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The goal of this paper is to reduce the total complexity of gradient-basedmethods for two classes of problems: affine-constrained composite convexoptimization and bilinear saddle-point structured non-smooth convexoptimization. Our technique is based on a double-loop inexact acceleratedproximal gradient (APG) method for minimizing the summation of a non-smooth butproximable convex function and two smooth convex functions with differentsmoothness constants and computational costs. Compared to the standard APGmethod, the inexact APG method can reduce the total computation cost if onesmooth component has higher computational cost but a smaller smoothnessconstant than the other. With this property, the inexact APG method can beapplied to approximately solve the subproblems of a proximal augmentedLagrangian method for affine-constrained composite convex optimization and thesmooth approximation for bilinear saddle-point structured non-smooth convexoptimization, where the smooth function with a smaller smoothness constant hassignificantly higher computational cost. Thus it can reduce total complexityfor finding an approximately optimal/stationary solution. This technique issimilar to the gradient sliding technique in the literature. The difference isthat our inexact APG method can efficiently stop the inner loop by using acomputable condition based on a measure of stationarity violation, while thegradient sliding methods need to pre-specify the number of iterations for theinner loop. Numerical experiments demonstrate significantly higher efficiencyof our methods over an optimal primal-dual first-order method and the gradientsliding methods.

Inexact accelerated proximal gradient method with line search and  reduced complexity for affine-constrained and bilinear saddle-point  structured convex problems