Monotone Optimisation with Learned Projections

2026-01-28

AI Summary

This work addresses the challenge of solving monotonic optimization problems when the objective or constraint functions are accessible only as black-box oracles, rendering traditional solvers—such as Polyblock outer approximation—that rely on explicit functional forms inapplicable. To overcome this limitation, the authors propose the HM-RI network, which directly learns the radial inverse function to replace computationally expensive bisection searches. The architecture is structurally designed to inherently enforce monotonicity and homogeneity. Building upon a theoretical characterization of the radial inverse, they further introduce a relaxed monotonicity condition compatible with the Polyblock algorithm, substantially reducing training overhead. Experimental results demonstrate that the proposed method significantly accelerates convergence across multiple monotonic optimization benchmarks while maintaining solution quality, outperforming baseline approaches that disregard monotonic structure.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Monotone Optimisation
• Polyblock Outer Approximation
• Radial Inverse
• Projection Primitive
• Data-driven Constraints

Innovation

• Monotone Optimization
• Radial Inverse
• Algorithm-Aware Learning
• Homogeneous-Monotone Networks
• Polyblock Outer Approximation