Learning to Optimize Across Problem Scales: Size-Agnostic Solver Selection for Accelerated Constrained Optimization

Standard numerical optimization typically fixes a single solver for the entire iterations, even though the local geometry of an optimization trajectory can shift across well-conditioned and ill-conditioned regimes with different computational tradeoffs. We formulate solver selection as a sequential decision process and learn a size-agnostic selection function that dynamically chooses between different solvers along a single trajectory. The selection function operates on a fixed-dimensional, problem-size-independent state representation, allowing one shared model to generalize across problem scales and structures without scale-specific retraining, in contrast to existing learning-to-optimize (L2O) methods that are usually scale-specific. We also propose an asynchronous selection mechanism to reduce the additional overhead of solver selection. Theoretically, we derive a critical condition number that governs local solver preference and show that heterogeneous condition-number regimes induce a positive solver-selection margin. We also characterize how selection errors under distribution shift translate into runtime regret and give a sufficient condition for improvement over static single-solver baselines. Experiments on nonconvex ACOPF and convex QP benchmarks show that our selector generalizes zero-shot across problem scales and unseen structures, while improving convergence reach and reducing wall-clock runtime by 15% on QP and 6% on ACOPF over existing L2O baselines.