Volume 14, Issue 2
Convergence of the Peaceman-Rachford Splitting Method for a Class of Nonconvex Programs

Miantao Chao, Deren HanXingju Cai

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 438-460.

Published online: 2021-01

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  • Abstract

In this paper, we analyze the convergence of the Peaceman-Rachford splitting method (PRSM) for a type of nonconvex and nonsmooth optimization with linear constraints, whose objective function is the sum of a proper lower semicontinuous function and a strongly convex differential function. When a suitable penalty parameter is chosen and the iterative point sequence is bounded, we show the global convergence of the PRSM. Furthermore, under the assumption that the associated function satisfies the Kurdyka-Łojasiewicz property, we prove the strong convergence of the PRSM. We also provide sufficient conditions guaranteeing the boundedness of the generated sequence. Finally, we present some preliminary numerical results to show the effectiveness of the PRSM and also give a comparison with the Douglas-Rachford splitting method.

  • Keywords

Kurdyka-Łojasiewicz inequality, Peaceman-Rachford splitting method, nonconvex, strongly convex, Douglas-Rachford splitting method.

  • AMS Subject Headings

90C26, 90C30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-438, author = {Chao , Miantao and Han , Deren and Cai , Xingju}, title = {Convergence of the Peaceman-Rachford Splitting Method for a Class of Nonconvex Programs}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {2}, pages = {438--460}, abstract = {

In this paper, we analyze the convergence of the Peaceman-Rachford splitting method (PRSM) for a type of nonconvex and nonsmooth optimization with linear constraints, whose objective function is the sum of a proper lower semicontinuous function and a strongly convex differential function. When a suitable penalty parameter is chosen and the iterative point sequence is bounded, we show the global convergence of the PRSM. Furthermore, under the assumption that the associated function satisfies the Kurdyka-Łojasiewicz property, we prove the strong convergence of the PRSM. We also provide sufficient conditions guaranteeing the boundedness of the generated sequence. Finally, we present some preliminary numerical results to show the effectiveness of the PRSM and also give a comparison with the Douglas-Rachford splitting method.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0063}, url = {http://global-sci.org/intro/article_detail/nmtma/18606.html} }
TY - JOUR T1 - Convergence of the Peaceman-Rachford Splitting Method for a Class of Nonconvex Programs AU - Chao , Miantao AU - Han , Deren AU - Cai , Xingju JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 438 EP - 460 PY - 2021 DA - 2021/01 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0063 UR - https://global-sci.org/intro/article_detail/nmtma/18606.html KW - Kurdyka-Łojasiewicz inequality, Peaceman-Rachford splitting method, nonconvex, strongly convex, Douglas-Rachford splitting method. AB -

In this paper, we analyze the convergence of the Peaceman-Rachford splitting method (PRSM) for a type of nonconvex and nonsmooth optimization with linear constraints, whose objective function is the sum of a proper lower semicontinuous function and a strongly convex differential function. When a suitable penalty parameter is chosen and the iterative point sequence is bounded, we show the global convergence of the PRSM. Furthermore, under the assumption that the associated function satisfies the Kurdyka-Łojasiewicz property, we prove the strong convergence of the PRSM. We also provide sufficient conditions guaranteeing the boundedness of the generated sequence. Finally, we present some preliminary numerical results to show the effectiveness of the PRSM and also give a comparison with the Douglas-Rachford splitting method.

Miantao Chao, Deren Han & Xingju Cai. (2021). Convergence of the Peaceman-Rachford Splitting Method for a Class of Nonconvex Programs. Numerical Mathematics: Theory, Methods and Applications. 14 (2). 438-460. doi:10.4208/nmtma.OA-2020-0063
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