Envelope-Gradient Optimization Layer

Use this as a specialized autograd path for a scalar decision-focused loss equal to the QP optimum $V(\phi)$, or to a weighted sum of independently solved QP values. The neural network outputs $\phi$; the forward layer calls a QP solver and caches $z^\star,\lambda^\star,\nu^\star$. In backward, compute vector-Jacobian products directly from $\frac12z^{\star\top}(dP)z^\star+(dq)^\top z^\star+dc+\lambda^{\star\top}((dA)z^\star-db)+\nu^{\star\top}((dG)z^\star-dh)$, then backpropagate through the inexpensive parameterization $\phi\mapsto(P,q,c,A,b,G,h)$. For example, if only $q(\phi)=W\phi+q_0$ varies, return $\partial V/\partial\phi=W^\top z^\star$; if only $h(\phi)=H\phi+h_0$ varies, return $\partial V/\partial\phi=-H^\top\nu^\star$. The solver supplies primal and dual values; the theorem and chain-rule contractions are taken from the paper, while gradient correctness should be checked empirically with finite differences. First test on an MLP forecasting 24-hour demand for a small economic-dispatch QP, comparing envelope differentiation with a generic differentiable-QP adjoint implementation and ordinary MSE training. Measure backward latency, peak memory, operating cost, and finite-difference relative error. The prediction is zero additional KKT linear solves for a scalar value loss, with relative gradient error below $10^{-3}$ while the active set is fixed; loss and gradient curves should show kinks only at active-set transitions, and the speedup should increase with QP dimension.