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

Hans van Rooij, Christof Vermeersch, Marie Deferme, Bart De Moor

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.

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

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.

Paper Structure

This paper contains 13 sections, 1 theorem, 37 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Pareto Fronts in Minimization
  3. Computational Elimination Approach
  4. Reformulation of the Pareto front as an eliminant
  5. Numerical elimination algorithm
  6. Geometric interpretation of the Pareto front
  7. Step-by-step solution of the motivational example
  8. Various Numerical Examples
  9. System Identification Problem
  10. Misfit-versus-latency modeling
  11. Relation to the Pareto front
  12. Applied numerical example
  13. Conclusions and Future Work

Key Result

Theorem 1

Let $w_i > 0$ be fixed for $i = 1, 2, \ldots m$ with $\sum_{i = 1}^m w_i = 1$. If $\bm{x}^\star$ is a minimum of the (convex) weighted minimization problem eq:weightedminimization, then $\bm{x}^\star$ is properly Pareto-efficient in eq:multiobjectiveminimization. $\blacktriangleleft$$\blacktrianglel

Figures (3)

  • Figure 1: Visualization of the Pareto front for the portfolio optimization problem in \ref{['ex:motivationalexample']}. The Pareto front (\ref{['plot:toyproblem:pareto']}) is obtained by considering the zero set of the eliminant polynomial (\ref{['plot:toyproblem:eliminant']}), which is computed via our novel elimination-based approach in \ref{['sec:motivationalexample']}. Only the points that correspond to nonnegative weights are contained in the Pareto front. A linear level curve (\ref{['plot:toyproblem:tangent']}) allows to recover the weight vector and decision variables that correspond to a specific point on the Pareto front ( \ref{['plot:toyproblem:point']} ).
  • Figure 2: Schematic representation of the misfit-latency model class: the measured data ($\bm{u}, \bm{y}$) is described by a latent input term ($\bm{e}$) and misfit term ($\widetilde{\bm{u}}, \widetilde{\bm{y}}$), such that the model-compliant data ($\widehat{\bm{u}}, \widehat{\bm{y}}$) satisfies a linear time-invariant model equation of the form $a(q) \widehat{\bm{y}} = b(q) \widehat{\bm{u}} + c(q) \bm{e}$.
  • Figure 3: Visualization of the Pareto front for the misfit-versus-latency modeling problem in \ref{['ex:mis_vs_lat']}. The Pareto front (\ref{['plot:misfitlatency:pareto']}) is a subset of the eliminant (\ref{['plot:misfitlatency:eliminant']}) obtained through the elimination approach described in \ref{['sec:numericalalgorithm']}, and it is validated by sampling Pareto-efficient points ( \ref{['plot:misfitlatency:sampled']} ). The two intersection points with the axes correspond to a pure autoregressive model without misfit ( \ref{['plot:misfitlatency:green']} ) and a pure autonomous model with output misfit ( \ref{['plot:misfitlatency:red']} ), which can also be obtained using dedicated identification algorithms.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Example 1: Motivational example
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • ...and 1 more