Geometrical Pareto Selection (GPS)

• Geometrical Pareto Selection (GPS) is a method that analyzes the geometric properties of Pareto fronts to resolve conflicting objectives in multi-objective optimization.
• It employs differential geometry and meta-modeling to construct efficient surrogates, guiding topology optimization and material screening with quantifiable efficiency ratios.
• GPS enables rapid design evaluation and identifies critical transition points, with applications spanning evolutionary dynamics, competitive market equilibria, and structural optimization.

Geometrical Pareto Selection (GPS) refers to a class of methodologies in multi-objective optimization and complex system design that leverage geometric properties of Pareto fronts in objective space to enable efficient analysis, modeling, and selection. The approach formalizes the structure of solution sets when multiple objectives conflict, analyzing their differential geometry to derive selection criteria, scaling laws, and links to critical phenomena. GPS has applications in topology optimization, material selection, combinatorial network design, evolutionary dynamics, and broader areas involving competing objectives (Duriez et al., 2022, Seoane et al., 2015).

1. Geometric Foundations of Pareto Fronts

The Pareto front is the image in objective space of the set of non-dominated solutions—those for which no other candidate improves all objectives simultaneously. Formally, for a design space $\Gamma$ and vector of target functions $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$, the Pareto set is

$$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$

Its image $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$ is a $(K-1)$-dimensional manifold: the Pareto front.

Key geometric descriptors include the tangent (direction of optimal trade-off), the normal (direction of steepest increase in all objectives), and the signed curvature $\kappa(s)$ (for $K=2$) which governs convexity, non-convexity, and the presence of critical points or phase-transition-like behavior (Seoane et al., 2015). For higher $K$, the second fundamental form and curvature tensors generalize these ideas.

2. GPS in Topology Optimization and Material Selection

Within structural topology optimization—minimizing compliance subject to a constraint on material usage—the compliance-volume fraction Pareto front, $C_{\mathrm{opt}}(V_f)$, possesses robust monotonic and convexity properties:

• $C_{\mathrm{opt}}(V_f)$ is strictly decreasing and smooth except for possible concave kinks.
• The reciprocal, optimal stiffness $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$0, is strictly increasing and (piecewise) concave.
• The derivative $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$1 is decreasing in $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$2.

A key metric is the efficiency-ratio

$T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$3

which quantifies the marginal gain per unit resource. Theorem 3.1 establishes $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$4, with $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$5 monotonic: $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$6 as $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$7; $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$8 as $T_f = \{t_1,...,t_K\}: \Gamma \to \mathbb{R}^K$9 (Duriez et al., 2022).

3. Universal Meta-Model Construction

Leveraging the bounded and monotonic efficiency ratio, GPS defines a two-parameter meta-model to fit the Pareto front from minimal data:

$$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$0

with $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$1, $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$2, $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$3, and $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$4. The parameters $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$5 are fit from just two values: $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$6 (full-solid compliance, analytic) and $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$7 (single topology optimization run). This meta-model uniformly approximates the full Pareto front with a maximum relative error bounded by 6.4% across diverse 2D structural examples (Duriez et al., 2022).

The practical GPS workflow for topology optimization is:

1. Choose $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$8; run a single SIMP topology optimization to obtain $$\Pi_\Gamma = \{ x \in \Gamma \mid \nexists\, y \in \Gamma,\; t_k(y) \leq t_k(x)\ \forall k,\;\exists\,k_0:t_{k_0}(y)<t_{k_0}(x) \}.$$9.
2. Compute $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$0 analytically.
3. Solve for $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$1; set $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$2.
4. Use $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$3 as a rapid Pareto front surrogate for any $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$4.

