Partition-Guided Distance Saliency: Bridging Decision and Objective Spaces in Many-Objective Optimization

# Partition-Guided Distance Saliency: Bridging Decision and Objective Spaces in Many-Objective Optimization
Source: [https://arxiv.org/html/2606.30836](https://arxiv.org/html/2606.30836)
11institutetext:A3Data, BH MG Brazil
11email:claudio\.lucio@a3data\.com\.br
[www\.a3data\.com\.br](https://arxiv.org/html/2606.30836v1/www.a3data.com.br)22institutetext:CEFET\-MG \- Centro Federal de Educação Tecnológica de Minas Gerais, BH MG Brazil
22email:flaviocruzeiro@cefetmg\.br33institutetext:Aston University, Birmingham, West Midlands, U\.K\.
33email:e\.wanner1@aston\.ac\.uk###### Abstract

Explainability in Many\-Objective Optimization \(MaO\) is currently hindered by the escalating complexity of the Pareto front, which renders the relationship between high\-dimensional decision variables and objective outcomes increasingly opaque\. As the number of objectives exceeds the limits of traditional visualization, decision\-makers encounter a “cognitive drought” in identifying relevant trade\-offs or specifying target regions withouta prioriknowledge\. To bridge this interpretability gap, we introduce the Partition\-Guided Distance Saliency \(PGDS\) framework, a novel XAI approach designed for continuous optimization landscapes\. Our framework automates the explanation process through a three\-stage pipeline that prioritizes geometric intuition over abstract rules\. First, we employ a surrogate model that learns how geometric distances in the decision space map to proximity in the objective space\. Second, to address the difficulty of manual target selection in high dimensions, the framework automatically partitions the objective landscape into distinct regions and identifies local “Dominating Points” to serve as automated targets for improvement\. Third, we quantify how sensitive a solution’s position is to each decision variable by measuring the distance shifts induced by perturbations to each variable\. This allows PGDS to categorize features as either “Drivers” which facilitate convergence toward preferred regions, or “Blockers” which represent geometric constraints hindering further progress\. Validation on 10\-objective benchmarks and a physics\-informed engineering problem \(Welded Beam\) demonstrates that PGDS provides differentiated, actionable insights that traditional visualization and rule\-based XAI methods fail to provide\.

## 1Introduction

In the realm of Multi\-Objective Optimization \(MOO\), the simultaneous optimization of conflicting criteria yields not a single optimal solution but a set of trade\-off solutions, known as the Pareto\-optimal set\. As the number of objectives increases beyond three, a scenario commonly referred to as Many\-Objective Optimization \(MaO\), the complexity of the problem space grows exponentially\. Evolutionary Algorithms \(EAs\), such as the Non\-dominated Sorting Genetic Algorithm \(NSGA\-III\)\[[4](https://arxiv.org/html/2606.30836#bib.bib4)\]and the Multi\-Objective Evolutionary Algorithm based on Decomposition \(MOEA/D\)\[[20](https://arxiv.org/html/2606.30836#bib.bib20)\], have proven highly effective in approximating these Pareto fronts111Pareto front represents the mapping of the Pareto\-optimal set onto the objective space\. However, finding an approximation of the Pareto front is merely the first step in a broader decision\-making process\. The ultimate goal is to assist a Decision Maker \(DM\) in selecting a single, most preferred solution from this potentially vast set\[[12](https://arxiv.org/html/2606.30836#bib.bib12)\]\.

In low\-dimensional objective spaces \(two or three objectives\), visualization techniques such as scatter plots or parallel coordinates can sufficiently aid the DM in understanding trade\-offs\. However, in many\-objective scenarios, these traditional visualization methods break down, rendering the relationship between thedecision space, the variables controlled by the DM, and theobjective spaceincreasingly opaque\[[19](https://arxiv.org/html/2606.30836#bib.bib19)\]\. A significant gap exists in providing the DM with intuitive, explainable insights intowhya specific solution resides in a particular region of the objective space andhowperturbations in the decision variables influence its position relative to desired targets\.

Current research in Explainable AI \(XAI\) for optimization often relies on rule\-based machine learning models to classify solutions or extract logical predicates\[[14](https://arxiv.org/html/2606.30836#bib.bib14),[13](https://arxiv.org/html/2606.30836#bib.bib13)\]\. While effective for discrete categorization, these methods frequently require discretizing continuous landscapes, thereby losing the geometrical fidelity necessary for precise engineering design or complex resource allocation, for example\. Furthermore, they often place a heavy cognitive burden on the DM to define specific “interesting” regions a priori, a task that is non\-trivial in high\-dimensional spaces\.

To bridge this gap, this paper addresses the following research questions:

RQ1:How can we bridge the interpretability gap between decision variables and objective outcomes in continuous, many\-objective landscapes without losing geometric fidelity?

RQ2:How can we automate the discovery of “regions of interest” in high\-dimensional objective spaces to guide the DM, rather than relying solely on manual target specification?

RQ3:Can a distance\-based surrogate model effectively quantify the sensitivity of objective space positioning to decision variable perturbations?

To answer these questions, we propose a novel framework: Partition\-Guided Distance Saliency \(PGDS\)\. This approach decouples the concept of explainability from rigid rule\-based classifiers\[[13](https://arxiv.org/html/2606.30836#bib.bib13)\], adopting instead a continuous, geometric perspective that respects the true geometry of the many\-objective landscape\. Our method employs the Minimal Learning Machine \(MLM\)\[[18](https://arxiv.org/html/2606.30836#bib.bib18)\]to construct a regression model based entirely on distance matrices\. This offers a computationally efficient mapping, scaling withΘ​\(K2​N\)\\Theta\(K^\{2\}N\), whereNNrepresents the total number of solutions in the archive andKKdenotes the subset of reference points used to anchor the geometric structure\. By integrating this geometric mapping with automated region discovery, we enable the mapping between the geometry of the input \(decision variables\) and output \(objective functions\) spaces\.

Central to our proposal is the adaptation of theDistance Explainermethodology, originally developed to interpret deep learning embeddings\[[11](https://arxiv.org/html/2606.30836#bib.bib11)\]\. We extend this concept to tabular decision variables in optimization, allowing us to generate saliency maps that quantify how specific decision variables influence a solution’s geometric proximity to a target solution in the objective space\. To address the difficulty of selecting targets in many\-objective spaces, we introduce an automated partitioning strategy based on K\-Dimensional Trees \(KD\-Trees\)\[[2](https://arxiv.org/html/2606.30836#bib.bib2)\]\. This strategy partitions the objective space into hyper\-rectangular regions and identifies “dominant points” within each block as automatic targets for explanation\.

The remainder of this paper is structured as follows: Section[2](https://arxiv.org/html/2606.30836#S2)reviews related work on interactive MOO/MaO and the current explainability approaches\. Section[3](https://arxiv.org/html/2606.30836#S3)details our theoretical framework, including the MLM formulation, KDTree partitioning, and the adaptation of the Distance Explainer\. Section[4](https://arxiv.org/html/2606.30836#S4)presents experimental validation on synthetic benchmarks \(DTLZ2, WFG3\) and a constrained engineering problem\. Finally, Section[5](https://arxiv.org/html/2606.30836#S5)concludes the paper and outlines future research directions\.

