Legendre Feasibility Layer

(1) Integration point: attach the layer immediately after a context encoder $z=f_\theta(x)$ and before the downstream decision loss. Store the fixed constraint matrix $A\in\mathbb{R}^{m\times n}$ and vector $b$ for $S=\{w:Aw<b\}$. Replace penalty terms, post-hoc clipping, or nondifferentiable Euclidean projection. The component is suitable for supervised prescriptive learning, constrained policy networks, and RL actors whose action must satisfy budgets or resource limits.

(2) Pseudocode:

z = encoder_theta(x)
w = initialize_strictly_feasible(A, b)
for t in 1..NewtonSteps:
    slack = b - A @ w
    g = z - eps * sum_rows(A / slack[:,None], axis=0)
    H = eps * A.T @ diag(1 / slack**2) @ A
    delta = solve(H + damping*I, g)
    alpha = largest_backtracking_step(w, delta, A, b)
    w = w - alpha * delta
return w

The gradient $g$ is the gradient of $\langle z,w\rangle+\varepsilon h(w)$ and $H$ is $\varepsilon\nabla^2h(w)$. During backward, either differentiate the unrolled Newton iterations or use the implicit Jacobian $J=-H^{-1}$; for upstream gradient $u=\partial L/\partial w$, return $\partial L/\partial z=J^\top u$. Add damping only to stabilize nearly singular systems.

(3) Computed from the mathematics: strict feasibility, barrier gradient and Hessian, Newton updates, backtracking, and the inverse-Hessian Jacobian. Estimated empirically: Newton iteration count, damping, backtracking factor, and the schedule for $\varepsilon$. Start with $\varepsilon=0.1$ and optionally anneal toward $0.01$; monitor the minimum slack because small $\varepsilon$ can create ill-conditioning. For simplex constraints, use the exact entropy form $h(w)=\sum_iw_i\log w_i$, which gives $T_\varepsilon(z)_i=\exp(-z_i/\varepsilon)/\sum_j\exp(-z_j/\varepsilon)$, as a correctness baseline. For equality constraints $Cw=d$, solve Newton systems in KKT form with matrix $\begin{bmatrix}H&C^\top\\C&0\end{bmatrix}$.

(4) First cheap experiment: use a 2-layer MLP on synthetic contextual resource allocation with $w\in\mathbb{R}^{20}$, constraints $w\ge0$, $\sum_iw_i=1$, and five additional linear capacity constraints. Train against a random linear-plus-quadratic downstream cost. Compare an unconstrained output plus penalty, Euclidean projection, and this Legendre layer at equal hidden size and optimizer. Measure constraint violations, downstream cost, gradient norms near active constraints, convergence steps, and wall-clock time. Success means zero violations up to numerical tolerance, smoother loss and gradient curves than projection, and lower test downstream cost at equal training steps; a useful target is at least 10% lower cost than the penalty baseline without degradation relative to projection.