4. Geometric Criteria for Multi-Objective Selection and Criticality

GPS analyzes not only modeling and interpolation but also the implications of Pareto front geometry for selection, transition phenomena, and ensemble behavior:

• In scalarized optimization, $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$5 picks points on $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$6 where the supporting hyperplane is tangent to the front. The ratio of slopes constraint:

$F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$7

• If $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$8 is linear (zero curvature), all convex combinations select the same segment—yielding a critical point with divergent response (analogous to thermodynamic criticality). Non-convexities ("cavities") correspond to first-order phase transitions: selection "jumps" between disconnected regions.

Pareto selective forces, whether evolutionary, ecological, or artificial (e.g., market equilibria or replication dynamics), can drive real systems to the critical points identified by GPS geometry (Seoane et al., 2015).

5. Applications: Material-Plus-Design Screening

The GPS meta-model, combined with geometric insights, enables nearly instantaneous Ashby-type material selection paired with design optimization:

• Express total mass $F = T_f(\Pi_\Gamma) \subset \mathbb{R}^K$9 under compliance constraint $(K-1)$0.
• Eliminate $(K-1)$1 using the inverted Pareto front, leading to a material index:

$(K-1)$2

• In practice, replace $(K-1)$3 with the closed-form inverse of $(K-1)$4.
• Screen candidate materials by: (1) $(K-1)$5 enforces only materials with $(K-1)$6 (minimum $(K-1)$7) can be optimal; (2) exclusion by $(K-1)$8 Pareto front.
• Compute $(K-1)$9 for the survivors. Select the lowest value and corresponding design variable.

In a benchmark MBB-beam case with four candidate materials, GPS correctly predicts shifts in the optimal material as load regime changes, and recovers total mass within 2% of the full Pareto front computation (Duriez et al., 2022).

Material	$\kappa(s)$0 (GPa)	$\kappa(s)$1 (kg/m³)	$\kappa(s)$2 ($\kappa(s)$3)	$\kappa(s)$4 (kg/m³)
Al 7475	70.8	2795	39.5	—
AISI 347	197	7915	40.2	—
Ti–6Al–4V	116	4400	37.9	100.1
Inconel 713	205	7900	38.5	101.4

Screening steps prune the set to Ti and Inconel; GPS yields Ti as optimal for moderate loading, with switch to Inconel under high load.

6. Mechanisms Implementing GPS and Broader Impact

GPS is realized in various algorithmic and natural mechanisms that construct, sample, or evolve along the Pareto front:

• Multi-objective evolutionary algorithms (e.g., NSGA-II) directly sample the non-dominated set, revealing its geometry and critical features.
• Highly Optimized Tolerance-type constrained optimization traverses the front by fixing one target and optimizing the other.
• Co-evolutionary predator-prey and game-theoretic replication (e.g., least-effort models for language) naturally drive populations or strategies toward specific Pareto front regions, often critical segments.
• Pareto geometry can also model competitive market equilibria, where criticality indicates maximum sensitivity to supply-demand trade-offs (Seoane et al., 2015).

A notable consequence is the unification of Self-Organized Criticality (SOC) and Highly Optimized Tolerance (HOT) within the geometric GPS framework.

7. Limitations, Extensions, and Open Directions

• Existence of a globally concave $\kappa(s)$5 and bounded efficiency ratio $\kappa(s)$6 is guaranteed only in linear, SIMP-type topology problems with high-quality global optima.
• The GPS meta-model is empirically validated for 2D cases; 3D problems and other objective pairs may require further assessment.
• No account for manufacturing, minimum member size, or buckling constraints; GPS may select unmanufacturable or unstable structures absent additional filtering.
• Extension to $\kappa(s)$7 objectives would necessitate analysis of higher-dimensional curvature tensors and more complex front geometries.
• Open areas include renormalization-group analysis of front curvature, empirical detection of criticality in evolving real-world systems, stochastic rounding of sharp front features, and applications in synthetic biology, communications, and socio-economic design (Seoane et al., 2015).

Geometrical Pareto Selection thus provides a rigorous geometric and analytic apparatus to unify theory and practice in multi-objective optimization, offering both rapid surrogate modeling for engineering design and a framework to interpret critical phenomena in biological, technological, and social systems.