Module implicit_diff 

Core implicit differentiation engine for optimization layers.

Given an optimal solution z* satisfying KKT conditions F(z*, θ) = 0, the implicit function theorem yields:

dz*/dθ = -(∂F/∂z)⁻¹ · (∂F/∂θ)

This module builds the KKT Jacobian ∂F/∂z, solves the resulting linear system, and supports an active-set variant that restricts differentiation to active inequality constraints.

Functions

adjust_complementarity_diagonal

Adjust the complementarity diagonal block with the rhs h.

compute_active_set_implicit_gradient

Compute implicit gradient using only active constraints.

compute_full_implicit_gradient

Compute the full implicit gradient dz*/dθ for a QP.

compute_kkt_jacobian

Build the KKT Jacobian matrix ∂F/∂z for the QP:

extract_active_constraints

Extract rows from G and h corresponding to the given active indices.

identify_active_constraints

Identify active inequality constraints: those where h_i - G_i x ≤ tol (i.e., the slack is near zero).

solve_implicit_system

Solve the linear system A x = rhs via Gaussian elimination with partial pivoting.

solve_implicit_system_multi

Solve the system mat * X = rhs_matrix where rhs_matrix has k columns. Returns the solution matrix (n × k) in row-major form.