Fix objective sensitivity when parameters appear in the objective (#340)

* fix parametric obj

* format

* split fd test

* add math

* typo

Author: klamike

src/NonLinearProgram/nlp_utilities.jl

@@ -489,9 +489,16 @@ function _compute_sensitivity(model::Model; tol = 1e-6)
     # Dual bounds upper
     ∂s[((num_w+num_cons+num_lower+1):end), :] *= -_sense_multiplier
 
-    # dual wrt parameter
+    grad = _compute_gradient(model)
+    # `grad` = [∇ₓf(x,p); ∇ₚf(x,p)] where `x` is the primal vars, `p` is the params,
+    #  and `f(x,p)` is the objective function. we extract the components
+    #  so we can form `∇ₚfᵒ(x,p) = ∇ₓf(x,p) * ∇ₚxᵒ(p) + ∇ₚf(x,p) * ∇ₚpᵒ(p)`
+    #  where `ᵒ` denotes "optimal". note that parameters are fixed, so
+    #  pᵒ(p) = p and ∇ₚpᵒ(p) = 𝐈ₚ.
     primal_idx = [i.value for i in model.cache.primal_vars]
-    df_dx = _compute_gradient(model)[primal_idx]
-    df_dp = df_dx'∂s[1:num_vars, :]
+    params_idx = [i.value for i in model.cache.params]
+    df_dx = grad[primal_idx]  # ∇ₓf(x,p)
+    df_dp_direct = grad[params_idx]  # ∇ₚf(x,p)
+    df_dp = df_dx'∂s[1:num_vars, :] + df_dp_direct'  # ∇ₚfᵒ(x,p) = ∇ₓf(x,p) * ∇ₚxᵒ(p) + ∇ₚf(x,p) * 𝐈ₚ
     return ∂s, df_dp
 end

test/nlp_program.jl

@@ -740,6 +740,40 @@ function test_ObjectiveSensitivity_model2()
     @test isapprox(dp, -1.5; atol = 1e-4)
 end
 
+function test_ObjectiveSensitivity_direct_param_contrib()
+    model = DiffOpt.nonlinear_diff_model(Ipopt.Optimizer)
+    set_silent(model)
+
+    p_val = 3.0
+    @variable(model, p ∈ MOI.Parameter(p_val))
+    @variable(model, x ≥ 1)
+    @objective(model, Min, p^2 * x^2)
+
+    optimize!(model)
+    @assert is_solved_and_feasible(model)
+
+    Δp = 0.1
+    DiffOpt.set_forward_parameter(model, p, Δp)
+    DiffOpt.forward_differentiate!(model)
+
+    df_dp = MOI.get(model, DiffOpt.ForwardObjectiveSensitivity())
+    @test isapprox(df_dp, 2 * p_val * Δp, atol = 1e-8)   # ≈ 0.6 for p=3
+
+    ε = 1e-6
+    df_dp_fdpos = begin
+        set_parameter_value(p, p_val + ε)
+        optimize!(model)
+        Δp * objective_value(model)
+    end
+    df_dp_fdneg = begin
+        set_parameter_value(p, p_val - ε)
+        optimize!(model)
+        Δp * objective_value(model)
+    end
+    df_dp_fd = (df_dp_fdpos - df_dp_fdneg) / (2ε)
+
+    @test isapprox(df_dp, df_dp_fd, atol = 1e-4)
+end
 function test_ObjectiveSensitivity_subset_parameters()
     # Model with 10 parameters, differentiate only w.r.t. 3rd and 7th
     model = Model(() -> DiffOpt.diff_optimizer(Ipopt.Optimizer))