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

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

arxiv: 2604.02806 · v1 · submitted 2026-04-03 · 🧮 math.OC

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

Hans van Rooij , Christof Vermeersch , Marie Deferme , Bart De Moor This is my paper

Pith reviewed 2026-05-13 19:08 UTC · model grok-4.3

classification 🧮 math.OC

keywords Pareto frontpolynomial eliminationmulti-objective optimizationalgebraic varietysystem identificationweighted sum

0 comments

The pith

Treating weights as extra variables in a weighted-sum formulation of polynomial objectives allows elimination to produce polynomial equations that exactly describe the Pareto front.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that when objectives and equality constraints are multivariate polynomials, the Pareto front can be recovered exactly by forming the weighted-sum scalarization, adjoining the weights as decision variables, and computing the elimination ideal in the objective values alone. A reader would care because the result is an explicit algebraic variety in objective space rather than a discrete set of sampled points. The method therefore supplies a geometric object that can be analyzed, factored, or sampled arbitrarily after the fact. The approach is illustrated on a system-identification example that trades model misfit against latency.

Core claim

When the objective functions and equality constraints are multivariate polynomials, the Pareto front is recovered by introducing the weights of the weighted-sum scalarization as additional variables, constructing the corresponding polynomial ideal, and eliminating all decision variables to obtain one or more polynomial relations that involve only the objective values; the real points of this elimination ideal contain the Pareto front.

What carries the argument

The elimination ideal in the objective values obtained by adjoining weights as variables to the weighted-sum system and removing all original decision variables.

If this is right

The Pareto front is represented as an algebraic variety in objective space without any predetermined number of sample points.
All efficient trade-off relations are encoded in a finite set of polynomial equations that can be studied symbolically or numerically.
The same elimination procedure directly yields the front for the system-identification problem that balances misfit and latency terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The resulting variety could be factored or decomposed further to reveal singularities or branches that correspond to distinct regimes of the trade-off.
The technique may apply to any algebraic optimization setting once the objectives are expressed or approximated by polynomials.
Numerical elimination algorithms would need to be paired with real-root isolation routines to extract only the relevant positive orthant segment of the front.

Load-bearing premise

That the real points of the elimination ideal obtained from the weighted-sum system contain precisely the Pareto front and no extraneous points.

What would settle it

For any low-dimensional polynomial example whose Pareto front is already known by direct inspection or grid search, the real roots of the computed eliminant must coincide exactly with those known efficient points.

Figures

Figures reproduced from arXiv: 2604.02806 by Bart De Moor, Christof Vermeersch, Hans van Rooij, Marie Deferme.

**Figure 1.** Figure 1: Visualization of the Pareto front for the portfolio optimization problem in Example 1. The Pareto front ( ) is obtained by considering the zero set of the eliminant polynomial ( ), which is computed via our novel elimination-based approach in Section 3.4. Only the points that correspond to nonnegative weights are contained in the Pareto front. A linear level curve ( ) allows to recover the weight vector an… view at source ↗

**Figure 2.** Figure 2: Schematic representation of the misfit-latency model class: the measured data (u, y) is described by a latent input term (e) and misfit term (ue, ye), such that the model-compliant data (ub, yb) satisfies a linear time-invariant model equation of the form a(q)yb = b(q)ub + c(q)e. latency introduced in the model, leading to the following objective function 1 2 α ∥ye∥ 2 2 + β ∥ue∥ 2 2 + γ ∥e∥ 2 2 , (13) … view at source ↗

**Figure 3.** Figure 3: Visualization of the Pareto front for the misfit-versus-latency modeling problem in Example 7. The Pareto front ( ) is a subset of the eliminant ( ) obtained through the elimination approach described in Section 3.2, and it is validated by sampling Pareto-efficient points ( ). The two intersection points with the axes correspond to a pure autoregressive model without misfit ( ) and a pure autonomous model … view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Elimination gives a polynomial whose zero set contains the Pareto front as a subset, but the paper must still show how to isolate the actual efficient points from the extra components.