## 2Related work

This section reviews the existing literature on multi\- and many\-objective optimization, the current state of Explainable AI within the optimization domain, and the emerging class of distance\-based learning methods\. We frame this discussion around the three core research questions elicited in Section[1](https://arxiv.org/html/2606.30836#S1)reproduced here for the reader’s convenience: \(RQ1\) bridging the interpretability gap without losing geometric fidelity, \(RQ2\) automating the discovery of regions of interest, and \(RQ3\) using distance\-based surrogates for sensitivity analysis\.

### 2\.1Complexity and Visualization in Many\-Objective Optimization

Multi\- and many\-objective optimization involves the simultaneous optimization of a vector of typically conflicting objectivesF​\(x\)=\(f1​\(𝒙\),…,fM​\(𝒙\)\)F\(x\)=\(f\_\{1\}\(\\bm\{x\}\),\\dots,f\_\{M\}\(\\bm\{x\}\)\)\. Fundamental to solving these problems is the concept of Pareto dominance: a solution𝒙𝒖\\bm\{x\_\{u\}\}is said to dominate another solution𝒙𝒗\\bm\{x\_\{v\}\}\(denoted as𝒙𝒖≺𝒙𝒗\\bm\{x\_\{u\}\}\\prec\\bm\{x\_\{v\}\}\) if𝒙𝒖\\bm\{x\_\{u\}\}is no worse than𝒙𝒗\\bm\{x\_\{v\}\}in all objectives and strictly better in at least one\[[12](https://arxiv.org/html/2606.30836#bib.bib12)\]\. The ultimate goal of an optimizer is to approximate the Pareto Front \(PF\), the set of all non\-dominated solutions in the objective space representing the optimal trade\-offs\.

For problems with two or three objectives, algorithms such as the Non\-dominated Sorting Genetic Algorithm II \(NSGA\-II\)\[[5](https://arxiv.org/html/2606.30836#bib.bib5)\]have established themselves as the standard, effectively maintaining diversity while converging toward the PF\. However, real\-world problems often scale to four or more objectives, a domain known as many\-objective optimization\. In such high\-dimensional spaces, the selection pressure of standard dominance\-based methods vanishes as nearly all solutions become non\-dominated\. Consequently, specialized algorithms like NSGA\-III\[[4](https://arxiv.org/html/2606.30836#bib.bib4)\]and MOEA/D\[[20](https://arxiv.org/html/2606.30836#bib.bib20)\]were developed to handle these landscapes using reference\-point adaptation and decomposition strategies, respectively\.

However, generating a well\-converged approximation of the PF addresses only the computational aspect; the decision\-making bottleneck remains\. As noted by Tušar and Filipič\[[19](https://arxiv.org/html/2606.30836#bib.bib19)\], traditional visualization techniques lose effectiveness beyond three dimensions\. To cope with this, practitioners often rely on dimension reduction techniques adapted for high\-dimensional Pareto fronts\. These include mapping solutions using Principal Component Analysis \(PCA\) to identify principal conflict directions\[[21](https://arxiv.org/html/2606.30836#bib.bib21)\], applying t\-Distributed Stochastic Neighbor Embedding \(t\-SNE\) to cluster trade\-off regions\[[22](https://arxiv.org/html/2606.30836#bib.bib22)\], or using Self\-Organizing Maps \(SOM\) to preserve the topological structure of the trade\-off surface in a lower\-dimensional grid\[[15](https://arxiv.org/html/2606.30836#bib.bib15)\]\. While useful for clustering, these methods inevitably introduce distortions when projecting high\-dimensional data into 2D\[[7](https://arxiv.org/html/2606.30836#bib.bib7)\]\. Such projections often fail to preserve global distance relationships or local topology, rendering the relationship between the decision variables \(design space\) and the resulting objective values \(performance space\) opaque\. This disconnection highlights the need for explainability methods that operate directly on the high\-dimensional geometry rather than on distorted projections\.

### 2\.2Explainable Optimization and the Target Selection Bottleneck

To bridge the gap between opaque algorithmic results and human understanding, the field of Explainable AI \(XAI\) has recently expanded into Evolutionary Computation\. While standard XAI methods such as SHAP\[[9](https://arxiv.org/html/2606.30836#bib.bib9)\]and LIME\[[17](https://arxiv.org/html/2606.30836#bib.bib17)\]successfully interpret prediction models by quantifying feature contributions, applying these concepts to optimization introduces distinct challenges\. Unlike classification, where the output is a discrete class, optimization involves selecting a decision vector from a continuous, potentially infinite feasible space\. Consequently, recent literature distinguishes betweenexplaining the optimization process, diagnosing algorithmic dynamics and operator contributions\[[1](https://arxiv.org/html/2606.30836#bib.bib1)\], andexplaining the solution, providing rationale for why a specific trade\-off is optimal or preferred\[[10](https://arxiv.org/html/2606.30836#bib.bib10)\]\.

Our work aligns with the latter category, focusing on the post\-hoc analysis of the final Pareto front\. Several data\-driven frameworks have emerged to address this\. In the context of Interactive Multi\-Objective Optimization, the R\-XIMO framework\[[14](https://arxiv.org/html/2606.30836#bib.bib14)\]adapts SHAP values to quantify how a Decision Maker’s aspiration levels \(reference points\) positively or negatively impact the achieved objective values\. Similarly, the XLEMOO framework\[[13](https://arxiv.org/html/2606.30836#bib.bib13)\]employs a learning mode to extract logical rules \(e\.g\., IF\-THEN predicates\) that characterize high\-quality regions of the search space\.

However, significant barriers remain when applying these methods to many\-objective problems \(addressing RQ1\)\. Rule\-based classifiers often require discretizing continuous variables, potentially sacrificing the fine\-grained geometric fidelity required for precision engineering\. Furthermore, interactive frameworks rely heavily on the DM to articulate initial preferences or select reference points\[[12](https://arxiv.org/html/2606.30836#bib.bib12)\]\. In high\-dimensional MaO scenarios \(M\>\>3M\>\>3\), DMs often encounter a “cognitive drought” unable to intuitively visualize the landscape or specify where to look\. This creates aTarget Selection Bottleneck: without a clear target, perturbation\-based explanations cannot be effectively anchored\. To support this \(addressing RQ2\), we propose moving beyond manual specification\. By employing KD\-Tree partitioning\[[2](https://arxiv.org/html/2606.30836#bib.bib2)\], our framework proactively segments the objective landscape, offering local “dominating points” as natural, data\-driven targets for explanation\.

