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The proximal augmented Lagrangian method with indefinite proximal regularization and its application in image restoration. (English) Zbl 1438.90266
Summary: In this paper, we generalize the proximal matrix in the proximal augmented Lagrangian method (PALM) from semi-definite to indefinite, and propose a proximal ALM with indefinite proximal regularization (PALM-IPR) for convex programming with linear constraints, which inherits the easily implementable property of the proximal ALM, and often has better numerical performance than the latter. Under mild assumptions, the global convergence of PALM-IPR is proved. Furthermore, we prove its worst-case \(\mathcal{O}(1/t)\) convergence rate in an ergodic sense under a more reasonable criterion than those adopted in [B. He and X. Yuan, SIAM J. Numer. Anal. 50, No. 2, 700–709 (2012; Zbl 1245.90084); Y. Xu, SIAM J. Optim. 27, No. 3, 1459–1484 (2017; Zbl 1373.90111)]. Finally, numerical results show that PALM-IPR is feasible and efficient for solving some problems in image reconstruction.
90C25 Convex programming
90C30 Nonlinear programming
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