PRISM: From Feasible Geometry to Routing Capacity in Learning to Optimize

Many learning-to-optimize methods struggle once the constraint set is curved, non-convex or disconnected: they either cannot guarantee hard feasibility there or lose solution quality. Feasibility alone is also not enough: a parametrisation can be feasible everywhere yet unable to represent the solution it is trained toward, and then no training closes the gap. For ray parametrisations, which place the output on a ray from an interior anchor, reachability is set by two budgets, anchors A and feasible segments K per ray: with one anchor and one segment, a second component or the far side of a hole is out of reach; each set admits a region of sufficient (A,K) budgets, and some are provably insufficient. PRISM realises such budgets: one shared network proposes ray candidates from an anchor bank, a parameter-free oracle marks each ray's feasible segments, and with valid anchors and an exact or certified oracle every hard-routed output is feasible. A measurement pass reads a data-supported (A,K) configuration off the feasible training labels and certifies its held-out coverage, without claiming the smallest budget. Under stated regularity assumptions we prove compact-set approximation, single-anchor obstructions, and a coverage-to-gap existence bound. On ten non-convex geometries the measured budgets cover at least 98% of held-out targets while single-anchor controls are starved on every disconnected set; on fifteen parametric problems PRISM meets each row's feasibility criterion and is best on twelve (the three legacy AC optimal power flow rows are coordinate-surrogate, box-only-residual evidence, not full-AC feasibility); on case30, conditional on a certified initial root, each accepted update certifies a continuous, strictly feasible root path with exact cost decrease.