### 2\.3Distance\-Based Learning and Perturbation Analysis

The third avenue of research \(RQ3\) addresses the mechanics of generating geometrically meaningful explanations\. Although standard perturbation methods such as SHAP\[[9](https://arxiv.org/html/2606.30836#bib.bib9)\]and LIME\[[17](https://arxiv.org/html/2606.30836#bib.bib17)\]are powerful for explaining scalar predictions or classification probabilities, they do not inherently model thespatial displacementof a solution within a multi\- or many\-objective landscape\. To capture this geometric sensitivity, we draw upon theDistance Explainermethodology proposed by Meijer and Bos\[[11](https://arxiv.org/html/2606.30836#bib.bib11)\]to interpret deep learning embeddings\.

A critical component of this methodology is the adaptation of the Randomized Input Sampling for Explanation \(RISE\) framework\[[16](https://arxiv.org/html/2606.30836#bib.bib16)\]\. Originally designed for image saliency, RISE estimates feature importance by probing a black\-box model with random binary masks and aggregating the resulting degradation in the model’s output\. In our context, we reframe this to measure the “shift in distance”: we define saliency as the degree to which masking a decision variable causes a solution to drift away from a target in the objective space\.

However, applying RISE\-based perturbations directly to expensive evaluation functions is computationally prohibitive\. This necessitates a surrogate model that is not only fast but also explicitly preserves the geometric structure of the data\. The Minimal Learning Machine \(MLM\)\[[18](https://arxiv.org/html/2606.30836#bib.bib18)\]serves as a bridge\. Unlike standard neural networks that act as opaque mappings, MLM constructs a regression model based entirely on distance matrices\. This unique property creates a direct mathematical alignment with the Distance Explainer, allowing us to rigorously quantify how perturbations in the decision space translate into movement in the objective space, thereby providing the robust foundation required for the Partition\-Guided Distance Saliency framework\.

## 3Methodology: The Partition\-Guided Distance Saliency \(PGDS\) Framework

The proposed framework enables a continuous, geometry\-aware explainability pipeline for many\-objective optimization\. It operates in three distinct stages: \(1\) training a distance\-preserving surrogate model \(MLM\) to map decision space geometry to objective space; \(2\) partitioning the objective landscape via KD\-Trees to identify automated “Regions of Interest” and their corresponding dominating targets; and \(3\) generating saliency maps via tabular perturbation to quantify the influence of decision variables on a solution’s proximity to these targets\.

### 3\.1Geometric Surrogate Modeling via Minimal Learning Machines

To explain the geometric displacement of a solution, we require a surrogate model that predicts distances rather than scalar objective values\. We employ the Minimal Learning Machine \(MLM\)\[[18](https://arxiv.org/html/2606.30836#bib.bib18)\], a supervised learning method that learns a mapping between input and output distance matrices\.

Let𝒳∈ℝn\\mathcal\{X\}\\in\\mathbb\{R\}^\{n\}be the decision space and𝒴∈ℝM\\mathcal\{Y\}\\in\\mathbb\{R\}^\{M\}be the objective space\. Given an archive ofNNsolutions, we randomly select a subset ofKKreference points from this set\. As established in the original MLM formulation\[[18](https://arxiv.org/html/2606.30836#bib.bib18)\], the set of reference points is drawn from the training data itself, and the parameterKKserves as a critical hyperparameter\. The method’s ability to accurately reconstruct the geometric manifold is sensitive to this choice; typically,KKis set as a function ofNN\(e\.g\.,K=⌈N⌉K=\\lceil\\sqrt\{N\}\\rceil\) to balance the regression error with the computational complexity of the subsequent matrix inversion\.

Considering the construction of the distance matrix, we compute the input distance matrixDx∈ℝN×KD\_\{x\}\\in\\mathbb\{R\}^\{N\\times K\}and the output distance matrixΔy∈ℝN×K\\Delta\_\{y\}\\in\\mathbb\{R\}^\{N\\times K\}, in which each entry represents the Euclidean distance between a data point and a reference point:

Dx​\(i,k\)=‖𝐱i−𝐫k‖2,Δy​\(i,k\)=‖𝐲i−𝐭k‖2,D\_\{x\(i,k\)\}=\|\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{r\}\_\{k\}\|\|\_\{2\},\\quad\\Delta\_\{y\(i,k\)\}=\|\|\\mathbf\{y\}\_\{i\}\-\\mathbf\{t\}\_\{k\}\|\|\_\{2\},\(1\)in which𝐫k\\mathbf\{r\}\_\{k\}and𝐭k\\mathbf\{t\}\_\{k\}are thekk\-th reference points in the decision and the objective space, respectively\.

The MLM assumes a linear mapping between these geometries:

Δy=Dx​𝐁\+𝐄\.\\Delta\_\{y\}=D\_\{x\}\\mathbf\{B\}\+\\mathbf\{E\}\.\(2\)The regression matrix𝐁∈ℝK×K\\mathbf\{B\}\\in\\mathbb\{R\}^\{K\\times K\}is estimated via Ordinary Least Squares \(OLS\), providing a closed\-form solution with complexityΘ​\(K2​N\)\\Theta\(K^\{2\}N\):

𝐁^=\(DxT​Dx\)−1​DxT​Δy\.\\hat\{\\mathbf\{B\}\}=\(D\_\{x\}^\{T\}D\_\{x\}\)^\{\-1\}D\_\{x\}^\{T\}\\Delta\_\{y\}\.\(3\)For a new query solution𝐱q​u​e​r​y\\mathbf\{x\}\_\{query\}, we compute its input distance vector𝐝i​n=\[‖𝐱q​u​e​r​y−𝐫1‖,…,‖𝐱q​u​e​r​y−𝐫K‖\]\\mathbf\{d\}\_\{in\}=\[\|\|\\mathbf\{x\}\_\{query\}\-\\mathbf\{r\}\_\{1\}\|\|,\\dots,\|\|\\mathbf\{x\}\_\{query\}\-\\mathbf\{r\}\_\{K\}\|\|\]\. The predicted output distances are given by𝜹^o​u​t=𝐝i​n​𝐁^\\hat\{\\boldsymbol\{\\delta\}\}\_\{out\}=\\mathbf\{d\}\_\{in\}\\hat\{\\mathbf\{B\}\}\. Finally, the coordinates in the objective space𝐲^\\hat\{\\mathbf\{y\}\}are reconstructed from𝜹^o​u​t\\hat\{\\boldsymbol\{\\delta\}\}\_\{out\}\. This process is intuitively similar to the operation of a Global Positioning System \(GPS\)\. In GPS, a receiver determines its location not by directly measuring the coordinates, but by knowing its distance to several satellites \(reference points\) and finding the intersection of the spheres centered at those satellites\.

