Certified Rank-Aware QP Layer

Integrate this at the output of a neural module whose prediction must obey linear constraints: for example, a portfolio allocation head with $x\ge0$ and $\mathbf1^\top x=1$, a control head with actuator bounds and rate inequalities, or a resource-allocation head with $Cx\ge b$. At each forward pass, form a strictly convex QP with $G=\operatorname{diag}(\operatorname{softplus}(g)+\epsilon)$ or a fixed SPD matrix, and let the network produce $a$. Initialize at the unconstrained minimizer $x_0=G^{-1}a$ and maintain an active set $\mathcal A$. Use this loop: (1) find the most violated constraint $i=\arg\min_j(C_j^\top x-b_j)$ and add it if its violation is below $-\tau$; (2) deduplicate and rank-reveal $C_{\mathcal A}$ using QR with threshold $\rho$, discarding dependent constraints or retaining one representative from each dependent group; (3) solve the equality-constrained KKT system $$\begin{bmatrix}G&-C_{\mathcal A}\\C_{\mathcal A}^\top&0\end{bmatrix}\begin{bmatrix}x\\\lambda_{\mathcal A}\end{bmatrix}=\begin{bmatrix}a\\b_{\mathcal A}\end{bmatrix};$$ (4) if any active multiplier is negative, remove the most negative one and repeat; (5) terminate only when the maximum violation is at most $ au$, the minimum multiplier is at least $- au$, and the stationarity residual $\|Gx-a-C\lambda\|_\infty$ is at most $ au$. Backpropagate through the final linear solve using implicit differentiation; for an initial implementation, stop gradients through active-set decisions and differentiate only the KKT solve. Compute rank and residual thresholds empirically, while the KKT solution, feasibility checks, and multiplier checks come directly from the paper's mathematics. The first cheap experiment should use a small MLP on a synthetic portfolio dataset and compare this layer with softmax-plus-penalty and projected gradient descent at equal forward cost. Measure constraint violation, objective gap against a trusted CVXPY solution, gradient stability, wall-clock time, and training loss. A successful result is near-machine-precision feasibility, lower objective gap, and fewer failed or oscillatory training steps, especially after duplicating or nearly duplicating constraint columns.