A Flexible and Generic Approach for Explainable Landscape Analysis and the pyXla Toolbox

Tony Ombaso; Anna S. Bosman; Arnaud Liefooghe; Katherine M. Malan; Sébastien Verel

cs.NEOct 5, 2026

A Flexible and Generic Approach for Explainable Landscape Analysis and the pyXla Toolbox

Authors: Tony Ombaso, Anna S. Bosman, Arnaud Liefooghe, Katherine M. Malan, Sébastien Verel

Organizations: Department of Computer Science, University of Pretoria, Pretoria, South Africa · African Institute for Data Science and Artificial Intelligence (AfriDSAI), University of Pretoria, Pretoria, South Africa · Univ. Littoral Cˆote d’Opale, UR 4491, LISIC, F-62100 Calais, France · Department of Decision Sciences, University of South Africa, Pretoria, South Africa

Abstract

Landscape analysis has been successfully applied to understand complex optimisation problems, gain insights into algorithm behaviour, and automate algorithm selection and configuration. Although many landscape analysis techniques have been developed over the last decades, it remains difficult for researchers and practitioners to decide which approaches are appropriate and to implement them in practice. Some tools are available, but these are either restricted to particular problem domains (e.g., unconstrained black-box continuous optimisation), or are limited in what they model and measure. In addition, output from landscape analysis is often not easily interpretable, especially when computed landscape features do not correspond with aspects of problems that practitioners are familiar with. In this paper, we introduce a principled approach for explainable landscape analysis (XLA) with an associated Python package called pyXla. The approach is generic in that it applies to problems with different representations (continuous or combinatorial), with single or multiple objectives, with or without constraints. The extent of analysis provided by the XLA framework depends on the data available, with richer analysis offered as additional information is provided by the user. We demonstrate the explainable output produced by pyXla on a selection of hand-crafted problems with diverse landscape characteristics.

Figures & tables

Figure 1: A high-level diagram of the proposed XLA framework. Data loading and sampling are separated from analysis.

Input	Description
X	Solutions in the sample
	Example headings for five decision variables: id x1 x2 x3 x4 x5
	If not available as input data, X can be generated by pyXla using: specification of the domain of each variable with sampling strategy.
F	Objective values of the sample of solutions
	Example headings for three objectives: id f1 f2 f3
	If not available as input data, F can be generated by pyXla using: specification of objective function(s) and X .

Table 1: Description of the input data in pyXla . The id in X , F , V , I , and id1 and id2 in D and N link the data between files.

Function	Description
sample_X	Takes a domain specification for each variable with a sampling strategy as input and generates an X file.
compute_F	Takes an objective function as input and generates an F file. The X file is required.
compute_V	Takes a violation function as input and generates a V file. The X file is required.
compute_D	Takes a distance metric function as input and generates a D file. The X file is required.
compute_N	Takes an arbitrary neighbourhood function that determines if two solutions are neighbours and generates an N file. The X file is required.

Table 2: Description of the auxiliary functions in pyXla .

Panel of Figure 2 — Figure 2: Comparison of the different sampling algorithms in 2D for 100 samples. Dots represent the sampled points. For ordered sampling methods, the connections between them indicate the sampling order. All samples lie within the bounding box [-5, 5]. Random walk ( Figure 2(c) ) and adaptive walk sampling ( Figure 2(d) ) are carried out with a step size of 0.5. The adaptive walk, which is guided by the objective function, is based on the sphere function ( Equation 1 ) in Figure 2(d) .

Figure 3: A diagrammatic summary of the landscape features currently available in pyXla . The transition diagram illustrates how richer analysis is provided as more input components are added. The Venn diagram depicts the various input component combinations, where the intersection between two circles indicates that both input components are present. Both the states of the transition diagram and sections of the Venn diagram are labelled with numbers. Each number label corresponds to a set of landscape features applicable to a given input component combination, as shown in the key, which is located to the right of the Venn diagram.

Feature	Requires	Description
distr_F	F	Name: Objective distribution
		Visual output: Histograms of function values and function ranks for each objective
		Numerical output: Descriptive statistics
distr_V	V	Name: Violation distribution
		Visual output: Histogram of violation values and violation ranks for each constraint
		Numerical output: Feasibility rate overall and per constraint, descriptive statistics

Table 3: Statistical landscape features included in pyXla . The symbols

\vee

and

\wedge

refer to conjunction and disjunction, respectively, of the presence of their operands, i.e., input types.