Similarly, the MLM has predicted how far the target solution should be from each of theKKreference points in the objective space\. We must therefore find the coordinate vector𝐲^∈ℝM\\hat\{\\mathbf\{y\}\}\\in\\mathbb\{R\}^\{M\}that best satisfies theseKKdistance constraints simultaneously\. This is formulated as a non\-linear optimization problem where we seek to minimize the discrepancy between the predicted distances and the actual Euclidean distances to the reference points:

𝐲^=arg⁡min𝐲∈ℝM​∑k=1K\(‖𝐲−𝐭k‖2−δ^o​u​t,k\)2\.\\hat\{\\mathbf\{y\}\}=\\arg\\min\_\{\\mathbf\{y\}\\in\\mathbb\{R\}^\{M\}\}\\sum\_\{k=1\}^\{K\}\\left\(\|\|\\mathbf\{y\}\-\\mathbf\{t\}\_\{k\}\|\|\_\{2\}\-\\hat\{\\delta\}\_\{out,k\}\\right\)^\{2\}\.\(4\)This multilateration problem is typically solved using the Levenberg\-Marquardt algorithm\[[18](https://arxiv.org/html/2606.30836#bib.bib18)\], which iteratively adjusts the coordinates to triangulate the solution’s precise position in the objective landscape\.

The selection of reference pointsℛ\\mathcal\{R\}introduces a stochastic component to the surrogate training\. However, the MLM is globally robust to this selection; providedKKis sufficiently large to capture the archive’s topology, the resulting saliency rankings remain stable across different random seeds\[[18](https://arxiv.org/html/2606.30836#bib.bib18)\]\.

Additionally, the MLM assumes a linear mapping between distance matrices; this does not imply a linear relationship between the decision variables and the objectives\. The choice to map distance matrices,Dx\{D\}\_\{x\}andΔy\{\\Delta\}\_\{y\}, rather than raw coordinate vectors, stems from the need to preserve the structure manifold of the Pareto front\. By modeling the relationship between distances, the MLM sidesteps the complexities of high\-dimensional coordinate transformations, which are often non\-linear and ill\-conditioned in Many\-Objective landscapes\. The regression matrix𝑩\\bm\{B\}acts as a linear geometric bridge\. It assumes that if two points are close in the decision space, their relative distances to a set of global “anchors” \(reference points\) should transform linearly into the objective space\. This allows the model to simultaneously predict the proximity of a query point to the entire archive, rather than predicting a single point\-to\-point coordinate\.

### 3\.2Automated Target Discovery via KD\-Tree Partitioning

In many\-objective scenarios, the Decision Maker \(DM\) often lacks the intuition to manually specify a target pointTT\. To automate this, we employ K\-Dimensional Tree \(KD\-Tree\) partitioning\[[2](https://arxiv.org/html/2606.30836#bib.bib2)\]to segment the objective space into hyper\-rectangular blocks\.

The archive𝒴\\mathcal\{Y\}is recursively split along the dimension with the highest variance\. The process terminates when a maximum depthdm​a​xd\_\{max\}is reached, or a block contains fewer thanNm​i​nN\_\{min\}points\. This creates a set of leaf nodes\{L1,…,LB\}\\\{L\_\{1\},\\dots,L\_\{B\}\\\}\.

For each leaf nodeLjL\_\{j\}, we define a localDominating PointPd​o​m\(j\)P\_\{dom\}^\{\(j\)\}, which serves as the automated target for that region\. Assuming without loss of generality that all objectives are formulated for minimization, this point is constructed from the minimum observed coordinates within the block boundaries:

Pd​o​m\(j\)=\[miny∈Lj⁡\(f1\),miny∈Lj⁡\(f2\),…,miny∈Lj⁡\(fM\)\]\.P\_\{dom\}^\{\(j\)\}=\[\\min\_\{y\\in L\_\{j\}\}\(f\_\{1\}\),\\min\_\{y\\in L\_\{j\}\}\(f\_\{2\}\),\\dots,\\min\_\{y\\in L\_\{j\}\}\(f\_\{M\}\)\]\.\(5\)
This point represents the “local utopia”, the best possible trade\-off available within that specific region of the objective space\. Importantly, these points serve as automated suggestions to facilitate exploration\. The framework presents these targets alongside the explicit partition rules defining their respective regions \(e\.g\.,R​e​g​i​o​n1:f1​<0\.5∧f2\>​0\.8Region\_\{1\}:f\_\{1\}<0\.5\\land f\_\{2\}\>0\.8\)\. This empowers the DM to inspect the logical boundaries of each block and proactively disregard regions or targets that do not align with their specific preferences or domain constraints\.

### 3\.3Saliency maps via Distance Explainers

With a trained surrogate and a defined targetTT\(either manually selected or automatically set toPd​o​m\(j\)P\_\{dom\}^\{\(j\)\}\), we quantify the variable importance using the Distance Explainer principle\[[11](https://arxiv.org/html/2606.30836#bib.bib11)\]\. We adapt the RISE framework\[[16](https://arxiv.org/html/2606.30836#bib.bib16)\]to generate tabular perturbations\.

Letm∈\{0,1\}nm\\in\\\{0,1\\\}^\{n\}be a binary mask vector in whichmi=0m\_\{i\}=0indicates a “masked” feature\. We define a baseline referencex¯\\bar\{x\}, representing the global mean of the archive\. The perturbed samplex~\\tilde\{x\}is generated as:

x~=xq​u​e​r​y⊙m\+x¯⊙\(1−m\)\.\\tilde\{x\}=x\_\{query\}\\odot m\+\\bar\{x\}\\odot\(1\-m\)\.\(6\)This operation effectively “neutralizes” the contribution of masked variables by reverting them to the population mean\.

Considering saliency calculation, we generateLLrandom masks\. For each mask𝐦\(l\)\\mathbf\{m\}^\{\(l\)\}, we compute the perturbed output distance to the targetTT:

ds​h​i​f​t\(l\)=‖MLM​\(x~\(l\)\)−T‖2\.d\_\{shift\}^\{\(l\)\}=\|\|\\text\{MLM\}\(\\tilde\{x\}^\{\(l\)\}\)\-T\|\|\_\{2\}\.\(7\)
The saliency scoreSiS\_\{i\}for the decision variablexix\_\{i\}is calculated as the correlation between the variable’s presence in the mask and its resulting proximity to the target\. A high score indicates that whenxix\_\{i\}is preserved \(unmasked\), the solution stays close to the target; when masked, it drifts away\.

The final saliency mapSSis derived by correlating the presence of each variable with the resulting distance shifts\. We distinguish between two types of influence:

- •Drivers: Variables that, when preserved \(unmasked\), significantlyreducethe distance to the target\. These are the levers the DM must pull to converge toward the region’s dominating point\.
- •Blockers: Variables that, when preserved,increaseor maintain a large distance from the target\. These indicate decision values that are currently hindering the solution from entering the optimal zone of the selected region\.

### 3\.4Algorithm Summary

The complete procedure for the Partition\-Guided Distance Saliency \(PGDS\) framework is detailed in Algorithm[1](https://arxiv.org/html/2606.30836#alg1)\.

