Constrained optimization as ecological dynamics with applications to random quadratic programming in high dimensions

TL;DR

This paper reveals a surprising duality between constrained optimization problems like quadratic programming and ecological dynamics, providing a new perspective on high-dimensional random QP through ecological models.

Contribution

It introduces a novel duality linking constrained optimization with ecological consumer-resource models and applies cavity solutions to analyze high-dimensional random QP.

Findings

Strong agreement between theory and numerical simulations

Deep connection established between optimization and ecology

New analytical tools for high-dimensional QP analysis

Abstract

Quadratic programming (QP) is a common and important constrained optimization problem. Here, we derive a surprising duality between constrained optimization with inequality constraints -- of which QP is a special case -- and consumer resource models describing ecological dynamics. Combining this duality with a recent `cavity solution', we analyze high-dimensional, random QP where the optimization function and constraints are drawn randomly. Our theory shows remarkable agreement with numerics and points to a deep connection between optimization, dynamical systems, and ecology.