A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems

We investigate an inertial forward–backward algorithm in connection with the minimization of the sum of a non-smooth and possibly non-convex and a non-convex differentiable function. The algorithm is formulated in the spirit of the famous FISTA method; however, the setting is non-convex and we allow...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 198; no. 1; pp. 387 - 427
Main Author: László, Szilárd Csaba
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2023
Springer Nature B.V
Subjects:
Algorithms
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Convergence
Critical point
Engineering
Global optimization
Image restoration
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Regularization
Sequences
Theory of Computation
Global optimization
Non-convex optimization
KL exponent
Abstract convergence theorem
90C30
Convergence rate
65K10
Inertial proximal-gradient algorithm
90C26
Kurdyka–Łojasiewicz inequality
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:We investigate an inertial forward–backward algorithm in connection with the minimization of the sum of a non-smooth and possibly non-convex and a non-convex differentiable function. The algorithm is formulated in the spirit of the famous FISTA method; however, the setting is non-convex and we allow different inertial terms. Moreover, the inertial parameters in our algorithm can take negative values too. We also treat the case when the non-smooth function is convex, and we show that in this case a better step size can be allowed. Further, we show that our numerical schemes can successfully be used in DC-programming. We prove some abstract convergence results which applied to our numerical schemes allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. Further, we obtain a general result that applied to our numerical schemes ensures convergence rates for the generated sequences and for the objective function values formulated in terms of the KL exponent of a regularization of the objective function. Finally, we apply our results to image restoration.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-023-02204-5