In this paper, we consider a class of structured fractional minimization problems where the numerator part of the objective is the sum of a convex function and a Lipschitz differentiable (possibly) nonconvex function, while the denominator part is a convex function. By exploiting the structure of the problem, we propose a first-order algorithm, namely, a proximal-gradient-subgradient algorithm with backtracked extrapolation (PGSA_BE) for solving this type of optimization problem. It is worth pointing out that there are a few differences between our backtracked extrapolation and other popular extrapolations used in convex and nonconvex optimization. One of such differences is as follows: if the new iterate obtained from the extrapolated iteration satisfies a backtracking condition, then this new iterate will be replaced by the one generated from the non-extrapolated iteration. We show that any accumulation point of the sequence generated by PGSA_BE is a critical point of the problem regarded. In addition, by assuming that some auxiliary functions satisfy the Kurdyka-Łojasiewicz property, we are able to establish global convergence of the entire sequence, in the case where the denominator is locally Lipschitz differentiable, or its conjugate satisfies the calmness condition. Finally, we present some preliminary numerical results to illustrate the efficiency of PGSA_BE.

在本文中，我们考虑一类结构化分数最小化问题，其中目标的分子部分是凸函数和 Lipschitz 可微（可能）非凸函数之和，而分母部分是凸函数。通过利用问题的结构，我们提出了一种一阶算法，即具有回溯外推的近端梯度次梯度算法（PGSA_BE）来解决此类优化问题。值得指出的是，我们的回溯外推与凸和非凸优化中使用的其他流行外推之间存在一些差异。其中一个差异如下：如果从外推迭代中获得的新迭代满足回溯条件，那么这个新的迭代将被非外推迭代生成的迭代替换。我们表明由 PGSA_BE 生成的序列的任何累积点都是所考虑问题的临界点。此外，通过假设一些辅助函数满足 Kurdyka-Łojasiewicz 性质，我们能够建立整个序列的全局收敛，在分母是局部 Lipschitz 可微的情况下，或者它的共轭满足平静条件。最后，我们提供了一些初步的数值结果来说明 PGSA_BE 的效率。我们能够建立整个序列的全局收敛，在分母是局部 Lipschitz 可微的情况下，或者它的共轭满足平静条件。最后，我们提供了一些初步的数值结果来说明 PGSA_BE 的效率。我们能够建立整个序列的全局收敛，在分母是局部 Lipschitz 可微的情况下，或者它的共轭满足平静条件。最后，我们提供了一些初步的数值结果来说明 PGSA_BE 的效率。