diff --git a/engine-proto/gram-test/Engine.jl b/engine-proto/gram-test/Engine.jl
index e6f0d97..10f4b02 100644
--- a/engine-proto/gram-test/Engine.jl
+++ b/engine-proto/gram-test/Engine.jl
@@ -1,10 +1,14 @@
 module Engine
 
 using LinearAlgebra
+using GenericLinearAlgebra
 using SparseArrays
 using Random
+using Optim
 
-export rand_on_shell, Q, DescentHistory, realize_gram
+export
+  rand_on_shell, Q, DescentHistory,
+  realize_gram_gradient, realize_gram_newton, realize_gram_optim, realize_gram
 
 # === guessing ===
 
@@ -76,8 +80,11 @@ end
 struct DescentHistory{T}
   scaled_loss::Array{T}
   neg_grad::Array{Matrix{T}}
+  base_step::Array{Matrix{T}}
+  hess::Array{Hermitian{T, Matrix{T}}}
   slope::Array{T}
   stepsize::Array{T}
+  positive::Array{Bool}
   backoff_steps::Array{Int64}
   last_line_L::Array{Matrix{T}}
   last_line_loss::Array{T}
@@ -85,13 +92,16 @@ struct DescentHistory{T}
   function DescentHistory{T}(
     scaled_loss = Array{T}(undef, 0),
     neg_grad = Array{Matrix{T}}(undef, 0),
+    hess = Array{Hermitian{T, Matrix{T}}}(undef, 0),
+    base_step = Array{Matrix{T}}(undef, 0),
     slope = Array{T}(undef, 0),
     stepsize = Array{T}(undef, 0),
+    positive = Bool[],
     backoff_steps = Int64[],
     last_line_L = Array{Matrix{T}}(undef, 0),
     last_line_loss = Array{T}(undef, 0)
   ) where T
-    new(scaled_loss, neg_grad, slope, stepsize, backoff_steps, last_line_L, last_line_loss)
+    new(scaled_loss, neg_grad, hess, base_step, slope, stepsize, positive, backoff_steps, last_line_L, last_line_loss)
   end
 end
 
@@ -101,7 +111,7 @@ function realize_gram_gradient(
   gram::SparseMatrixCSC{T, <:Any},
   guess::Matrix{T};
   scaled_tol = 1e-30,
-  target_improvement = 0.5,
+  min_efficiency = 0.5,
   init_stepsize = 1.0,
   backoff = 0.9,
   max_descent_steps = 600,
@@ -152,7 +162,7 @@ function realize_gram_gradient(
       improvement = loss_last - loss
       push!(history.last_line_L, L)
       push!(history.last_line_loss, loss / scale_adjustment)
-      if improvement >= target_improvement * stepsize * slope
+      if improvement >= min_efficiency * stepsize * slope
         history.backoff_steps[end] = backoff_steps
         break
       end
@@ -201,7 +211,7 @@ function realize_gram_newton(
   scale_adjustment = sqrt(T(length(constrained)))
   tol = scale_adjustment * scaled_tol
   
-  # use newton's method
+  # use Newton's method
   L = copy(guess)
   for step in 0:max_steps
     # evaluate the loss function
@@ -229,8 +239,10 @@ function realize_gram_newton(
       deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj)
       hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
     end
+    hess = Hermitian(hess)
+    push!(history.hess, hess)
     
-    # compute the newton step
+    # compute the Newton step
     step = hess \ reshape(neg_grad, total_dim)
     L += rate * reshape(step, dims)
   end
@@ -239,4 +251,168 @@ function realize_gram_newton(
   L, history
 end
 