read the letter

The paper's main contribution is to set up the weighted-sum scalarization, treat the weights as extra variables, and eliminate all other variables to obtain a polynomial relation in the objective values alone. For polynomial objectives and equality constraints this produces an explicit algebraic variety in objective space instead of a cloud of sample points. The system-identification example, trading off misfit against latency, is a natural fit and shows the method can be applied to a concrete problem without choosing a fixed number of points in advance. That is useful when you want a geometric object you can interrogate symbolically or plot exactly for low degrees. The construction itself rests on standard elimination theory, so the algebra is not circular. The soft spot is exactly the one the stress-test flags: the resulting ideal vanishes on the image of all critical points, including those reached only with negative weights or at local maxima, so the zero set is typically larger than the Pareto front. The abstract says the polynomial “describes the Pareto front,” yet the text does not appear to supply an explicit selection step or a proof that the front coincides with a particular real component. Without that, or without degree bounds and stability checks for the elimination, a user still has to do extra work to extract the efficient points. Numerical examples are shown, but they are small. This is worth sending to referees for readers in algebraic optimization and control who already work with Groebner bases or resultants; they can judge whether the extra filtering step is easy to add and whether the system-ID case justifies the approach. A serious editor should let it go to review rather than desk-reject.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes an algebraic method to compute the Pareto front for multi-objective optimization problems in which the objective functions and equality constraints are multivariate polynomials. The approach augments the weighted-sum scalarization by treating the weights as additional decision variables, forms the corresponding first-order stationarity conditions, and applies polynomial elimination (via Groebner bases or resultants) to remove all variables except the objective values, yielding one or more polynomial equations whose zero set is asserted to describe the Pareto front exactly. The method is illustrated on low-dimensional numerical examples and applied to a system-identification problem that trades off model misfit against latency.

Significance. If the elimination step isolates precisely the Pareto front (rather than a superset), the technique would supply an explicit algebraic representation of the front as a variety in objective space, replacing point-wise sampling with a closed-form geometric object. This is potentially valuable for exact trade-off analysis in polynomial system identification, where the resulting polynomial could be used for further symbolic or certified-numeric post-processing. The paper correctly invokes standard elimination theory and supplies a concrete application, but the strength of the contribution hinges on whether the derived ideal coincides with the Pareto set.

major comments (2)

[§3] §3 (Elimination procedure) and the abstract: the claim that the eliminated polynomial 'describes the Pareto front' is not justified. The elimination ideal consists of all polynomials vanishing on the image of the critical points over all weights (including negative weights); its real zero set is therefore the Zariski closure and typically contains extraneous components (local maxima, saddle points, or dominated points). The manuscript must either prove that the front coincides with the entire real variety or supply an additional semi-algebraic selection step (e.g., via sign conditions on the weights or dominance checks) that is currently absent.
[§4] §4 (System-identification application): no degree bounds, complexity estimates, or numerical-stability analysis are given for the elimination step. For the misfit-latency polynomials arising from measured data, the degree of the resulting eliminant can grow rapidly; without such bounds or a discussion of Groebner-basis heuristics, it is unclear whether the method remains practical beyond the low-dimensional examples shown.

minor comments (3)

[§2] Notation for the weight vector w and the normalization constraint is introduced inconsistently between the abstract, §2, and §3; a single, clearly labeled definition would improve readability.
Figure captions for the numerical examples should explicitly state which polynomial is plotted and whether the displayed curve is the full eliminant or the selected Pareto portion.
The reference list omits several standard works on real algebraic geometry and multi-objective polynomial optimization (e.g., on the use of critical-point methods for Pareto fronts); adding them would strengthen the positioning.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important clarifications needed regarding the precise relationship between the elimination ideal and the Pareto front, as well as the need for complexity and stability analysis in the application. We address each point below and will incorporate the suggested revisions.

read point-by-point responses

Referee: [§3] §3 (Elimination procedure) and the abstract: the claim that the eliminated polynomial 'describes the Pareto front' is not justified. The elimination ideal consists of all polynomials vanishing on the image of the critical points over all weights (including negative weights); its real zero set is therefore the Zariski closure and typically contains extraneous components (local maxima, saddle points, or dominated points). The manuscript must either prove that the front coincides with the entire real variety or supply an additional semi-algebraic selection step (e.g., via sign conditions on the weights or dominance checks) that is currently absent.

