Polynomial-Time Approximation for Nonconvex Optimization Problems with an L1-Constraint

TL;DR

This paper demonstrates that nonconvex optimization problems with an L1-constraint can be approximately solved in polynomial time, providing new algorithms and a PTAS for continuous and mixed-integer cases.

Contribution

It introduces a polynomial-time approximation scheme for nonconvex L1-constrained problems, including nonlinear integer and polynomial integer programs.

Findings

Nonconvex L1-constrained problems are solvable in polynomial time.

Polynomial integer programs have polynomial complexity in dimension and constraints.

A PTAS is provided for Lipschitz continuous functions with L1-constraints.

Abstract

Nonconvex optimization problems with an L1-constraint are ubiquitous, and are found in many application domains including: optimal control of hybrid systems, machine learning and statistics, and operations research. This paper shows that nonconvex optimization problems with an L1-constraint can be approximately solved in polynomial time. We first show that nonlinear integer programs with an L1-constraint can be solved in a number of oracle steps that is polynomial in the dimension of the decision variable, for each fixed radius of the L1-constraint. When specialized to polynomial integer programs, our result shows that such problems have a time complexity that is polynomial in simultaneously both the dimension of the decision variables and number of constraints, for each fixed radius of the L1-constraint. We prove this result using a geometric argument that leverages ideas from stochastic process theory and from the theory of convex bodies in high-dimensional spaces. We conclude by providing an additive polynomial time approximation scheme (PTAS) for continuous optimization of Lipschitz functions subject to Lipschitz constraints intersected with an L1-constraint, and we sketch a generalization to mixed-integer optimization.