+LinearAlgebra.eigen!(A::Symmetric{BigFloat, Matrix{BigFloat}}; sortby::Nothing) =
+  eigen!(Hermitian(A))
+
+function convertnz(type, mat)
+  J, K, values = findnz(mat)
+  sparse(J, K, type.(values))
+end
+
+function realize_gram_optim(
+  gram::SparseMatrixCSC{T, <:Any},
+  guess::Matrix{T}
+) where T <: Number
+  # find the dimension of the search space
+  dims = size(guess)
+  element_dim, construction_dim = dims
+  total_dim = element_dim * construction_dim
+  
+  # list the constrained entries of the gram matrix
+  J, K, _ = findnz(gram)
+  constrained = zip(J, K)
+  
+  # scale the loss function
+  scale_adjustment = length(constrained)
+  
+  function loss(L_vec)
+    L = reshape(L_vec, dims)
+    Δ_proj = proj_diff(gram, L'*Q*L)
+    dot(Δ_proj, Δ_proj) / scale_adjustment
+  end
+  
+  function loss_grad!(storage, L_vec)
+    L = reshape(L_vec, dims)
+    Δ_proj = proj_diff(gram, L'*Q*L)
+    storage .= reshape(-4*Q*L*Δ_proj, total_dim) / scale_adjustment
+  end
+  
+  function loss_hess!(storage, L_vec)
+    L = reshape(L_vec, dims)
+    Δ_proj = proj_diff(gram, L'*Q*L)
+    indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim]
+    for (j, k) in indices
+      basis_mat = basis_matrix(T, j, k, dims)
+      neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat
+      neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained)
+      deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj) / scale_adjustment
+      storage[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
+    end
+  end
+  
+  optimize(
+    loss, loss_grad!, loss_hess!,
+    reshape(guess, total_dim),
+    Newton()
+  )
+end
+
+# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
+# explicit entry of `gram`. use gradient descent starting from `guess`
+function realize_gram(
+  gram::SparseMatrixCSC{T, <:Any},
+  guess::Matrix{T};
+  scaled_tol = 1e-30,
+  min_efficiency = 0.5,
+  init_rate = 1.0,
+  backoff = 0.9,
+  reg_scale = 1.1,
+  max_descent_steps = 200,
+  max_backoff_steps = 110
+) where T <: Number
+  # start history
+  history = DescentHistory{T}()
+  
+  # find the dimension of the search space
+  dims = size(guess)
+  element_dim, construction_dim = dims
+  total_dim = element_dim * construction_dim
+  
+  # list the constrained entries of the gram matrix
+  J, K, _ = findnz(gram)
+  constrained = zip(J, K)
+  
+  # scale the tolerance
+  scale_adjustment = sqrt(T(length(constrained)))
+  tol = scale_adjustment * scaled_tol
+  
+  # initialize variables
+  grad_rate = init_rate
+  L = copy(guess)
+  
+  # use Newton's method with backtracking and gradient descent backup
+  Δ_proj = proj_diff(gram, L'*Q*L)
+  loss = dot(Δ_proj, Δ_proj)
+  for step in 1:max_descent_steps
+    # stop if the loss is tolerably low
+    if loss < tol
+      break
+    end
+    
+    # find the negative gradient of loss function
+    neg_grad = 4*Q*L*Δ_proj
+    
+    # find the negative Hessian of the loss function
+    hess = Matrix{T}(undef, total_dim, total_dim)
+    indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim]
+    for (j, k) in indices
+      basis_mat = basis_matrix(T, j, k, dims)
+      neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat
+      neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained)
+      deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj)
+      hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
+    end
+    hess = Hermitian(hess)
+    push!(history.hess, hess)
+    
+    # regularize the Hessian
+    min_eigval = minimum(eigvals(hess))
+    push!(history.positive, min_eigval > 0)
+    if min_eigval <= 0
+      hess -= reg_scale * min_eigval * I
+    end
+    base_step = reshape(hess \ reshape(neg_grad, total_dim), dims)
+    push!(history.base_step, base_step)
+    
+    # store the current position, loss, and slope
+    L_last = L
+    loss_last = loss
+    push!(history.scaled_loss, loss / scale_adjustment)
+    push!(history.neg_grad, neg_grad)
+    push!(history.slope, norm(neg_grad))
+    
+    # find a good step size using backtracking line search
+    push!(history.stepsize, 0)
+    push!(history.backoff_steps, max_backoff_steps)
+    empty!(history.last_line_L)
+    empty!(history.last_line_loss)
+    rate = one(T)
+    step_success = false
+    for backoff_steps in 0:max_backoff_steps
+      history.stepsize[end] = rate
+      L = L_last + rate * base_step
+      Δ_proj = proj_diff(gram, L'*Q*L)
+      loss = dot(Δ_proj, Δ_proj)
+      improvement = loss_last - loss
+      push!(history.last_line_L, L)
+      push!(history.last_line_loss, loss / scale_adjustment)
+      if improvement >= min_efficiency * rate * dot(neg_grad, base_step)
+        history.backoff_steps[end] = backoff_steps
+        step_success = true
+        break
+      end
+      rate *= backoff
+    end
+    
+    # if we've hit a wall, quit
+    if !step_success
+      return L_last, false, history
+    end
+  end
+  
+  # return the factorization and its history
+  push!(history.scaled_loss, loss / scale_adjustment)
+  L, true, history
+end
+
 end
\ No newline at end of file
diff --git a/engine-proto/gram-test/circles-in-triangle.jl b/engine-proto/gram-test/circles-in-triangle.jl
index b031711..12975c2 100644
--- a/engine-proto/gram-test/circles-in-triangle.jl
+++ b/engine-proto/gram-test/circles-in-triangle.jl
@@ -81,16 +81,28 @@ guess = hcat(
 =#
 
