Inexact Accelerated Proximal Gradient Method Revisit: An Economical Variant via Shadow Points

Abstract

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function via the inexact accelerated proximal gradient (APG) method. A key limitation of existing inexact APG methods is their reliance on feasible approximate solutions of the subproblems, which is often computationally expensive or even unrealistic to obtain in practice. To overcome this limitation, we develop a shadow-point enhanced inexact APG method (SpinAPG), which relaxes the feasibility requirement by allowing the computed iterates to be potentially infeasible, while introducing an auxiliary feasible shadow point solely for error control without requiring its explicit computation. This design decouples feasibility enforcement from the algorithmic updates and leads to a flexible and practically implementable inexact framework. Under suitable summable error conditions, we show that SpinAPG preserves all desirable convergence properties of the APG method, including the iterate convergence and an $o(1/k^2)$ convergence rate for the objective function values. These results complement and extend existing convergence analyses of inexact APG methods by demonstrating that the accelerated convergence can be retained even in the presence of controlled infeasibility. Numerical experiments on sparse quadratic programming problems illustrate the practical advantages of SpinAPG, showing that it can substantially reduce computational overhead by avoiding explicit computations of feasible points.