An accelerated preconditioned primal-dual gradient algorithm for structured nonconvex optimization problems
•An novel accelerated preconditioned primal-dual gradient algorithm for solving nonconvex optimization problems by the conjugate duality theory of nonconvex functions.•Our algorithm only needs to calculate the proximal mapping of the conjugate function which is always convex and lower semicontinuous...
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| Vydáno v: | Communications in nonlinear science & numerical simulation Ročník 153; s. 109480 |
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| Jazyk: | angličtina |
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Elsevier B.V
01.02.2026
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| ISSN: | 1007-5704 |
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| Abstract | •An novel accelerated preconditioned primal-dual gradient algorithm for solving nonconvex optimization problems by the conjugate duality theory of nonconvex functions.•Our algorithm only needs to calculate the proximal mapping of the conjugate function which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. the computation load may be significantly reduced.•Global convergence under Kurdyka-Lojasiewicz condition.•Numerical results illustrate that the proposed algorithm is quite competitive with some existing algorithms.
For a nonconvex problem, the computation of the proximal operator of the nonconvex function is difficult in general. In this paper, based on the conjugate duality theory of nonconvex functions, we present an accelerated preconditioned primal-dual gradient algorithm for a class of nonconvex optimization problems. Compared with the existing algorithms, our algorithm only needs to calculate the proximal mapping of the conjugate function which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. Hence, the computation load may be significantly reduced. We prove that the sequence generated by the proposed algorithm globally converges to a critical point under Kurdyka-Łojasiewicz framework. Furthermore, we derive the convergence rate of the proposed algorithm. Finally, numerical results on signal recovery, image denoising and sparse principal component analysis illustrate that the proposed algorithm is quite competitive with some existing algorithms. |
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| AbstractList | •An novel accelerated preconditioned primal-dual gradient algorithm for solving nonconvex optimization problems by the conjugate duality theory of nonconvex functions.•Our algorithm only needs to calculate the proximal mapping of the conjugate function which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. the computation load may be significantly reduced.•Global convergence under Kurdyka-Lojasiewicz condition.•Numerical results illustrate that the proposed algorithm is quite competitive with some existing algorithms.
For a nonconvex problem, the computation of the proximal operator of the nonconvex function is difficult in general. In this paper, based on the conjugate duality theory of nonconvex functions, we present an accelerated preconditioned primal-dual gradient algorithm for a class of nonconvex optimization problems. Compared with the existing algorithms, our algorithm only needs to calculate the proximal mapping of the conjugate function which is always convex and lower semicontinuous and it does not need to calculate the proximal mapping of nonconvex functions. Hence, the computation load may be significantly reduced. We prove that the sequence generated by the proposed algorithm globally converges to a critical point under Kurdyka-Łojasiewicz framework. Furthermore, we derive the convergence rate of the proposed algorithm. Finally, numerical results on signal recovery, image denoising and sparse principal component analysis illustrate that the proposed algorithm is quite competitive with some existing algorithms. |
| ArticleNumber | 109480 |
| Author | Gou, Zhun Long, Xian-Jun Li, Gao-Xi Nie, Jia-Lin Sun, Xiang-Kai |
| Author_xml | – sequence: 1 givenname: Xian-Jun surname: Long fullname: Long, Xian-Jun email: xianjunlong@ctbu.edu.cn – sequence: 2 givenname: Jia-Lin surname: Nie fullname: Nie, Jia-Lin email: niejialin00@163.com – sequence: 3 givenname: Zhun surname: Gou fullname: Gou, Zhun email: 577298380@qq.com – sequence: 4 givenname: Xiang-Kai surname: Sun fullname: Sun, Xiang-Kai email: sunxk@ctbu.edu.cn – sequence: 5 givenname: Gao-Xi surname: Li fullname: Li, Gao-Xi email: ligaoxicn@126.com |
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| Cites_doi | 10.1137/16M1064064 10.1007/s10898-020-00943-7 10.1287/moor.1100.0449 10.1137/18M1190689 10.1137/14098435X 10.1007/s10589-022-00364-0 10.1109/TIT.2006.871582 10.1016/j.apnum.2023.03.014 10.1016/j.ejor.2025.04.034 10.1137/S0363012998338806 10.1007/s10107-011-0484-9 10.1007/s10851-015-0565-0 10.1109/TCBB.2017.2756628 10.1016/j.apnum.2024.05.006 10.1287/moor.2017.0900 10.1137/140952363 10.1007/s10898-019-00819-5 10.1137/130942954 10.1109/MSP.2018.2877582 10.1007/s10107-013-0701-9 10.1007/s12532-018-0153-6 10.1137/16M1107863 10.1007/s10898-022-01176-6 |
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| Keywords | Structured nonconvex and nonsmooth problem Global convergence Kurdyka-Łojasiewicz property 90C30 90C33 Primal-dual gradient algorithm 90C26 Conjugate duality theory |
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| SubjectTerms | Conjugate duality theory Global convergence Kurdyka-Łojasiewicz property Primal-dual gradient algorithm Structured nonconvex and nonsmooth problem |
| Title | An accelerated preconditioned primal-dual gradient algorithm for structured nonconvex optimization problems |
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