Monotone Optimisation with Learned Projections

Abstract: Monotone optimisation problems admit specialised global solvers such as the Polyblock Outer Approximation (POA) algorithm, but these methods typically require explicit objective and constraint functions. In many applications, these functions are only available through data, making POA difficult to apply directly. We introduce an algorithm-aware learning approach that integrates learned models into POA by directly predicting its projection primitive via the radial inverse, avoiding the costly bisection procedure used in standard POA. We propose Homogeneous-Monotone Radial Inverse (HM-RI) networks, structured neural architectures that enforce key monotonicity and homogeneity properties, enabling fast projection estimation. We provide a theoretical characterisation of radial inverse functions and show that, under mild structural conditions, a HM-RI predictor corresponds to the radial inverse of a valid set of monotone constraints. To reduce training overhead, we further develop relaxed monotonicity conditions that remain compatible with POA. Across multiple monotone optimisation benchmarks (indefinite quadratic programming, multiplicative programming, and transmit power optimisation), our approach yields substantial speed-ups in comparison to direct function estimation while maintaining strong solution quality, outperforming baselines that do not exploit monotonic structure.