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Minimizing smooth Kurdyka-{\L}ojasiewicz functions via generalized descent methods: Convergence rate and complexity
Masoud Ahookhosh, Susan Ghaderi, Alireza Kabgani, Morteza Rahimi
https://arxiv.org/abs/2511.10414 https://arxiv.org/pdf/2511.10414 https://arxiv.org/html/2511.10414
arXiv:2511.10414v1 Announce Type: new
Abstract: This paper addresses the generalized descent algorithm (DEAL) for minimizing smooth functions, which is analyzed under the Kurdyka-{\L}ojasiewicz (KL) inequality. In particular, the suggested algorithm guarantees a sufficient decrease by adapting to the cost function's geometry. We leverage the KL property to establish the global convergence, convergence rates, and complexity. A particular focus is placed on the linear convergence of generalized descent methods. We show that the constant step-size and Armijo line search strategies along a generalized descent direction satisfy our generalized descent condition. Additionally, for nonsmooth functions by leveraging the smoothing techniques such as forward-backward and high-order Moreau envelopes, we show that the boosted proximal gradient method (BPGA) and the boosted high-order proximal-point (BPPA) methods are also specific cases of DEAL, respectively. It is notable that if the order of the high-order proximal term is chosen in a certain way (depending on the KL exponent), then the sequence generated by BPPA converges linearly for an arbitrary KL exponent. Our preliminary numerical experiments on inverse problems and LASSO demonstrate the efficiency of the proposed methods, validating our theoretical findings.
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On fundamental properties of high-order forward-backward envelope
Alireza Kabgani, Masoud Ahookhosh
https://arxiv.org/abs/2511.10421 https://arxiv.org/pdf/2511.10421 https://arxiv.org/html/2511.10421
arXiv:2511.10421v1 Announce Type: new
Abstract: This paper studies the fundamental properties of the high-order forward-backward splitting mapping (HiFBS) and its associated forward-backward envelope (HiFBE) through the lens of high-order regularization for nonconvex composite functions. Specifically, we (i) establish the boundedness and uniform boundedness of HiFBS, along with the H\"older and Lipschitz continuity of HiFBE; (ii) derive an explicit form for the subdifferentials of HiFBE; and (iii) investigate necessary and sufficient conditions for the differentiability and weak smoothness of HiFBE under suitable assumptions. By leveraging the prox-regularity of $g$ and the concept of $p$-calmness, we further demonstrate the local single-valuedness and continuity of HiFBS, which in turn guarantee the differentiability of HiFBE in neighborhoods of calm points. This paves the way for the development of gradient-based algorithms tailored to nonconvex composite optimization problems.
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