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.

• The paper introduces an elimination-based approach to compute the Pareto front of polynomial multi-objective optimization problems.
• It leverages Macaulay matrices and variable elimination to derive an algebraic description of the trade-off set.
• Application to system identification demonstrates its capacity to capture all optimal misfit-latency pairs with low residual errors.

Algebraic Characterization of the Pareto Front in Polynomial Multi-Objective Optimization

Introduction

The paper "Computing the Pareto Front by Polynomial Elimination, With an Application From System Identification" (2604.02806) introduces a numerical framework for the exact computation of Pareto fronts in multi-objective optimization problems characterized by multivariate polynomial objectives and polynomial equality constraints. Traditional approaches to Pareto front computation, predominantly based on scalarization and sampling, are limited in their ability to capture the entirety and geometric structure of the Pareto front, especially in the presence of non-convex or intricate front geometries. The proposed elimination-based methodology leverages algebraic properties of polynomial systems and numerical elimination algorithms to yield an implicit but comprehensive algebraic description of the Pareto front in the objective space.

Problem Formalization and Limitations of Conventional Methods

The paper considers multi-objective minimization problems of the form: $$\min_{x \in \mathbb{R}^n} \quad \left( f_1(x), f_2(x), ..., f_m(x) \right)^\top \quad \text{subject to} \quad g_k(x) = 0, \ k=1,\ldots,n_g$$ with all $f_i$ and $g_k$ polynomials in $x$. Due to inherent conflicts among $f_i$, the set of minima is replaced by the set of Pareto-efficient solutions, characterized by non-dominance properties.

Standard scalarization techniques, such as the convex weighted sum, sample the Pareto front pointwise by varying weights $w_i$ in a convex combination. These methods suffer from two principal drawbacks: (1) uniform weight sampling rarely corresponds to uniform front coverage, and (2) non-convex regions of the front remain undiscovered. Additionally, sampling-based strategies are inherently computationally redundant if one ultimately seeks an algebraic rather than sampled representation of the trade-off set.

Elimination-Based Approach

Reformulation via Polynomial Elimination

The core innovation is the direct computation of the Pareto front as a positive-dimensional algebraic variety by variable elimination from the KKT system. Introducing auxiliary variables $s_i = f_i(x)$ for $i=1,\ldots,m$, the optimality system becomes a polynomial system in $(x, w, \lambda, s)$, where $w$ are the (now variable) weights and $f_i$0 the Lagrange multipliers for the constraints. Elimination of all variables except $f_i$1 produces a system of polynomial equations in the Pareto front coordinates, thus capturing the entire set in an implicit form.

Numerical Elimination Algorithm

Leveraging Macaulay matrices, the procedure constructs polynomial multiples of the original equations up to degree $f_i$2; the Macaulay matrix rows encode the coefficients of all such polynomials. Elimination reduces to identifying the intersection of two subspaces: the row span of the Macaulay matrix (encoding the ideal generated by the system) and the subspace admitting nonzero coefficients only for monomials in $f_i$3. This is achieved by computing the left nullspace of the submatrix populated by monomials containing eliminated variables. After appropriate degree lifting to ensure nontrivial intersection, the resulting linear algebra yields the desired eliminant—the system of polynomials vanishing on the projected solutions.

Geometric and Computational Interpretation

The zeros of the resultant eliminant system in the $f_i$4-coordinates contain the image of the Pareto front under the objectives. However, not all points in the variety correspond to properly Pareto-efficient solutions, as positivity constraints on $f_i$5 must hold. The tangent and normal spaces—computed via the Jacobian of the eliminant system—encode the necessary hyperplanes for the supporting direction at each front point, linking the algebraic and geometric fronts directly. The approach is computationally intensive due to rapid growth in the Macaulay matrix size with problem dimension and polynomial degree.

Numerical Validation and Application to System Identification

Three canonical polynomial test cases validate the algorithm: unconstrained tri-objective, constrained bi-objective with high-degree polynomials, and a low-dimensional but multi-objective scenario producing an ideal defined by multiple generators. Across all, the method returns low residuals when verified against sampled or analytically derived front points.

The methodology is then applied to system identification within the misfit-versus-latency model structure. Here, the weights modulate trade-offs between misfit (difference between measured and model output) and latency (unmodeled or latent inputs). The elimination framework yields an explicit algebraic relation describing all optimal misfit-latency pairs, exposing the full continuum of admissible models, including classical special cases (pure misfit, pure latency). Numerical experiments with low data cardinality illustrate the ability to recover and certify all theoretically admissible trade-offs, as well as associated model parameters for each front point.

Implications and Future Directions

By providing an algebraic (as opposed to sampled) representation of the Pareto front, the presented elimination technique gives a detailed, global view of the objective trade-off space. Practically, this enables comprehensive model selection and hyperparameter exploration, particularly in applications where Pareto-optimality, rather than optimality with respect to a fixed scalarization, is desired. Theoretically, the approach highlights the link between multi-objective optimality and real algebraic geometry, suggesting that further results may be imported from polynomial systems theory.

Future directions outlined by the authors include (1) development of scalable hierarchical or approximate elimination procedures to manage the combinatorial explosion in Macaulay matrix size for high-dimensional or high-degree systems, and (2) extension of the elimination concept beyond polynomial objectives, possibly through polynomial approximations or hybrid symbolic-numeric techniques.

Conclusion

This work establishes a numerically tractable pathway to compute, in an exact algebraic sense, the Pareto front of polynomial multi-objective optimization problems via systematic elimination. The approach circumvents the limitations of weight-sampling and scalarization, offering a geometric and algebraic description of the efficient set. The application to system identification shows the potential for broad adoption in engineering domains where multi-objective trade-offs are ubiquitous. The main current bottleneck is computational; ongoing research will determine the practical applicability to larger and more realistic problem instances (2604.02806).