The algorithm accepts the final optimization archive, comprising decision vectorsXXand objective vectorsYY, along with the specific query solutionxqx\_\{q\}to be explained\. Three hyperparameters control the process222The hyperparametersKK\(anchors\) andLL\(sampling\) represent a trade\-off between resolution and cost\.KKdetermines the rigidity of the geometric surrogate model, whileLLensures the statistical convergence of the saliency scores\. In our experiments, as will be seen in Section[4](https://arxiv.org/html/2606.30836#S4), these were tuned to ensureR2\>0\.99R^\{2\}\>0\.99, ensuring that the generated explanations are anchored in a high\-fidelity geometric mapping\. Additionally, the KD\-Tree depth D is a user\-specified parameter that dictates the granularity of the objective space partitioning\. This allows the DM to adapt the “Regions of Interest” to their specific exploration goals and the desired solution density within each partitioned block\.KKdefines the number of geometric anchors for the surrogate;DDcontrols the granularity of the KD\-Tree partition; andLLsets the number of random masks used for sampling\. The procedure returns two key artifacts: the automatedtargetTT\(the geometric point of reference\) and thesaliency mapSS, which quantifies the influence of each decision variable on the solution’s proximity to that target\.

Algorithm 1Partition\-Guided Distance Saliency \(PGDS\)1:Input:Archive

\(X,Y\)\(X,Y\), Query

xqx\_\{q\}, max depth

DD, References

KK, Masks

LL\.

2:Step 1: Surrogate Training \(MLM\)

3:Select set of

KKreference points

ℛ=\{𝐫1,…,𝐫K\}\\mathcal\{R\}=\\\{\\mathbf\{r\}\_\{1\},\\dots,\\mathbf\{r\}\_\{K\}\\\}randomly from Archive

XX
4:Compute Distance Matrices

DxD\_\{x\}and

Δy\\Delta\_\{y\}
5:Solve for Regression Matrix:

B^=\(DxT​Dx\)−1​DxT​Δy\\hat\{B\}=\(D\_\{x\}^\{T\}D\_\{x\}\)^\{\-1\}D\_\{x\}^\{T\}\\Delta\_\{y\}
6:Step 2: Automated Target Identification

7:Build KD\-Tree on

YYwith depth

DD
8:Identify leaf block

LjL\_\{j\}containing

xqx\_\{q\}
9:Set Target

T←Pd​o​m\(j\)T\\leftarrow P\_\{dom\}^\{\(j\)\}\(min coords of

LjL\_\{j\}\)

10:Step 3: Saliency Map Generation \(Distance Explainer\)

11:for

l=1l=1to

LLdo

12:Generate random binary mask

m\(l\)m^\{\(l\)\}
13:Perturb Input:

x~←xq⊙m\(l\)\+x¯⊙\(1−m\(l\)\)\\tilde\{x\}\\leftarrow x\_\{q\}\\odot m^\{\(l\)\}\+\\bar\{x\}\\odot\(1\-m^\{\(l\)\}\)
14:Predict Position:

15:Compute input distances:

di​n←‖x~−ℛ‖d\_\{in\}\\leftarrow\|\|\\tilde\{x\}\-\\mathcal\{R\}\|\|
16:Predict output distances:

δ^o​u​t←di​n​B^\\hat\{\\delta\}\_\{out\}\\leftarrow d\_\{in\}\\hat\{B\}
17:Multilaterate:

y^←miny​∑\(‖y^−Tr​e​f​s‖−δ^o​u​t\)2\\hat\{y\}\\leftarrow\\min\_\{y\}\\sum\(\|\|\\hat\{y\}\-T\_\{refs\}\|\|\-\\hat\{\\delta\}\_\{out\}\)^\{2\}
18:Compute Distance Shift:

d​i​s​t\(l\)←‖y^−T‖dist^\{\(l\)\}\\leftarrow\|\|\\hat\{y\}\-T\|\|
19:endfor

20:Compute Saliency

SSvia correlation of

mmand

d​i​s​tdist
21:Return:Saliency Map

SS, Target

TT

The process begins with the surrogate training step \(Lines 2\-5\)\. Here, the Minimal Learning Machine is initialized by randomly selectingKKreference points from the archiveXXto serve as geometric anchors\. The input and output distance matrices \(DxD\_\{x\}andΔy\\Delta\_\{y\}\) are calculated, and the regression matrixB^\\hat\{B\}is solved via Ordinary Least Squares \(Line 5\), creating the global topological mapping\.

The second step constitutes the automated target identification \(Lines 6\-9\), addressing the decision\-making bottleneck\. A KD\-Tree is constructed on the objective vectorsYYto partition the space \(Line 7\)\. For a given query solutionxqx\_\{q\}, the algorithm identifies the specific leaf blockLjL\_\{j\}containing it and automatically sets the targetTTto be the “Dominating Point” \(Pd​o​m\(j\)P\_\{dom\}^\{\(j\)\}\) of that region \(Line 9\), defined by the minimum coordinates within the block boundaries\.

The final step represents the saliency map generation \(Lines 10\-18\), which adapts the RISE framework\. The algorithm iterates throughLLrandomly generated binary masks \(Lines 11\-12\), perturbing the query solution by replacing masked features with the population meanx¯\\bar\{x\}\(Line 13\)\.

For each perturbed sample, the position is predicted via the MLM surrogate chain: calculating input distances, projecting them to output distances, and reconstructing the coordinates via multilateration \(Lines 14\-17\)\.

Finally, the shift in distance relative to the targetTTis recorded \(Line 18\), and the variable saliency is computed as the correlation between the mask presence and the proximity to the target \(Line 20\)\.

An implemented version in Python is available at[PGDS](https://github.com/ClaudioLucioLopes/PGDS)\.

## 4Experiments

### 4\.1Visualizing the PGDS Framework: A Step\-by\-Step Walkthrough

To demonstrate the practical utility of the PGDS framework, we present a walkthrough using the DTLZ7 benchmark problem\[[8](https://arxiv.org/html/2606.30836#bib.bib8)\]\. DTLZ7 is chosen for its disconnected, multi\-modal Pareto front, which presents a significant challenge for traditional visualization\. The process is illustrated in Figure[2](https://arxiv.org/html/2606.30836#S4.F2), which maps the algorithmic steps from Section[3\.4](https://arxiv.org/html/2606.30836#S3.SS4)to the DM’s perspective\.

Before any explanation can be generated, the system must learn to predict the problem\. As detailed in Algorithm[1](https://arxiv.org/html/2606.30836#alg1), this process relies on the hyperparameterKK\(Reference Points\), which determines the number of geometric anchors randomly sampled from the archive\. In this specific validation,KKwas set to half the total number of solutions \(K=N/2K=N/2\)\. As shown in Figure[1](https://arxiv.org/html/2606.30836#S4.F1), the MLM model is trained on the optimization archive\. The plot overlays the original objective vectors, represented by red crosses, with the MLM’s predictions represented by black dots\. The high coefficient of determination,R2\>0\.99R^\{2\}\>0\.99, and negligible Mean Squared Error, MSE≈0\.0002\\approx 0\.0002, across all three objectives confirm that the surrogate has successfully captured the manifold’s geometry\. This high\-fidelity mapping is the prerequisite for reliable distance\-based explanations\.

