An adaptive forward-backward-forward splitting algorithm for solving pseudo-monotone inclusions
Flavia Chorobura, Ion Necoara, Jean-Christophe Pesquet
TL;DR
This work addresses finding zeros of a sum of three operators $A+B+C$ where $A$ is continuous, $B$ is Lipschitz, and $C$ is maximally monotone, a broad formulation covering constrained optimization and fractional programming. The authors introduce an adaptive forward-backward-forward (AFBF) splitting method with two novel stepsize rules that obviate costly line searches while ensuring convergence under pseudo-monotonicity and, under uniform pseudo-monotonicity, provide linear or sublinear rates. Theoretical results establish global convergence and rates, with detailed convergence proofs and residual bounds, and the method is validated on convex QCQPs, multiple kernel learning in SVM, and fractional programming, showing competitive or superior performance to state-of-the-art algorithms. The approach offers a flexible and scalable framework for broad inclusion problems, enabling efficient solution of complex constrained and nonconvex settings with adaptive stepsizes.
Abstract
In this paper, we propose an adaptive forward-backward-forward splitting algorithm for finding a zero of a pseudo-monotone operator which 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 stepsize 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 to some existing optimization methods and software.