# Monotone Optimisation with Learned Projections

arXiv: https://arxiv.org/abs/2601.20983 · HTML: https://mlsanity.com/paper/2601.20983/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Builds on (most cited first, 14 of 14)

- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Certified Monotonic Neural Networks](https://mlsanity.com/paper/2011.10219/) (2020)
- [Constrained Monotonic Neural Networks](https://mlsanity.com/paper/2205.11775/) (2022)
- [Optimization with Constraint Learning: A Framework and Survey](https://mlsanity.com/paper/2110.02121/) (2021)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [General Heuristics for Nonconvex Quadratically Constrained Quadratic Programming](https://mlsanity.com/paper/1703.07870/) (2017)
- [Deep Equilibrium Learning of Explicit Regularizers for Imaging Inverse Problems](https://mlsanity.com/paper/2303.05386/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Optimizing Objective Functions from Trained ReLU Neural Networks via Sampling](https://mlsanity.com/paper/2205.14189/) (2022)
- [Monotone Function Estimation for Computer Experiments](https://mlsanity.com/paper/1309.3802/) (2013)
- [An Overview of Load Balancing in HetNets: Old Myths and Open Problems](https://mlsanity.com/paper/1307.7779/) (2013)
- [Neuro-Optimization: Learning Objective Functions Using Neural Networks](https://mlsanity.com/paper/1905.10079/) (2019)
- [Optimization over Trained Neural Networks: Going Large with Gradient-Based Algorithms](https://mlsanity.com/paper/2512.24295/) (2025)
- [CRRM: A 5G system-level simulator](https://mlsanity.com/paper/2511.02692/) (2025)

## Abstract (arXiv)

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.