A Bregman proximal linearized ADMM for fractional programming with nonlinear coupling constraints

Abstract: This paper considers a nonconvex fractional optimization problem with composite structure and nonlinear equality constraints. We reformulate the problem as a non-fractional min-max problem with corresponding optimal solutions. By linearizing the nonlinear constraint terms in the augmented Lagrangian and incorporating Bregman proximal linearization and a relaxation factor in $(0, 2)$ into the alternating direction method of multipliers (ADMM), we develop a Bregman proximal linearized ADMM with guaranteed subsequential convergence to a lifted critical point. Convergence of the entire sequence is established under the Kurdyka--Łojasiewicz (KL) property. We further derive convergence rates under either a Hölderian value proximity error bound condition or the KL property, with the corresponding exponent in $[0, 1)$. In particular, we establish superlinear convergence for exponents in $(0, 1/2)$, together with finite, linear, and sublinear convergence in the remaining cases. Numerical experiments demonstrate the effectiveness of the proposed algorithm.