Feature	Requires	Description
distr_Par	F $~\vee~$ V	Name: Pareto rank distribution
		Visual output: Histogram of Pareto ranks ( Goldberg, 1989 ) for objective values, violations, and the combination (treating violations as objectives)
		Numerical output: Descriptive statistics
distr_Deb	F $~\wedge~$ V	Name: Deb’s feasibility rule ( Deb, 2000 ) ranks distribution
		Visual output: Histogram of feasibility rules’ ranking
		Numerical output: Descriptive statistics

Table 4: Rank-based landscape features included in pyXla . The symbols

\vee

and

\wedge

refer to conjunction and disjunction, respectively, of the presence of their operands, i.e., input types.

Feature	Requires	Description
FDC	F $~\wedge~$ D	Name: objective-distance correlation
		Visual output: Scatter plots of objective values against distance to the nearest best solution in sample per objective, for all solutions or for feasible solutions only
		Numerical output: Spearman’s correlation coefficients per objective
VDC	V $~\wedge~$ D	Name: violation-distance correlation
		Visual output: Scatter plots of violation values against distance to the nearest feasible solution in sample per constraint, for infeasible solutions only
		Numerical output: Spearman’s correlation coefficients per violation

Table 5: Distance-based landscape features included in pyXla . The symbols

\vee

and

\wedge

refer to conjunction and disjunction, respectively, of the presence of their operands, i.e., input types.

Feature	Requires	Description
NFC	F $~\wedge~$ N	Name: Neighbouring solutions’ objective values correlation
		Visual output: A scatter plot of objective values between neighbours with a regression line for all solutions or feasible solutions only. The plot is divided by a broken line through the origin such that, assuming minimisation, points above the line are deteriorating neighbours, those below the line are improving neighbours, and those on the line are neutral neighbours.
		Numerical output: Spearman’s correlation coefficients
NVC	V $~\wedge~$ N	Name: Neighbouring solutions’ violation values correlation
		Visual output: A scatter plot of violation values between neighbours for each constraint, for infeasible solutions only, with a regression line. The plot is divided by a broken line through the origin such that points above the line are deteriorating neighbours, those below are improving neighbours, and those on the line are neutral neighbours.
		Numerical output: Spearman’s correlation coefficients

Table 6: Neighbourhood-based landscape features included in pyXla . The symbols

\vee

and

\wedge

refer to conjunction and disjunction, respectively, of the presence of their operands, i.e., input types.

Problem	Domain	Objective(s)	Constraint(s)	Sampling
Sphere ( $n\!=\!1$ )	Cont.	eq. 1	eqs. ( 2 ), ( 3 )	Hilbert curve
Deceptive ( $n\!=\!1$ )	Cont.	eq. 4	eq. 5	Hilbert curve
Easom ( $n\!=\!2$ )	Cont.	eq. 6	None	Hilbert curve
Easy knapsack ( $n\!=\!10$ )	Combin.	Profit	Weight	Exhaustive
Hard knapsack ( $n\!=\!10$ )	Combin.	Profit	Weight	Exhaustive
NK ( $n\!=\!14,k\!=\!1$ )	Combin.	$g$	None	Exhaustive

Table 7: Selected problem instances, with

n

denoting the problem size. For NK landscape instances,

g

are random lookup tables that draw real numbers uniformly from the range [0, 1], see Section 4.1.5 for details. For Hilbert curve sampling, the sample size is

100

Panel of Figure 4 — Figure 4: Line plots of the constrained sphere problem objective and constraints violation values.

Panel of Figure 5 — Figure 5: Line plots of the constrained deceptive problem’s objective ( $f0$ ) and constraint ( $v0$ ).

Figure 6: Surface plot of the Easom function.

Figure 7: Neural network simulating the XOR gate. The weights and biases are labelled as $w_{i}$ and $b_{i}$ , respectively. The variables $in_{1}$ and $in_{2}$ correspond to columns Input A and Input B in Table 8 . The biases are multiplied by a constant input of 1.

Input $A$	Input $B$	Output ( $A\oplus B$ )
False	False	False
False	True	True
True	False	True
True	True	False

Table 8: XOR gate truth table.

Panel of Figure 8 — Figure 8: Output of distr_F : histogram of objective values for the constrained sphere problem and the Easom function.

Statistic	Sphere	Easom
Minimum	0.0015	-1.5046
Maximum	25.0	0.0277
Mean	8.7426	-0.0606
Median	6.5597	6.5705e-33
Q1	1.8607	-1.4481e-16
Q3	14.873	1.0137e-08

Table 9: Numerical output of distr_F for the constrained sphere problem and the Easom function.

Panel of Figure 9 — Figure 9: Output of distr_F : histogram of objective values for simulating the XOR gate using a neural network: ( 9(a) ) corresponds to the distribution of loss values for an SGD trajectory in parameter space, ( 9(b) ) shows the distribution of loss values when the neural network parameters are randomly sampled.

