TY - JOUR

T1 - A proximal algorithm with backtracked extrapolation for a class of structured fractional programming

AU - Li, Qia

AU - Shen, Lixin

AU - Zhang, Na

AU - Zhou, Junpeng

N1 - Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2022/1

Y1 - 2022/1

N2 - 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.

AB - 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.

KW - Backtracked extrapolation

KW - Fractional programming

KW - Proximal algorithm

KW - Sparse recovery

UR - http://www.scopus.com/inward/record.url?scp=85113403674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85113403674&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2021.08.004

DO - 10.1016/j.acha.2021.08.004

M3 - Article

AN - SCOPUS:85113403674

SN - 1063-5203

VL - 56

SP - 98

EP - 122

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

ER -