Authors: We agree that the elimination ideal yields the Zariski closure of the image of all critical points (including those for non-positive weights), and that this closure may contain extraneous real components not belonging to the Pareto front. The original manuscript implicitly assumes that the relevant real positive branch corresponds to the front in the examples, but does not provide a general selection procedure. In the revision we will add an explicit post-processing step: after computing the real points of the variety, we retain only those points that are non-dominated and arise from strictly positive weights (verified by back-substitution into the stationarity conditions). We will also include a short theoretical remark stating the precise conditions under which the entire real variety coincides with the Pareto front (e.g., when all objectives are convex). revision: yes
Referee: [§4] §4 (System-identification application): no degree bounds, complexity estimates, or numerical-stability analysis are given for the elimination step. For the misfit-latency polynomials arising from measured data, the degree of the resulting eliminant can grow rapidly; without such bounds or a discussion of Groebner-basis heuristics, it is unclear whether the method remains practical beyond the low-dimensional examples shown.

Authors: We acknowledge that the current version lacks explicit degree bounds and complexity discussion. In the revised manuscript we will insert a new subsection that (i) recalls the standard multi-homogeneous Bézout bound on the degree of the eliminant in terms of the degrees of the objective and constraint polynomials and the number of variables, (ii) applies this bound to the concrete misfit-latency polynomials of the system-identification example, and (iii) comments on the practical performance of modern Gröbner-basis implementations (e.g., F4/F5 algorithms and monomial-order heuristics) together with floating-point stability considerations when the coefficients come from measured data. While we cannot give a universal complexity guarantee that would cover arbitrary data sets, the added analysis will make the practical scope of the method clearer. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies standard polynomial elimination to weighted-sum stationarity conditions

full rationale

The paper forms the stationarity conditions of the weighted-sum scalarization (with weights as extra variables), then eliminates all variables except the objective values y to obtain polynomial generators in the y-space. This is a direct, one-way application of Gröbner-basis or resultant-based elimination in polynomial rings; the resulting ideal is computed from the input polynomials without any parameter fitting, self-referential definition, or load-bearing self-citation. No step renames a fitted quantity as a prediction, imports uniqueness from prior author work, or smuggles an ansatz via citation. The derivation chain is therefore self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method relies on classical results of elimination theory in multivariate polynomial rings; no free parameters are introduced and no new entities are postulated.

axioms (1)

standard math Multivariate polynomial rings admit effective elimination of variables via Groebner bases or resultants, yielding an ideal in the remaining variables.
Invoked when the weighted system is reduced to an equation in objective values alone.

pith-pipeline@v0.9.0 · 5481 in / 1178 out tokens · 80764 ms · 2026-05-13T19:08:10.800542+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The Pareto front can then be described by multivariate polynomial equations and is, therefore, contained in a (positive-dimensional) algebraic variety.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

A geometrical approach to finding multivairate approximate LCMs and GCDs.Linear Algebra and its Applications, 438(9):3618–3628, 2013

Kim Batselier, Philippe Dreesen, and Bart De Moor. A geometrical approach to finding multivairate approximate LCMs and GCDs.Linear Algebra and its Applications, 438(9):3618–3628, 2013

work page 2013
[2]

Hermann Grassmann and the creation of linear algebra

Desmond Fearnley-Sander. Hermann Grassmann and the creation of linear algebra. The American Mathematical Monthly, 86(10):809–817, 1979

work page 1979
[3]

Geoffrion

Arthur M. Geoffrion. Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22:618–63, 1968

work page 1968
[4]

Exact solution to the least squares realization problem as multiparameter eigenvalue problem.Automatica, 185:112792, 2026

Sibren Lagauw, Lukas Vanpoucke, and Bart De Moor. Exact solution to the least squares realization problem as multiparameter eigenvalue problem.Automatica, 185:112792, 2026

work page 2026
[5]

Misfit versus latency.Automatica, 37(12):2057–2067, 2001

Philippe Lemmerling and Bart De Moor. Misfit versus latency.Automatica, 37(12):2057–2067, 2001

work page 2057
[6]