Figure 1:Surrogate model validation on DTLZ7\. The alignment between original data \(red crosses\) and MLM predictions \(black dots\) demonstrates the model’s ability to preserve the complex, disconnected geometry of the Pareto front with high precision \(R2≈0\.99R^\{2\}\\approx 0\.99\)\.Once the model is ready, the interaction begins, Figure[2](https://arxiv.org/html/2606.30836#S4.F2),top\-left\. The KD\-Tree algorithm partitions the disconnected search space into distinct hyper\-rectangular blocks \(blue transparent cuboids\)\. The system automatically identifies a “Dominating Point” \(small red square\) for each region\. Instead of facing a “cognitive drought” and staring at a cloud of points, the DM is presented with structured options\. The DM simply selects a solution of interest, the blue circle, within one of these regions\.

Figure 2:The user journey through PGDS\.Top\-left: the space is partitioned into cuboids, offering automated Dominating Points \(red squares\)\.Top\-tight: a trajectory is defined from the user’s selection to the local target\.Bottom\-left: the constraints of the region are visualized\.Bottom\-right: the saliency map reveals that variablex2x\_\{2\}is the primary driver for reaching the target in this specific region\.The framework then contextualizes the selected solution, Figure[2](https://arxiv.org/html/2606.30836#S4.F2),top\-right\. It draws a geometric vector, represented by the red dashed line, connecting the user’s selection, given by the blue circle, to the region’s automated dominating point, represented by the red square\. This vector defines the “direction of improvement” specific to that local trade\-off region\. It answers the implicit question: “What is the best I can theoretically achieve if I stay within this specific configuration of constraints?”

A key advantage of the PGDS framework is the ability to visualize the “why” behind a region\. TheRegion constraints plot, Figure[2](https://arxiv.org/html/2606.30836#S4.F2),bottom\-left, directly visualizes the boundary rules of the leaf blockLjL\_\{j\}identified in Algorithm[1](https://arxiv.org/html/2606.30836#alg1)\(Line 8\)\.Gray Area, the rules:This shaded band represents the min/max bounds of the current region \(LjL\_\{j\}\) for every objective\. It answers the question: “What defines this region?”\.Trajectory:The plot shows the gap between the User’s selection \(blue circle\) and the region’s dominating point \(red cross\)\. This allows the DM to instantly see which specific objective is hitting the region’s boundary\.

Finally, the system explainshowto traverse the trajectory toward the target, Figure[2](https://arxiv.org/html/2606.30836#S4.F2),bottom\-right\. The Saliency Map explicitly categorizes the decision variables based on the correlation logic defined in Section[3\.3](https://arxiv.org/html/2606.30836#S3.SS3):

- •Green bars, theDrivers: Variables likex2x\_\{2\}show a strong negative impact on distance, meaning they are the primary levers to pull to move the solution closer to the target\.
- •Red bars, theBlockers: Variables likex1x\_\{1\}show a positive impact, indicating that their current values or perturbations might be hindering progress toward the target\.

This transforms the abstract geometric distance into actionable engineering insight: “To reach improvements in this region using this target, adjustx1x\_\{1\}to remove the bottleneck while carefully maintaining the driverx2x\_\{2\}\.”

Figure 3:Validating the “Blocker” Hypothesis\. Variablex1x\_\{1\}was identified as a blocker \(red bar in saliency map\)\. Manually increasingx1x\_\{1\}results in a9\.4%9\.4\\%improvement in the solution’s proximity to the target \(Global View, green vector\), confirming that the variable was indeed hindering convergence\.To verify the actionable insights generated by the Saliency Map, we perform a sensitivity analysis by aligning the identified high\-saliency variables with the configuration of the region’s Dominating Point\. Instead of arbitrary tuning, we calculate the precise delta \(Δi=xt​a​r​g​e​t,i−xc​u​r​r​e​n​t,i\\Delta\_\{i\}=x\_\{target,i\}\-x\_\{current,i\}\) required to shift the variable from its current state to the optimal value observed in the target solution\. We then apply this calculated shift and observe the resulting impact on the geometric distance in the objective space\.

First, we address the identifiedBlocker, variablex1x\_\{1\}\(Figure[3](https://arxiv.org/html/2606.30836#S4.F3)\)\. The system calculates thatx1x\_\{1\}must be increased by0\.180\.18to match the target configuration\. Applying this specific adjustment results in a9\.4%9\.4\\%improvement \(decrease\) in the Euclidean distance to the target, confirming that the variable indeed hindered convergence\.

Second, we test the identifiedDriver, variablex2x\_\{2\}\(Figure[4](https://arxiv.org/html/2606.30836#S4.F4)\)\. The system identifies thatx2x\_\{2\}is critical\. When we deviate from the calculated target trajectory by decreasingx2x\_\{2\}by0\.050\.05, the distance to the target worsens by2\.7%2\.7\\%\. This degradation confirms thatx2x\_\{2\}plays a critical role in maintaining proximity to the optimal front and must be preserved\.

Figure 4:Validating the “Driver” Hypothesis\. Variablex2x\_\{2\}was identified as a driver \(green bar\)\. Decreasingx2x\_\{2\}leads to a2\.7%2\.7\\%degradation \(distance increase\) in performance\. This sensitivity confirms thatx2x\_\{2\}is a critical component of the solution’s success and must be carefully managed\.
### 4\.2Physics\-Informed Validation \(Welded Beam\)

The second experiment addresses RQ3, regarding sensitivity, and RQ1, interpretability, by applying PGDS to the Welded Beam Design problem\[[6](https://arxiv.org/html/2606.30836#bib.bib6)\]\. This problem is chosen because the relationships among decision variables \(weld thicknesshh, lengthll, heighttt, widthbb\) and constraints \(shear stressτ\\tau, bending stressσ\\sigma, buckling loadPcP\_\{c\}\) are governed by known physical laws\. This allows us to verify whether the mathematically derived blockers correspond to actual physical limitations or not\.

The engineering objective is to design a welded beam that minimizes the fabrication cost \(f1f\_\{1\}\) and the end deflection \(f2f\_\{2\}\)\. The problem consists of four decision variablesx→=\(h,l,t,b\)\\vec\{x\}=\(h,l,t,b\)and is subject to four non\-linear constraints, as formulated in\[[6](https://arxiv.org/html/2606.30836#bib.bib6)\]:

