An Inexact Variable Metric Proximal Gradient-subgradient Algorithm for a Class of Fractional Optimization Problems

In this paper, we study a class of fractional optimization problems, in which the numerator of the objective is the sum of a convex function and a differentiable function with a Lipschitz continuous gradient, while the denominator is a nonsmooth convex function. This model has broad applicability and encompasses several important optimization problems in the literature. To address these problems, we propose an inexact variable metric proximal gradient-subgradient algorithm (iVPGSA), which, to our knowledge, is the first inexact proximal algorithm specifically designed for solving such type of fractional problems. By incorporating a variable metric proximal term and allowing for inexact solutions to the subproblem under a flexible error criterion, the proposed algorithm is highly adaptable to a broader range of problems while achieving favorable computational efficiency. Under mild assumptions, we establish that any accumulation point of the sequence generated by the iVPGSA is a critical point of the target problem. Moreover, we develop an improved Kurdyka-{\L}ojasiewicz (KL)-based analysis framework to prove the global convergence of the entire sequence and characterize its convergence rate, \textit{without} requiring a strict sufficient descent property. Our results offer detailed insights into how the KL exponent and inexactness influence the convergence rate. The proposed analysis framework also has the potential to serve as a theoretical tool for studying the convergence rates of a wide range of inexact algorithms beyond the iVPGSA. Finally, some numerical experiments on the $\ell_1/\ell_2$ Lasso problem and the constrained $\ell_1/\ell_2$ sparse optimization problem are conducted to show the superior performance of the iVPGSA in comparison to existing algorithms.