Inexact DC Algorithms in Hilbert Spaces with Applications to PDE-Constrained Optimization

P. D. Khanh; V. V. H. Khoa; B. S. Mordukhovich; D. B. Tran; N. V. Vo

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arXiv · 2601.06622

Inexact DC Algorithms in Hilbert Spaces with Applications to PDE-Constrained Optimization

P. D. Khanh·V. V. H. Khoa·B. S. Mordukhovich·D. B. Tran·N. V. Vo

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Abstract

In this paper, we design and apply novel inexact adaptive algorithms to deal with minimizing difference-of-convex (DC) functions in Hilbert spaces. We first introduce I-ADCA, an inexact adaptive counterpart of the well-recognized DCA (difference-of-convex algorithm), that allows inexact subgradient evaluations and inexact solutions to convex subproblems while still guarantees global convergence to stationary points. Under a Polyak-Lojasiewicz type property for DC objectives, we obtain explicit convergence rates for the proposed algorithm. Our main application addresses elliptic optimal control problems with control constraints and nonconvex $L^{1-2}$ sparsity-enhanced regularizers admitting a DC decomposition. Employing I-ADCA and appropriate versions of finite element discretization leads us to an efficient procedure for solving such problems with establishing its well-posedness and error bound estimates confirmed by numerical experiments.

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P. D. Khanh, V. V. H. Khoa, B. S. Mordukhovich, D. B. Tran, N. V. Vo. 2026-01-10. Inexact DC Algorithms in Hilbert Spaces with Applications to PDE-Constrained Optimization. https://arxiv.org/abs/2601.06622

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