diff --git a/src/shiftedGroupNormL2Binf.jl b/src/shiftedGroupNormL2Binf.jl
index 162e4e85..4ae4a312 100644
--- a/src/shiftedGroupNormL2Binf.jl
+++ b/src/shiftedGroupNormL2Binf.jl
@@ -33,7 +33,7 @@ end
 
 function (ψ::ShiftedGroupNormL2Binf)(y)
   @. ψ.xsy = ψ.sj + y
-  indball_val = IndBallLinf(1.1 * ψ.Δ)(ψ.xsy)
+  indball_val = IndBallLinf(1.01 * ψ.Δ)(ψ.xsy)
   ψ.xsy .+= ψ.xk
   return ψ.h(ψ.xsy) + indball_val
 end
@@ -78,40 +78,42 @@ function prox!(
   V2 <: AbstractVector{R},
 }
   ψ.sol .= q .+ ψ.xk .+ ψ.sj
-  ϵ = 1 ## sasha's initial guess
+
   softthres(x, a) = sign.(x) .* max.(0, abs.(x) .- a)
   l2prox(x, a) = max(0, 1 - a / norm(x)) .* x
+  linfproj(x) = max.(-ψ.Δ, min.(ψ.Δ, x))
   for (idx, λ) ∈ zip(ψ.h.idx, ψ.h.lambda)
     σλ = λ * σ
-    ## find root for each block
     froot(n) =
       n - norm(
-        σ .* softthres(
-          (ψ.sol[idx] ./ σ .- (n / (σ * (n - σλ))) .* ψ.xk[idx]),
-          ψ.Δ * (n / (σ * (n - σλ))),
-        ) .- ψ.sol[idx],
+        (n / (n - σλ)) .* (
+          ψ.xk[idx] .+ linfproj(
+            ((n - σλ)/n) .* ψ.sol[idx] .- ψ.xk[idx]
+          )
+        )
       )
-    lmin = σλ * (1 + eps(R)) # lower bound
-    fl = froot(lmin)
-
-    ansatz = lmin + ϵ #ansatz for upper bound
-    step = ansatz / (σ * (ansatz - σλ))
-    zlmax = norm(softthres((ψ.sol[idx] ./ σ .- step .* ψ.xk[idx]), ψ.Δ * step))
-    lmax = norm(ψ.sol[idx]) + σ * (zlmax + abs((ϵ - 1) / ϵ + 1) * λ * norm(ψ.xk[idx]))
-    fm = froot(lmax)
-    if fl * fm > 0
+    xlength = length(ψ.xk[idx])
+    xnorminf = ψ.χ(ψ.xk[idx])
+    xnorm = norm(ψ.xk[idx])
+    qnorm = norm(ψ.sol[idx])
+    qproj = sum(linfproj(ψ.sol[idx] .- ψ.xk[idx]) - ψ.xk[idx])
+    τ = 1e4*eps(R)
+    n = NaN
+    if xnorminf > ψ.Δ #case 1
+      n = find_zero(froot, (σλ + xnorminf  - ψ.Δ, σλ + xnorm + ψ.Δ*√xlength), Roots.A42())
+    elseif qnorm > σλ &&  xnorminf < ψ.Δ #case 2a
+      n = find_zero(froot, (σλ + τ, qnorm + xnorm + ψ.Δ*√xlength), Roots.A42())
+    elseif qnorm > σλ && xnorm ≈ ψ.Δ && qproj ≈ 0.0 #case 4a
+      n = find_zero(froot, (σλ + τ, min(σλ + xnorm + ψ.Δ*sqrt(n), qnorm)), Roots.A42())
+    end
+    if isnan(n)
       y[idx] .= 0
     else
-      n = fzero(froot, lmin, lmax)
       step = n / (σ * (n - σλ))
-      if abs(n - σλ) ≈ 0
-        y[idx] .= 0
-      else
-        y[idx] .= l2prox(
-          ψ.sol[idx] .- σ .* softthres((ψ.sol[idx] ./ σ .- step .* ψ.xk[idx]), ψ.Δ * step),
-          σλ,
-        )
-      end
+      y[idx] .= l2prox(
+        ψ.sol[idx] .- σ .* softthres((ψ.sol[idx] ./ σ .- step .* ψ.xk[idx]), ψ.Δ * step),
+        σλ,
+      )
     end
     y[idx] .-= (ψ.xk[idx] + ψ.sj[idx])
   end