Efficient Convexification of Kolmogorov-Arnold Networks with Polynomial Functional Forms Via a Continuous Graham Scan Approach

TL;DR

This paper introduces a novel continuous convexification method for Kolmogorov-Arnold Networks with polynomial components, enabling tighter relaxations and improved efficiency in global optimization tasks.

Contribution

It develops a continuous Graham Scan approach to compute exact convex envelopes of univariate polynomials within KANs, enhancing relaxation strength without discretization.

Findings

Relaxations are often orders of magnitude tighter than state-of-the-art solvers.

The method is computationally efficient and robust across various problems.

Exact convex envelopes improve the overall solution quality in global optimization.

Abstract

Deterministic global optimization of nonlinear models is important in many scientific and engineering applications. This framework typically involves repeatedly solving convex relaxations of the nonconvex problem, meaning that the strength of the relaxations and the cost of computing them directly determine overall efficiency and solution quality. In this work, we develop a tailored continuous convexification framework for Kolmogorov-Arnold Networks in which the univariate components are polynomial functions. By exploiting the additive separable structure of this architecture, the relaxation problem reduces to computing tight convex envelopes of univariate polynomials. We propose a continuous variant of the classical Graham Scan that constructs these envelopes exactly by identifying the bitangents of the polynomial convex hull without discretization or factorable reformulations. We establish the correctness of the algorithm and characterize its computational complexity, and show how these envelopes can be combined to construct strong convex relaxations for polynomial KANs. Computational results demonstrate that the proposed relaxations are both strong and robust, often producing bounds that are comparable, or even orders of magnitude tighter than relaxations of state-of-the-art global optimization solvers while remaining computationally efficient.