Critical-Region LP Policy Layer

Integrate this as a custom differentiable optimization layer between a neural policy and the downstream loss. The policy network receives state $s_t$ and outputs LP parameters, for example $h_t=\operatorname{softplus}(g_\psi(s_t))+\epsilon$ and optionally $c_t=g^c_\psi(s_t)$. The forward pass solves the LP with a standard primal or primal-dual solver and records the active constraint indices, the basis matrix $B_t$, the primal action $a_t$, primal slacks, and dual variables. In the backward pass, first identify the $n$ active independent rows; solve linear systems with $B_t$ rather than explicitly forming $B_t^{-1}$. Compute $A_t=B_t^{-1}(h_{t,\theta}+h_{t,s}S_t)$ and propagate $S_{t+1}=\phi_sS_t+\phi_aA_t+\phi_\theta$. Backpropagate the final loss through $S_t$ into the neural parameters using ordinary autodiff. If the LP objective coefficients are learned, include the reduced-cost derivative using the recorded dual solution; if a parameter changes $G$, use the general term $-B_t^{-1}(B_{t,x}a_t)$. Mark a sample as unsafe when the smallest active slack margin, reduced-cost margin, or basis condition number is below a threshold; for unsafe samples, fall back to solver autodiff, a damped gradient, or a straight-through action while logging the event. The mathematical quantities are the basis sensitivity and recurrent chain rule; Jacobians of $\phi$ and the network are estimated by autodiff. First test a small inventory or transportation policy: a 2-layer MLP controlling a differentiable min-cost-flow LP on randomly generated demand. Compare analytic basis backpropagation against unrolled solver differentiation and finite differences at equal forward solves. Success means lower gradient variance, faster loss reduction at equal wall-clock time, and no degradation near—but not exactly on—basis boundaries.