 # complete the gram matrix using gradient descent followed by Newton's method
+#=
 L, history = Engine.realize_gram_gradient(gram, guess, scaled_tol = 0.01)
 L_pol, history_pol = Engine.realize_gram_newton(gram, L, rate = 0.3, scaled_tol = 1e-9)
 L_pol2, history_pol2 = Engine.realize_gram_newton(gram, L_pol)
+=#
+L, success, history = Engine.realize_gram(gram, guess)
 completed_gram = L'*Engine.Q*L
 println("Completed Gram matrix:\n")
 display(completed_gram)
+#=
 println(
   "\nSteps: ",
   size(history.scaled_loss, 1),
   " + ", size(history_pol.scaled_loss, 1),
   " + ", size(history_pol2.scaled_loss, 1)
 )
-println("Loss: ", history_pol2.scaled_loss[end], "\n")
\ No newline at end of file
+println("Loss: ", history_pol2.scaled_loss[end], "\n")
+=#
+if success
+  println("\nTarget accuracy achieved!")
+else
+  println("\nFailed to reach target accuracy")
+end
+println("Steps: ", size(history.scaled_loss, 1))
+println("Loss: ", history.scaled_loss[end], "\n")
\ No newline at end of file
diff --git a/engine-proto/gram-test/overlapping-pyramids.jl b/engine-proto/gram-test/overlapping-pyramids.jl
index 51606e1..ee4c1fc 100644
--- a/engine-proto/gram-test/overlapping-pyramids.jl
+++ b/engine-proto/gram-test/overlapping-pyramids.jl
@@ -47,17 +47,30 @@ guess = hcat(
   Engine.rand_on_shell(fill(BigFloat(-1), 2))
 )
 
-# complete the gram matrix using gradient descent followed by Newton's method
+# complete the gram matrix
+#=
 L, history = Engine.realize_gram_gradient(gram, guess, scaled_tol = 0.01)
 L_pol, history_pol = Engine.realize_gram_newton(gram, L)
+=#
+L, success, history = Engine.realize_gram(gram, guess)
 completed_gram = L'*Engine.Q*L
 println("Completed Gram matrix:\n")
 display(completed_gram)
+#=
 println("\nSteps: ", size(history.scaled_loss, 1), " + ", size(history_pol.scaled_loss, 1))
 println("Loss: ", history_pol.scaled_loss[end], "\n")
+=#
+if success
+  println("\nTarget accuracy achieved!")
+else
+  println("\nFailed to reach target accuracy")
+end
+println("Steps: ", size(history.scaled_loss, 1))
+println("Loss: ", history.scaled_loss[end], "\n")
 
 # === algebraic check ===
 
+#=
 R, gens = polynomial_ring(AbstractAlgebra.Rationals{BigInt}(), ["x", "t₁", "t₂", "t₃"])
 x = gens[1]
 t = gens[2:4]
@@ -85,3 +98,4 @@ x_constraint = 25//16 * to_univariate(S, evaluate(rank_constraints[1], [2], [ind
 t₂_constraint = 25//16 * to_univariate(S, evaluate(rank_constraints[3], [2], [indep_val]))
 x_vals = PolynomialRoots.roots(x_constraint.coeffs)
 t₂_vals = PolynomialRoots.roots(t₂_constraint.coeffs)
+=#
\ No newline at end of file
diff --git a/engine-proto/gram-test/sphere-in-tetrahedron.jl b/engine-proto/gram-test/sphere-in-tetrahedron.jl
index dc5588c..273d3ad 100644
--- a/engine-proto/gram-test/sphere-in-tetrahedron.jl
+++ b/engine-proto/gram-test/sphere-in-tetrahedron.jl
@@ -33,10 +33,18 @@ guess = sqrt(1/BigFloat(3)) * BigFloat[
   1  1  1  1  1
 ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5))
 
-# complete the gram matrix using Newton's method
+# complete the gram matrix
+#=
 L, history = Engine.realize_gram_newton(gram, guess)
+=#
+L, success, history = Engine.realize_gram(gram, guess)
 completed_gram = L'*Engine.Q*L
 println("Completed Gram matrix:\n")
 display(completed_gram)
-println("\nSteps: ", size(history.scaled_loss, 1))
+if success
+  println("\nTarget accuracy achieved!")
+else
+  println("\nFailed to reach target accuracy")
+end
+println("Steps: ", size(history.scaled_loss, 1))
 println("Loss: ", history.scaled_loss[end], "\n")
\ No newline at end of file