NEURAL-REFINE: A Learn-and-Refine Framework for Scalable Constrained Optimization

Learning-based solvers can amortize the cost of repeatedly solving related constrained optimization problems, but a neural prediction is useful only if it is both feasible and competitive in objective value. Existing predict-then-correct approaches often train the predictor largely independently of the numerical procedure used later to repair or improve its output, so the learned component need not discover starting points that are especially useful to that downstream optimizer. We introduce \method, a closed-loop learn-and-refine framework in which the numerical backend used at deployment also generates training feedback, without differentiating through the solver. The proposer produces multiple candidate starting points and, when useful, compact auxiliary information used by the numerical refiner. Periodically during training, the backend refines these candidates; the best feasible refined solution found so far for each instance is retained as a stop-gradient self-elite target for later proposer updates. At deployment, the trained proposer, deterministic refiner, and a common feasibility-first selector are executed once. Across all nine benchmark tasks formed by QP, QCQP, and SOCP in convex, nonconvex, and nonsmooth settings, \method passes the $10^{-4}$ feasibility criterion on all nine tasks, attains mean $10^{-6}$ feasible rate $0.843$ versus $0.024$ for FSNet, and gives the best feasible objective among learning-based methods on six tasks. It also substantially reduces online solution time relative to CPU IPOPT on the tasks where IPOPT satisfies the same feasibility criterion. In a separate matched-GPU scaling study, numerical solvers are faster at small dimensions, while \method exhibits a crossover as problem size grows. With separate training at each target dimension, a sparse realization remains operational up to $10^9$ variables, extending the same learn-and-refine design from standard benchmark sizes to the billion-variable scale.