We　consider　a　class　of　nonsmooth　convex　optimization　problems　where　the　objective　function　is　the　composition　of　a　strongly　convex　differentiable　function　with　a　linear　mapping,　regularized　by　the　sum　of　both　1-norm　and　2-norm　of　the　optimization　variables.　This　class　of　problems　arise　naturally　from　applications　in　sparse　group　Lasso,　which　is　a　popular　technique　for　variable　selection.　An　effective　approach　to　solve　such　problems　is　by　the　Proximal　Gradient　Method　(PGM).　In　this　paper　we　prove　a　local　error　bound　around　the　optimal　solution　set　for　this　problem　and　use　it　to　establish　the　linear　convergence　of　the　PGM　method　without　assuming　strong　convexity　of　the　overall　objective　function.　©　2013　Operations　Research　Society　of　China,　Periodicals　Agency　of　Shanghai　University,　and　Springer-Verlag　Berlin　Heidelberg.