Monotone Optimisation with Learned Projections

Optimisation algorithms typically assume oracle access to objectives and constraints, yet in many applications these functions are available only through data. To recover them, a common approach learns a surrogate for each function and optimises over it, but this can be unnecessarily expensive and may fail to preserve the structure required by the optimisation algorithm. We study this problem for monotone optimisation, a class of non-convex problems which can be solved globally using the Polyblock Outer Approximation (POA) algorithm. Our key observation is that POA does not require the constraint functions themselves, but only a scalar quantity we call the radial inverse. Learning this quantity directly replaces POA's inner bisection loop over a learned constraint with a single radial inverse evaluation. We characterise exactly which maps arise as radial inverses of monotone constraints, and relate their approximation error to perturbations of the feasible set and optimal objective value. Building on these results, we introduce Homogeneous-Monotone Radial Inverse (HM-RI) networks, structured min-max models that enforce the monotonicity and positive-homogeneity of radial inverses by construction. Across synthetic monotone optimisation problems and a cellular transmit-power application, HM-RI improves solution quality over direct constraint surrogates and local optimisation baselines by up to 58.6 percentage points relative to the Oracle reference objective, while achieving a $3.0\text{textendash}4.7\times$ speed-up over bisection-based POA with learned constraint surrogates, with the largest solution-quality gains occurring in data-limited and piecewise-constant settings.