Remove second SDP relaxation due to unreliable numerics.

docs/src/tutorials/applications/optimal_power_flow.jl

@@ -345,13 +345,10 @@ better_lower_bound
 E(k, n) = SparseArrays.sparse([k], [n], 1, N, N);
 
 # Of course, we've shifted the nonlinearity into the equality constraint
-# ``W = V V^*``: it is this constraint we will now relax using two different
-# semidefinite programming approaches, one with the complex voltage variables
-# ``V`` and the other one without.
+# ``W = V V^*``: it is this constraint we will now relax using a
+# semidefinite programming approach.
 
-# ### ``W`` and ``V`` relaxation
-
-# In the first instance, we will make use of complex voltages and relax
+# We will make use of complex voltages and relax
 # ``W = V V^*`` to;
 # ```math
 # W \succeq V V^*,
@@ -419,12 +416,12 @@ optimize!(model)
 
 #-
 
-first_relaxation_lower_bound = round(objective_value(model); digits = 2)
+sdp_relaxation_lower_bound = round(objective_value(model); digits = 2)
 Test.@test termination_status(model) in (OPTIMAL, ALMOST_OPTIMAL) #src
 Test.@test primal_status(model) in (FEASIBLE_POINT, NEARLY_FEASIBLE_POINT) #src
-Test.@test isapprox(first_relaxation_lower_bound, 2753.04; rtol = 1e-3) #src
+Test.@test isapprox(sdp_relaxation_lower_bound, 2753.04; rtol = 1e-3) #src
 println(
- "Objective value (W & V relax. lower bound): $first_relaxation_lower_bound",
+ "Objective value (W & V relax. lower bound): $sdp_relaxation_lower_bound",
 )
 
 # We can more easily see solution values by rounding out noisy data:
@@ -441,75 +438,10 @@ DataFrames.DataFrame(;
 
 # For further information on exploiting sparsity see Jabr (2012).
 
-# ### ``W`` only relaxation
-
-# In the second instance, we will relax ``W = V V^*`` to
-# ```math
-# W \succeq 0
-# ```
-# where we now introduce ``W`` as a new Hermitian matrix of decision variables.
-# This enables us to write down a relaxation without complex voltage variables.
-
-# With this modification we get the following semidefinite relaxation of the
-# AC-OPF problem:
-
-model = Model(Clarabel.Optimizer)
-set_attribute(model, "tol_gap_rel", 0.01)
-@variable(
- model,
- S_G[i in 1:N] in ComplexPlane(),
- lower_bound = P_Gen_lb[i] + Q_Gen_lb[i] * im,
- upper_bound = P_Gen_ub[i] + Q_Gen_ub[i] * im,
-)
-@variable(model, W[1:N, 1:N] in HermitianPSDCone())
-@constraint(model, [i in 1:N], 0.9^2 <= real(W[i, i]) <= 1.1^2)
-@constraint(
- model,
- [i in 1:N],
- S_G[i] - S_Demand[i] == LinearAlgebra.tr((conj(Y) * E(i, i)) * W),
-)
-P_G = real(S_G)
-@objective(
- model,
- Min,
- (0.11 * P_G[1]^2 + 5 * P_G[1] + 150) +
- (0.085 * P_G[2]^2 + 1.2 * P_G[2] + 600) +
- (0.1225 * P_G[3]^2 + P_G[3] + 335),
-)
-optimize!(model)
-
-#-
-
-second_relaxation_lower_bound = round(objective_value(model); digits = 2)
-println(
- "Objective value (W relax. lower bound): $second_relaxation_lower_bound",
-)
-
-Test.@test termination_status(model) in (OPTIMAL, ALMOST_OPTIMAL) #src
-Test.@test primal_status(model) in (FEASIBLE_POINT, NEARLY_FEASIBLE_POINT) #src
-Test.@test isapprox(second_relaxation_lower_bound, 2753.04; rtol = 1e-3) #src
-
-# We can more easily see the solution by filtering out the noisy data
-# arising from solver tolerances:
-
-W_2 = SparseArrays.sparse(round.(value.(W); digits = 1))
-Test.@test iszero(imag(LinearAlgebra.diag(W_2))) #src
-
-# with corresponding voltage magnitude estimates:
-
-DataFrames.DataFrame(;
- Bus = 1:N,
- Magnitude = round.(real.(sqrt.(LinearAlgebra.diag(W_2))); digits = 2),
-)
-
-# The first relaxation has the advantage that we can work directly with complex
+# This relaxation has the advantage that we can work directly with complex
 # voltages to extend the formulation, strengthen the relaxation and gain
 # additional approximate information about the voltage variables.
 
-# The second relaxation has the advantage of compactness in variables and
-# constraints while giving the same objective lower bound as the first
-# relaxation.
-
 # ## References and further resources
 
 # **Bukhsh**, W. A., Grothey, A., McKinnon, K. I., & Trodden, P. A.