Minimizef1​\(x→\)=1\.10471​h2​l\+0\.04811​t​b​\(14\.0\+l\),\\displaystyle f\_\{1\}\(\\vec\{x\}\)=1\.10471h^\{2\}l\+0\.04811tb\(14\.0\+l\),\(8\)Minimizef2​\(x→\)=2\.1952t3​b,\\displaystyle f\_\{2\}\(\\vec\{x\}\)=\\frac\{2\.1952\}\{t^\{3\}b\},\(9\)subject to:

g1​\(x→\)\\displaystyle g\_\{1\}\(\\vec\{x\}\)≡13,600−τ​\(x→\)≥0,\(Shear Stress\)\\displaystyle\\equiv 13,600\-\\tau\(\\vec\{x\}\)\\geq 0,\\quad\(\\text\{Shear Stress\}\)\(10\)g2​\(x→\)\\displaystyle g\_\{2\}\(\\vec\{x\}\)≡30,000−σ​\(x→\)≥0,\(Bending Stress\)\\displaystyle\\equiv 30,000\-\\sigma\(\\vec\{x\}\)\\geq 0,\\quad\(\\text\{Bending Stress\}\)\(11\)g3​\(x→\)\\displaystyle g\_\{3\}\(\\vec\{x\}\)≡b−h≥0,\(Weld Geometry\)\\displaystyle\\equiv b\-h\\geq 0,\\quad\(\\text\{Weld Geometry\}\)\(12\)g4​\(x→\)\\displaystyle g\_\{4\}\(\\vec\{x\}\)≡Pc​\(x→\)−6,000≥0,\(Buckling Load\)\\displaystyle\\equiv P\_\{c\}\(\\vec\{x\}\)\-6,000\\geq 0,\\quad\(\\text\{Buckling Load\}\)\(13\)0\.125\\displaystyle 0\.125≤b,h≤5\.0,\\displaystyle\\leq b,h\\leq 5\.0,\\quad\(14\)0\.1\\displaystyle 0\.1≤l,t≤10\.0\.\\displaystyle\\leq l,t\\leq 10\.0\.\\quad\(15\)
For brevity, we present the high\-level constraint definitions above\. The detailed constitutive equations governing the physical mechanics involve complex interactions between the beam geometry and applied forces\. The complete mathematical derivations and parameter constants are fully detailed in\[[6](https://arxiv.org/html/2606.30836#bib.bib6)\]\.

To test the framework’s sensitivity, we utilized NSGA\-II\[[3](https://arxiv.org/html/2606.30836#bib.bib3)\]to generate an approximation of the Pareto front \(200 solutions\) and selected a specific “Constraint Trap” solution\.

Figure 5:Physics\-Informed Validation on Welded Beam\. Left plot is the “Trap Solution” \(red Circle\), that is stuck at a low\-cost optimum\. PGDS identifies that to reach the high\-performance target \(green star\), the solution must escape the buckling constraint\. Right plot presents the saliency map, which identifies beam width \(bb\) as the primary “Blocker” \(highest red bar\), correctly signaling that width is one of the physical bottlenecks preventing improvement\.While Algorithm[1](https://arxiv.org/html/2606.30836#alg1)provides an automated target, the PGDS framework also supports manual target selection\. As detailed in Figure[5](https://arxiv.org/html/2606.30836#S4.F5), left plot, the selected solution \(red circle\) is located in the low\-cost region of the Pareto front \(c​o​s​t≈2\.50cost\\approx 2\.50\)\. However, this efficiency comes at a price: the solution is physically precarious\. An analysis of its constraint values reveals it is positioned almost exactly on the boundary of the buckling constraint \(g4≈−0\.0001g\_\{4\}\\approx\-0\.0001\), meaning the beam is on the verge of structural failure\.

We executed the PGDS with a distinct goal: rather than a local dominating point, we set the target to the best deflection point found in the archive \(green star in Figure[5](https://arxiv.org/html/2606.30836#S4.F5)\)\. This evaluates whether the method can steer a low\-cost design towards a high\-performance configuration\.

The resulting saliency map \(Figure[5](https://arxiv.org/html/2606.30836#S4.F5), right plot, identifies the variablebb\(Width\) as the overwhelming “Blocker” with a normalized saliency value of1\.01\.0\. While other variables lengthllshow some blocking influence, the framework signals thatbbis the critical variable preventing the solution from moving toward the target\.

To validate this diagnosis, we performed a sensitivity analysis by perturbing the identified blockerbb\(Width\)\. As shown in Table[1](https://arxiv.org/html/2606.30836#S4.T1), blindly adjusting other variables while keepingbbfixed at its trap value \(0\.2360\.236\) would likely result in an infeasible design due to the active buckling constraint\. However, relaxingbb\(increasing it to0\.2590\.259\) immediately relieves the pressure on the buckling constraint, shiftingg3g\_\{3\}from a critical−0\.0001\-0\.0001to a safe−0\.0049\-0\.0049\.

This confirms that PGDS successfully detected the “wall” created by the buckling equationPc​\(x\)P\_\{c\}\(x\)without explicit knowledge of the formula\. By flaggingbb\(Width\) as a blocker, the framework correctly informed the DM that no significant improvement in deflection is possible without first increasing the beam’s width\.

Table 1:Physics\-Informed Verification of the “Blocker” Hypothesis
### 4\.3Experiment 3: Scalability to Many\-Objective Optimization \(WFG3\)

The final experiment examines RQ2, concerning automated discovery, and RQ1, concerning interpretability, in the setting of many\-objective optimization\. As the number of objectives increases beyond three, traditional scatter plots become unintelligible, making it difficult for the DM to visually identify interesting trade\-offs\. We utilize the PGDS framework to help navigate and explain a 10\-objective landscape\.

We selected the WFG3\[[8](https://arxiv.org/html/2606.30836#bib.bib8)\]benchmark problem configured with 10 objectives and 20 decision variables\. WFG3 is particularly challenging because it features a degenerate Pareto front \(a lower\-dimensional curve embedded in high\-dimensional hyperspace\), making it an ideal test for the geometric sensitivity of the PGDS\.

The experimental protocol proceeded as follows:

1. 1\.Optimization: We employed NSGA\-III to generate an approximation of the Pareto front\. The algorithm ran for 400 generations, producing a final archive of 108 non\-dominated solutions\.
2. 2\.Blind Targeting: Unlike previous experiments where targets were visually verified, here we relied entirely on the KD\-Tree to partition the 10\-dimensional objective space into semantic blocks\.
3. 3\.Region Selection Logic: While PGDS automatically partitions the space into hyper\-rectangular semantic blocks, we applied standard decision\-making heuristics to select specific regions for explanation\. This simulates a DM prioritizing regions based on common goals: - •TheKnee Region: The block containing the solution with the minimum Euclidean distance to the ideal point \(the origin\)\. - •TheExtreme Region: The block containing the solution with the best value for objectivef10f\_\{10\}, regardless of other trade\-offs\.

Figure 6:Region selection discovery in 10 Dimensions\. The Parallel Coordinate Plot reveals the contrasting geometries identified by the KD\-Tree\. Thekneeregion \(Blue\) represents balanced solutions, while theextremeregion \(Red\) captures solutions optimizing specific objectives \(f10f\_\{10\}\) at the expense of others \(f9f\_\{9\}\)\. The bold lines are the dominating solutions in each region\.The PGDS framework successfully distinguished the two regions without human intervention\. Figure[6](https://arxiv.org/html/2606.30836#S4.F6)presents a Parallel Coordinate Plot \(PCP\) visualizing the Knee Region \(Blue\), which exhibits a balanced profile across all 10 objectives, whereas the Extreme Region \(Red\) shows aggressive trade\-offs\.

