Fix Lagrangian Hessian prototype dimensions in OptimizationDIExt

The Lagrangian Hessian prototype was incorrectly sized as (num_constraints × num_variables)
instead of (num_variables × num_variables). This caused a `BoundsError` when computing the
Lagrangian Hessian with more variables than constraints, as the prototype was used as a
buffer for the n×n Hessian matrix.

Changes:
- Fix lag_hess_prototype initialization to use zeros(Bool, length(x), length(x))
  in both in-place and out-of-place function instantiation
- Add tests to verify prototype dimensions

This ensures the Lagrangian Hessian is always correctly sized as n×n regardless of the
number of constraints, matching the mathematical definition of the Lagrangian Hessian
as the second derivative with respect to the decision variables.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

Author: SebastianM-C

lib/OptimizationBase/src/OptimizationDIExt.jl

@@ -198,7 +198,7 @@ function instantiate_function(
         lag_prep = prepare_hessian(
             lagrangian, soadtype, x, Constant(one(eltype(x))),
             Constant(ones(eltype(x), num_cons)), Constant(p))
-        lag_hess_prototype = zeros(Bool, num_cons, length(x))
+        lag_hess_prototype = zeros(Bool, length(x), length(x))
 
         function lag_h!(H::AbstractMatrix, θ, σ, λ)
             if σ == zero(eltype(θ))
@@ -457,7 +457,7 @@ function instantiate_function(
         lag_prep = prepare_hessian(
             lagrangian, soadtype, x, Constant(one(eltype(x))),
             Constant(ones(eltype(x), num_cons)), Constant(p))
-        lag_hess_prototype = zeros(Bool, num_cons, length(x))
+        lag_hess_prototype = zeros(Bool, length(x), length(x))
 
         function lag_h!(θ, σ, λ)
             if σ == zero(eltype(θ))

lib/OptimizationBase/test/adtests.jl

@@ -144,6 +144,17 @@ optprob.cons_h(H3, x0)
     optprob.lag_h(H4, x0, σ, μ)
     @test H4≈σ * H2 + μ[1] * H3[1] rtol=1e-6
 
+    # Test that the AD-generated lag_hess_prototype has correct dimensions
+    @test !isnothing(optprob.lag_hess_prototype)
+    @test size(optprob.lag_hess_prototype) == (length(x0), length(x0))  # Should be n×n, not num_cons×n
+
+    # Test that we can actually use it as a buffer
+    if !isnothing(optprob.lag_hess_prototype)
+        H_proto = similar(optprob.lag_hess_prototype, Float64)
+        optprob.lag_h(H_proto, x0, σ, μ)
+        @test H_proto ≈ σ * H2 + μ[1] * H3[1] rtol=1e-6
+    end
+
     G2 = Array{Float64}(undef, 2)
     H2 = Array{Float64}(undef, 2, 2)