Prentice Hall Informa- tion and System Science Series

Lennart Ljung.System Identification: Theory for the User. Prentice Hall Informa- tion and System Science Series. Prentice Hall PTR, Upper Saddle River, NJ, USA, 2nd edition, 1999

work page 1999
[7]

Portfolio selection.The Journal of Finance, 7(1):77–91, 1952

Harry Markowitz. Portfolio selection.The Journal of Finance, 7(1):77–91, 1952

work page 1952
[8]

Marler and Jasbir S

Tim R. Marler and Jasbir S. Arora. Survey of multi-objective optimization methods for engineering.Structural and Multidisciplinary Optimization, 26:369–395, 2004

work page 2004
[9]

Kluwer Academic Pub- lishers, Dordrecht, The Netherlands, 1st edition, 1999

Kaisa Miettinen.Nonlinear Multiobjective Optimization. Kluwer Academic Pub- lishers, Dordrecht, The Netherlands, 1st edition, 1999

work page 1999
[10]

Wright.Numerical Optimization

Jorge Nocedal and Stephen J. Wright.Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY, USA, 2nd edition, 2006

work page 2006
[11]

Vilfredo Pareto.Cours D’ ´Economie Politique. F. Rouge, Lausanne, Switzerland,

work page
[12]

Springer, New York, NY, USA, 1988

Wolfram Stadler, editor.Multicriteria Optimization in Engineering and in the Sci- ences, volume 1 ofMathematical Concepts and Methods in Science and Engineering. Springer, New York, NY, USA, 1988. 15

work page 1988
[13]

Numerical elimination theory via the Macaulay matrix

Hans van Rooij and Bart De Moor. Numerical elimination theory via the Macaulay matrix. Technical report, KU Leuven, Leuven, Belgium, 2026.https://ftp.esat. kuleuven.be/stadius/hvanrooi/numerical_elimination_theory.pdf

work page 2026
[14]

PhD thesis, KU Leuven, Leuven, Belgium, 2023

Christof Vermeersch.The (Block) Macaulay Matrix. PhD thesis, KU Leuven, Leuven, Belgium, 2023

work page 2023
[15]

Globally optimal least-squares ARMA model identification is an eigenvalue problem.IEEE Control Systems Letters, 3(4):1062–1067, 2019

Christof Vermeersch and Bart De Moor. Globally optimal least-squares ARMA model identification is an eigenvalue problem.IEEE Control Systems Letters, 3(4):1062–1067, 2019

work page 2019
[16]

Solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems with MacaulayLab

Christof Vermeersch and Bart De Moor. Solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems with MacaulayLab. Technical report, KU Leuven, Leuven, Belgium, 2025

work page 2025
[17]

Multivariate polyno- mial optimization in complex variables is a (rectangular) multiparameter eigenvalue problem

Christof Vermeersch, Sibren Lagauw, and Bart De Moor. Multivariate polyno- mial optimization in complex variables is a (rectangular) multiparameter eigenvalue problem. InProc. of the 62nd IEEE Conference on Decision and Control (CDC), pages 7305–7311, Marina Bay Sands, Singapore, 2023

work page 2023
[18]

Jan C. Willems. From time series to linear systems — Part I. finite dimensional linear time invariant systems.Automatica, 22(5):561–580, 1986

work page 1986
[19]

A numerical elimination method for polynomial computations

Zhonggang Zeng. A numerical elimination method for polynomial computations. Theoretical Computer Science, 409(2):318–331, 2008. 16

work page 2008

[1] [1]

A geometrical approach to finding multivairate approximate LCMs and GCDs.Linear Algebra and its Applications, 438(9):3618–3628, 2013

Kim Batselier, Philippe Dreesen, and Bart De Moor. A geometrical approach to finding multivairate approximate LCMs and GCDs.Linear Algebra and its Applications, 438(9):3618–3628, 2013

work page 2013

[2] [2]

Hermann Grassmann and the creation of linear algebra

Desmond Fearnley-Sander. Hermann Grassmann and the creation of linear algebra. The American Mathematical Monthly, 86(10):809–817, 1979

work page 1979

[3] [3]

Geoffrion

Arthur M. Geoffrion. Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22:618–63, 1968

work page 1968

[4] [4]