To resolve the “cognitive drought” inherent in many\-objective visualization, we distinguish between thegeometrical potentialof a region and theactual solutionanalyzed\. Figure[6](https://arxiv.org/html/2606.30836#S4.F6)visualizes the “Dominating Points” \(local utopias\) that define the boundaries of each partition\. In contrast, Figure[7](https://arxiv.org/html/2606.30836#S4.F7)illustrates the specific representative solutions from Table[2](https://arxiv.org/html/2606.30836#S4.T2)\. This distinction explains the trade\-off profile: the bold red solution in Figure[7](https://arxiv.org/html/2606.30836#S4.F7)achieves a superior, lower value atf10f\_\{10\}compared to the knee solution, which sacrificesf10f\_\{10\}to maintain global balance across the other nine objectives\.

Table 2:Automated “Blind” Targets identified by PGDS in WFG3Figure 7:Representative solutions selected for saliency analysis in WFG3\-10\. Unlike Figure[6](https://arxiv.org/html/2606.30836#S4.F6), which highlights the local utopian boundaries \(dominating points\) of the KD\-Tree partitions, this plot visualizes the actual solutions identified in Table[2](https://arxiv.org/html/2606.30836#S4.T2), the balanced Knee Point \(blue bold\) and the Extreme Point \(red bold\) optimized forf10f\_\{10\}\.Table[2](https://arxiv.org/html/2606.30836#S4.T2)details the specific automated targets found\. Note that while the Knee point maintains a middle ground \(all objectives≈1\.5−2\.0\\approx 1\.5\-2\.0\), the Extreme point achieves a superiorf10f\_\{10\}\(0\.580\.58\) but sacrificesf9f\_\{9\}\(7\.527\.52\)\.

The most critical finding using the saliency analysis is the explanation, illustrated in Figure[8](https://arxiv.org/html/2606.30836#S4.F8)\.Knee Region: in the left plot, maintaining the balanced knee position, the system identifies the variablex18\{x\_\{18\}\}as the primary driver \(Score:−1\.00\-1\.00\)\. The variablex16x\_\{16\}also plays a supporting driver role\. This suggests thatx18x\_\{18\}is the keystone variable for global balance in WFG3\.Extreme Region: In the right plot, the landscape changes completely\. Here, the variablex15\{x\_\{15\}\}emerges as a massive blocker \(score:\+1\.00\+1\.00\)\. This indicates thatx15x\_\{15\}is the distinct “constraint” preventing further optimization off10f\_\{10\}or causing the severe trade\-off withf9f\_\{9\}\.

The identified saliency forx18x\_\{18\}\(Knee\) andx15x\_\{15\}\(Extreme\) provides actionable insights rooted in the benchmark’s structure\. In WFG3, these specific variables act as keystone position parameters that control the underlying manifold’s curvature\. By flaggingx15x\_\{15\}as a “Blocker” in the Extreme region, PGDS correctly identifies the specific decision variable whose current configuration prevents the solution from sliding further down thef10f\_\{10\}boundary without degrading other objectives\.

Figure 8:Context\-aware saliency comparison\. In the Knee Region, the left one, variablex18x\_\{18\}is the primary driver \(green\) facilitating convergence\. In the right plot, the extreme region of the variablex15x\_\{15\}becomes a dominantBlocker\(Red\), hindering performance\. This proves that variable importance is not static but dependent on the region of the Pareto front\.The experiment confirms that PGDS scales effectively to high\-dimensional problems\. Even without visual aids, the framework provided differentiated, actionable insights: telling the DM to focus onx18x\_\{18\}for a balanced solution, but to manipulatex15x\_\{15\}if minimizingf10f\_\{10\}is the priority\. This capability effectively bridges the interpretability gap in Many\-Objective Optimization\.

In many\-objective optimization, traditional visualization represents a “failing baseline” in decision and objective explainability, because a standard PCP, the gray lines in Figures[6](https://arxiv.org/html/2606.30836#S4.F6)and[7](https://arxiv.org/html/2606.30836#S4.F7), for example, provides no guidance on where to look or which variables to adjust\. PGDS overcomes this by overlaying automated targets and providing the saliency map, as in Figure[8](https://arxiv.org/html/2606.30836#S4.F8), which transforms the intelligible set of solutions into a directed set of engineering instructions—identifying exactly which levers \(x18x\_\{18\}\) to pull for balance\.

## 5Conclusion

This work introduced the Partition\-Guided Distance Saliency \(PGDS\) framework, a methodology engineered to bridge the interpretability gap between high\-dimensional decision spaces \(𝒳\\mathcal\{X\}\) and complex, many\-objective landscapes \(𝒴\\mathcal\{Y\}\)\. By synergizing the geometric learning capabilities of Minimal Learning Machines \(MLM\) with the automated spatial decomposition of KD\-Trees, PGDS effectively resolves the “cognitive drought” often experienced by DMs when traditional visualization techniques fail\.

Our experimental validation confirms the framework’s robustness across some perspectives: convergence, diversity, and scalability\. Collectively, these results establish PGDS as a diagnostic instrument\. It empowers DMs to navigate the Pareto front withouta prioritarget knowledge, automating the identification of “Dominating Points” within local regions and rigorously quantifying the directional influence of decision variables through distance\-based saliency\.

A direct quantitative comparison with existing rule\-based XAI methods \(such as XLEMOO or R\-XIMO\) remains due to the differing nature of their outputs—logical predicates versus continuous saliency maps\. Future research should focus on developing standardized explanation quality indicators to benchmark distance\-based saliency against classification\-based attribution in high\-dimensional spaces\.

A systematic sensitivity analysis of theKK,DD, andLLhyperparameters is required to rigorously define the bounds of the framework’s robustness and provide practitioners with automated tuning heuristics for varying archive sizes\.

While PGDS successfully classifies variables asDriversorBlockersbased on their geometric influence, the current iteration operates as a diagnostic rather than a prescriptive framework\. The methodology relies on sensitivity analysis to estimate the impact of perturbations; however, it does not yet mathematically invert the surrogate model to yield the precise decision vector𝐱∗\\mathbf\{x\}^\{\*\}required to reach a specific targetTT\. This limitation stems from the inherent difficulty of the inverse problem, where the mapping from a high\-dimensional decision space \(ℝn\\mathbb\{R\}^\{n\}\) to another dimensional objective space \(ℝm\\mathbb\{R\}^\{m\}\) can be ill\-posed and involve significant information loss\.

Another future research will focus on evolving PGDS from a descriptive tool into a fully prescriptive engine\. Our primary objective is to address the inversion problem inherent ton≫mn\\gg mmappings by employing the differentiable properties of the MLM to guide gradient\-based search strategies\. Furthermore, we intend to validate the framework on a broader suite of real\-world problems characterized by varying degrees of constraint complexity, extending beyond standard benchmarks to deployment in industrial engineering applications\.

Ultimately, PGDS represents a fundamental paradigm shift in Many\-Objective Optimization\. By moving beyond opaque, black\-box optimization and providing geometrically grounded, context\-aware explanations, we pave the way for transparent and trustworthy decision\-making in high\-dimensional spaces\.

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