# A Method for Convex Black-Box Integer Global Optimization

## TL;DR

This paper introduces a new convex underestimator method for integer black-box optimization that does not require gradient information, enabling global minimization on finite integer domains.

## Contribution

It proposes a novel secant-based underestimator and an algorithm that alternates between updating this underestimator and evaluating the objective, with proven convergence to the global minimum.

## Key findings

- The method effectively underestimates the objective in disconnected regions.
- The algorithm converges to the global minimum on the feasible set.
- Comparative results show computational advantages over existing methods.

## Abstract

We study the problem of minimizing a convex function on a nonempty, finite subset of the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to (sub)gradients of the objective but, rather, uses secant linear functions that interpolate the objective function at previously evaluated points. These linear mappings are shown to underestimate the objective in disconnected portions of the domain. Therefore, the union of these conditional cuts provides a nonconvex underestimator of the objective. We propose an algorithm that alternates between updating the underestimator and evaluating the objective function. We prove that the algorithm converges to a global minimum of the objective function on the feasible set. We present two approaches for representing the underestimator and compare their computational effectiveness. We also compare implementations of our algorithm with existing methods for minimizing functions on a subset of the integer lattice. We discuss the difficulty of this problem class and provide insights into why a computational proof of optimality is challenging even for moderate problem sizes.