Panel of Figure 10 — Figure 10: Output of distr_V : histogram of violation values (top row) and ranks (bottom row) for the easy and hard knapsack instances.

Figure 11: Output of corr : scatter plots of pairs of objectives for an NK instance with 3 objectives

Figure 12: Output of corr_ranks : scatter plots of pairs of ranks for the constrained sphere problem, split by feasibility; feas. is an abbreviation for feasible.

Figure 13: (Continued) Output of corr_ranks : scatter plots of pairs of ranks for the constrained sphere problem, split by feasibility; feas. is an abbreviation for feasible.

Panel of Figure 14 — Figure 14: Output of X_imp for scenarios simulating the XOR gate using a neural network: ( 14(a) ) corresponds to the SGD trajectory in parameter space, and ( 14(b) ) corresponds to the scenario where neural network parameters are randomly sampled. The bar charts in the top row show the correlation values between each variable and the objective. The bar charts in the bottom row show the permutation feature importance scores.

Panel of Figure 15 — Figure 15: Visual output of the X_imp feature applied to the NK problem instances. The bar charts in the top row show the correlation values between each variable and the objective. The bar charts in the bottom row show the permutation feature importance scores.

Panel of Figure 16 — Figure 16: Histogram of Pareto ranks of objective and violation functions.

Table 10: Output of distr_Par : descriptive statistics of the Pareto rank of objective and violation values for the constrained sphere problem.

Statistic	Value
$paretoFV$ minimum	1
$paretoFV$ maximum	100
$paretoFV$ mean	50.5
$paretoFV$ median	50.5
$paretoFV$ Q1	25.25
$paretoFV$ Q3	75.75

Table 11: Output of distr_Par : descriptive statistics of the Pareto rank of objective and violation values for the constrained deceptive problem.

Panel of Figure 17 — Figure 17: Histograms of Deb’s feasibility rule ranks.

Panel of Figure 18 — Figure 18: Histograms of objective ranks.

Statistic	Easy knapsack instance	Hard knapsack instance
Minimum	1	1
Maximum	463	928
Mean	205.0576	464.5
Median	197.0	464.5
Q1	126.0	238.0
Q3	274.0	691.0

Table 12: Output of distr_Deb Descriptive statistics of Deb’s feasibility rule ranks for the easy and hard knapsack instances.

Panel of Figure 19 — Figure 19: FDC visual output for the constrained sphere and constrained deceptive problems.

Panel of Figure 20 — Figure 20: FDC visual output for NK landscape instances.

Panel of Figure 21 — Figure 21: Output of FDC for scenarios simulating the XOR gate using a neural network: ( 21(a) ) corresponds to the SGD trajectory in parameter space, and ( 21(b) ) corresponds to the scenario where neural network parameters are randomly sampled.

Figure 22: Visual output of the VDC feature for each constraint of the constrained sphere problem, where

v0

and

v1

are Constraints ( 2 ) and ( 3 ).

Figure 23: Visual output of the RDC feature for each rank computed for the constrained sphere problem.

Figure 24: Visual output of the PDC feature for the constrained sphere problem.

Figure 25: Visual output of the PDC feature for the constrained deceptive problem.

Figure 26: Visual output of the PDC feature for the Easom problem.

Value/Rank space	Sphere
Objective space( $F$ )	0.4393
Violation space ( $V$ )	0.4116
Equation 2 constraint	0.4566
Equation 3 constraint	0.0098
Objective rank	0.3433
Violation rank	0.4233

Table 13: Numerical output of PDC for the constrained sphere problem.

Value/Rank space	Deceptive
Objective space ( $F$ )	0.6251
Violation space ( $V$ )	0.3155
Objective rank	0.6413
Violation rank	0.3274
$Deb$ rank	0.6413
$paretoFV$	0.6413

Table 14: Numerical output of PDC for the constrained deceptive problem.

Figure 27: Visual output of disp_best for the constrained sphere problem; disp. is an abbreviation for dispersion.

Figure 28: Visual output of disp_best for the constrained deceptive problem; disp. is an abbreviation for dispersion

Panel of Figure 29 — Figure 29: Visual output NFC .

Figure 30: Visual output of the NVC feature for the constrained sphere problem.

Figure 31: Visual output of NRC for the constrained sphere problem.

Figure 32: Visual output of NRC for the Easom function problem.

Panel of Figure 33 — Figure 33: Bar chart showing output of NCF for the constrained sphere problem.

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Multi-Objective Optimization Saliency Interpretability Bridging Language Distance

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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 without a priori knowledge. 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.

Abstract

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