An Adaptive Forward-Backward-Forward Splitting Algorithm for Solving Pseudo-Monotone Inclusions

In this paper, we propose an adaptive forward-backward-forward splitting algorithm for finding a zero of a pseudo-monotone operator that is split as a sum of three operators: the first is continuous single-valued, the second is Lipschitzian, and the third is maximally monotone. This setting covers, in particular, constrained minimization scenarios, such as problems having smooth and convex functional constraints (e.g., quadratically constrained quadratic programs) or problems with a pseudo-convex objective function minimized over a simple closed convex set (e.g., quadratic over linear fractional programs). For the general problem, we design a forward-backward-forward splitting type method based on novel adaptive step-size strategies. Under an additional generalized Lipschitz property of the first operator, sublinear convergence rate is derived for the sequence generated by our adaptive algorithm. Moreover, if the sum is uniformly pseudo-monotone, linear/sublinear rates are derived depending on the parameter of uniform pseudo-monotonicity. Preliminary numerical experiments demonstrate the good performance of our method when compared with some existing optimization methods and software. Funding: The research leading to these results has received funding from project TraDE-OPT funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skolodowska-Curie grant agreement [Grant 861137]; Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii, Romania [Grant PN-III-P4-PCE-2021-0720] under project L2O-MOC, nr. 70/2022; and GAR2023 funded by the Patrimony Foundation of Romanian Academy, nr. 260/2023.