Computing the Pareto Front by Polynomial Elimination, With an Application From System Identification

TL;DR

This paper introduces a new algebraic elimination method to compute the Pareto front in polynomial multi-objective optimization, providing explicit relations instead of point-wise approximations.

Contribution

The authors develop an elimination-based approach that derives polynomial equations describing the Pareto front for multivariate polynomial problems, including a system identification application.

Findings

The method produces explicit algebraic relations for the Pareto front.

It outperforms sampling-based methods by avoiding point-wise approximation.

Applied successfully to a system identification problem involving trade-offs.

Abstract

We propose a novel numerical approach to compute the Pareto front in multivariate polynomial multi-objective optimization problems. When the objective functions and (equality) constraints are multivariate polynomials, the Pareto front, which describes the efficient points of the multiple (often conflicting) objective functions, can be interpreted as a subset of a positive-dimensional algebraic variety. By combining the objective functions with weights and considering the weights as additional decision variables, we can eliminate all variables except the objective values and obtain one (or multiple) polynomial equation(s) that describes the Pareto front. Unlike sampling-based methods that approximate the Pareto front point-wise, our elimination-based approach yields an explicit algebraic relation between the objective values, representing the Pareto front as a geometric object in the objective space without requiring a predetermined number of sample points. Besides numerical examples illustrating the elimination-based approach, we use elimination on a challenging application that originates from system identification, in which we analyze the trade-off between misfit and latency terms when determining the optimal model parameters from measured data.