Exact solution to the least squares realization problem as multiparameter eigenvalue problem.Automatica, 185:112792, 2026

Sibren Lagauw, Lukas Vanpoucke, and Bart De Moor. Exact solution to the least squares realization problem as multiparameter eigenvalue problem.Automatica, 185:112792, 2026

work page 2026

[5] [5]

Misfit versus latency.Automatica, 37(12):2057–2067, 2001

Philippe Lemmerling and Bart De Moor. Misfit versus latency.Automatica, 37(12):2057–2067, 2001

work page 2057

[6] [6]

Prentice Hall Informa- tion and System Science Series

Lennart Ljung.System Identification: Theory for the User. Prentice Hall Informa- tion and System Science Series. Prentice Hall PTR, Upper Saddle River, NJ, USA, 2nd edition, 1999

work page 1999

[7] [7]

Portfolio selection.The Journal of Finance, 7(1):77–91, 1952

Harry Markowitz. Portfolio selection.The Journal of Finance, 7(1):77–91, 1952

work page 1952

[8] [8]

Marler and Jasbir S

Tim R. Marler and Jasbir S. Arora. Survey of multi-objective optimization methods for engineering.Structural and Multidisciplinary Optimization, 26:369–395, 2004

work page 2004

[9] [9]

Kluwer Academic Pub- lishers, Dordrecht, The Netherlands, 1st edition, 1999

Kaisa Miettinen.Nonlinear Multiobjective Optimization. Kluwer Academic Pub- lishers, Dordrecht, The Netherlands, 1st edition, 1999

work page 1999

[10] [10]

Wright.Numerical Optimization

Jorge Nocedal and Stephen J. Wright.Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY, USA, 2nd edition, 2006

work page 2006

[11] [11]

Vilfredo Pareto.Cours D’ ´Economie Politique. F. Rouge, Lausanne, Switzerland,

work page

[12] [12]

Springer, New York, NY, USA, 1988

Wolfram Stadler, editor.Multicriteria Optimization in Engineering and in the Sci- ences, volume 1 ofMathematical Concepts and Methods in Science and Engineering. Springer, New York, NY, USA, 1988. 15

work page 1988

[13] [13]

Numerical elimination theory via the Macaulay matrix

Hans van Rooij and Bart De Moor. Numerical elimination theory via the Macaulay matrix. Technical report, KU Leuven, Leuven, Belgium, 2026.https://ftp.esat. kuleuven.be/stadius/hvanrooi/numerical_elimination_theory.pdf

work page 2026

[14] [14]

PhD thesis, KU Leuven, Leuven, Belgium, 2023

Christof Vermeersch.The (Block) Macaulay Matrix. PhD thesis, KU Leuven, Leuven, Belgium, 2023

work page 2023

[15] [15]

Globally optimal least-squares ARMA model identification is an eigenvalue problem.IEEE Control Systems Letters, 3(4):1062–1067, 2019

Christof Vermeersch and Bart De Moor. Globally optimal least-squares ARMA model identification is an eigenvalue problem.IEEE Control Systems Letters, 3(4):1062–1067, 2019

work page 2019

[16] [16]

Solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems with MacaulayLab

Christof Vermeersch and Bart De Moor. Solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems with MacaulayLab. Technical report, KU Leuven, Leuven, Belgium, 2025

work page 2025

[17] [17]

Multivariate polyno- mial optimization in complex variables is a (rectangular) multiparameter eigenvalue problem

Christof Vermeersch, Sibren Lagauw, and Bart De Moor. Multivariate polyno- mial optimization in complex variables is a (rectangular) multiparameter eigenvalue problem. InProc. of the 62nd IEEE Conference on Decision and Control (CDC), pages 7305–7311, Marina Bay Sands, Singapore, 2023

work page 2023

[18] [18]

Jan C. Willems. From time series to linear systems — Part I. finite dimensional linear time invariant systems.Automatica, 22(5):561–580, 1986

work page 1986

[19] [19]

A numerical elimination method for polynomial computations

Zhonggang Zeng. A numerical elimination method for polynomial computations. Theoretical Computer Science, 409(2):318–331, 2008. 16

work page 2008