective function relative to the solution of the subproblem. This approach enhances the performance of DCA. However, if the first DC component is nondifferentiable, the BDCA direction may become an ascent direction, rendering the monotone linesearch ineffective. To address this, we propose an Inexact nonmonotone Boosted Difference of Convex Algorithm (InmBDCA). This algorithm incorporates two main features of inexactness: First, the subproblem therein is solved approximately, allowing us to a controlled relative error tolerance in defining the linesearch direction. Second, an inexact nonmonotone linesearch scheme is used to determine the step size for the next iteration. Due to the difficulty in computing the subdifferential of certain nondifferentiable functions, InmBDCA can, however, become conceptual when the first DC component g is nondifferentiable and its subdifferential is challenging to compute. Under suitable assumptions, we demonstrate that InmBDCA is well-defined, with any accumulation point of the sequence generated by InmBDCA being a critical point of the problem. We also provide iteration-complexity bounds for the algorithm. To complement the theoretical development, we include numerical illustrations aimed at verifying that the inexact solutions obtained by the BDCA and Nonmonotone Boosted DC Algorithm (nmBDCA) satisfy the conditions required by InmBDCA. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:In this paper, we introduce an inexact approach to the Boosted Difference of Convex Functions Algorithm (BDCA) for solving nonconvex and nondifferentiable problems involving the difference of two convex functions (DC functions). Specifically, when the first DC component is differentiable and the second may be nondifferentiable, BDCA utilizes the solution from the subproblem of the DC Algorithm (DCA) to define a descent direction for the objective function. A monotone linesearch is then performed to find a new point that improves the objective function relative to the solution of the subproblem. This approach enhances the performance of DCA. However, if the first DC component is nondifferentiable, the BDCA direction may become an ascent direction, rendering the monotone linesearch ineffective. To address this, we propose an Inexact nonmonotone Boosted Difference of Convex Algorithm (InmBDCA). This algorithm incorporates two main features of inexactness: First, the subproblem therein is solved approximately, allowing us to a controlled relative error tolerance in defining the linesearch direction. Second, an inexact nonmonotone linesearch scheme is used to determine the step size for the next iteration. Due to the difficulty in computing the subdifferential of certain nondifferentiable functions, InmBDCA can, however, become conceptual when the first DC component g is nondifferentiable and its subdifferential is challenging to compute. Under suitable assumptions, we demonstrate that InmBDCA is well-defined, with any accumulation point of the sequence generated by InmBDCA being a critical point of the problem. We also provide iteration-complexity bounds for the algorithm. To complement the theoretical development, we include numerical illustrations aimed at verifying that the inexact solutions obtained by the BDCA and Nonmonotone Boosted DC Algorithm (nmBDCA) satisfy the conditions required by InmBDCA. [ABSTRACT FROM AUTHOR]
ISSN:00223239
DOI:10.1007/s10957-025-02895-y