An Adaptive Proximal Inexact Gradient Framework and Its Application to Per-Antenna Constrained Joint Beamforming and Compression Design

TL;DR

This work introduces the Adaptive Proximal Inexact Gradient (APIG) framework for nonsmooth composite optimization with inexact function and gradient information, featuring a novel line-search that adaptively handles errors while preserving convergence guarantees. It proves that APIG attains an $\epsilon$-stationary point in $\mathcal{O}(\epsilon^{-2})$ iterations for nonconvex objectives and an $\epsilon$-optimal solution in $\mathcal{O}(\epsilon^{-1})$ iterations for convex ones, under general inexactness conditions. The framework is specialized to the joint beamforming and compression problem (JBCP) with per-antenna power constraints (PAPCs) by leveraging SDR tightness to obtain a differentiable dual, and solving it via an APIG-FP algorithm that uses fixed-point iterations to compute function and gradient approximations. Numerical results show APIG-FP significantly outperforms state-of-the-art benchmarks in computational efficiency, validating the approach and its applicability to other signal processing problems with function and gradient errors.

Abstract

In this paper, we propose an adaptive proximal inexact gradient (APIG) framework for solving a class of nonsmooth composite optimization problems involving function and gradient errors. Unlike existing inexact proximal gradient methods, the proposed framework introduces a new line search condition that jointly adapts to function and gradient errors, enabling adaptive stepsize selection while maintaining theoretical guarantees. Specifically, we prove that the proposed framework achieves an $ε$-stationary point within $\mathcal{O}(ε^{-2})$ iterations for nonconvex objectives and an $ε$-optimal solution within $\mathcal{O}(ε^{-1})$ iterations for convex cases, matching the best-known complexity in this context. We then custom-apply the APIG framework to an important signal processing problem: the joint beamforming and compression problem (JBCP) with per-antenna power constraints (PAPCs) in cooperative cellular networks. This customized application requires careful exploitation of the problem's special structure such as the tightness of the semidefinite relaxation (SDR) and the differentiability of the dual. Numerical experiments demonstrate the superior performance of our custom-application over state-of-the-art benchmarks for the JBCP.

Figures (1)

• Figure 1: The average CPU time and relative average CPU time versus the SINR target $\overline{\gamma}$ for different algorithms.

Theorems & Definitions (21)

• Lemma 1
• proof
• Proposition 1
• Lemma 2
• proof
• Lemma 3: Controlled Descent Property
• proof
• Theorem 1: Iteration Complexity and Convergence Rate
• proof
• Proposition 2