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

Szilárd Csaba László ()
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Szilárd Csaba László: Technical University of Cluj-Napoca

Journal of Optimization Theory and Applications, 2023, vol. 198, issue 1, No 15, 387-427

Abstract: Abstract 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.

Keywords: Global optimization; Inertial proximal-gradient algorithm; Non-convex optimization; Abstract convergence theorem; Kurdyka–Łojasiewicz inequality; KL exponent; Convergence rate; 90C26; 90C30; 65K10 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02204-5

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