Feasible by Tolerance or by Geometry? Radial Retractions in Neural Solvers for Constrained Optimization

Many neural solvers for parametric constrained programs correct the network's prediction with a differentiable projection or feasibility iteration, which approaches the feasible set from outside and stops at a small residual. Whether the corrected output passes a fixed feasibility check then depends on how this iteration was stopped. Evaluating all methods under one strict rule, we find that the strict feasibility rate of two recent solvers moves from 0% to 100% as the iteration budget, the floating-point precision or the stopping tolerance changes, while the trained network stays fixed. We propose RRA (radial retraction from an anchor), which reaches the boundary from inside instead. A differentiable Gauss–Newton interior map moves the network's proposal to a strictly feasible anchor in the basin the network selected, and a radial retraction stops at the first boundary contact on the ray from this anchor toward the proposal. The exit parameter of this ray is a closed-form root for affine, convex-quadratic, box and second-order-cone constraints, and a scan brackets it from below for trigonometric constraints. The returned point is then feasible for any strictly feasible anchor, whatever the tolerance, depth or convergence of the map that placed it. RRA passes the strict rule on every test instance and training seed of ten families of convex and nonconvex QP, QCQP and SOCP, and its optimality gap is lower than that of FSNet on all ten. It is on par with the most accurate correction baseline while training 5 to 14 times faster. Longer training, which still takes at most 55% of that baseline's training time, moves RRA ahead of it. RRA also solves the two nonconvex second-order-cone families on which that baseline returns almost no feasible point.