Draft version of HET-NSGA-III / MFE-NSGA-III

Code for two companion papers on surrogate-assisted evolutionary multi- and many-objective optimization with heterogeneous evaluation times:

1 (HET-NSGA-III): Balija Santoshkumar and Kalyanmoy Deb, "Handling Objectives with Heterogeneous Evaluation Times in Surrogate-Assisted Evolutionary Multi-Objective Optimization," IEEE Transactions on Evolutionary Computation, 2025. [Link]

2 (MFE-NSGA-III): Balija Santoshkumar and Kalyanmoy Deb, "Handling of Objectives and Constraints with Heterogeneous Evaluation Times for Surrogate-Assisted Evolutionary Multi- and Many-Objective Optimization," Swarm and Evolutionary Computation, vol. 100, 2026. [Link]

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Overview

In most surrogate-assisted evolutionary multi-objective optimization (SA-EMO) algorithms, every infill solution is evaluated with high-fidelity models for all objectives and constraints. This is wasteful when evaluation times differ across objectives and constraints, and when a solution is largely infeasible or its surrogate prediction error is small.

This repository implements a mixed-fidelity evaluation (MFE) framework that decides, for each population member individually, which objectives and constraints need high-fidelity evaluation. The decision is based on four factors:

1. Whether the solution is largely feasible, largely infeasible, or close to a constraint boundary
2. The probability of the solution being non-dominated within its reference vector neighborhood
3. The relative high-fidelity evaluation time of each objective and constraint
4. The surrogate model uncertainty for each objective

MFE-NSGA-III extends HET-NSGA-III to constrained problems. When no constraints are present, MFE-NSGA-III degenerates to HET-NSGA-III.

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Repository Structure

HET-NSGA-III/
├── pyheterogeneous/
│   ├── algorithms/
│   │   ├── he_rho_generalized_cons_first.py       # HECF  — MFE-NSGA-III 
│   │   ├── he_rho_generalized_cons_first_block.py # HECFB — MFE-NSGA-III with block evaluation
│   │   ├── Util_func.py                           # Selection utilities 
│   │   └── probability_independent.py             # P_s^m and PG_s computation 
│   └── core/
│       ├── evaluator.py
│       ├── scheduler.py
│       ├── problem.py
│       ├── display.py
│       ├── termination.py
│       └── util.py
├── AddProblems/                   # Additional test problems (MaF, engineering problems)
├── pymoo/                         # pymoo dev 0.5.1 (bundled)
├── pydacefit/                     # Bundled dependency
├── ezmodel/                       # Bundled dependency
├── pysamoo/                       # Surrogate modeling framework (bundled)
├── evaluator_data_args.py         # Run script for HECF (MFE-NSGA-III)
├── evaluator_data_args_block.py   # Run script for HECFB (block evaluation)
└── requirements.txt

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Algorithm Description

Mixed-Fidelity Selection (MFS) Metric — Eq. (2)

For every feasible solution s and each objective m, the MFS metric is:

$$\rho_s^m = \left(1 + \widehat{ET}^m\right)^\alpha \cdot P_s^m \cdot \left(1 + \left(\frac{\sigma_s^m}{|\Delta f_m|}\right)^{1/\eta}\right)$$

where P_s^m is the probability of solution s being non-dominated among its reference vector neighbors, σ_s^m is the surrogate uncertainty, |Δf_m| is the range of objective m in the population, η = 20, and α adapts over time per Eq. (4).

Constraint Classification — Eq. (1)

Every solution s is classified using:

$$Z_s^j = \frac{-\mu_s^j}{\sigma_s^j}$$

where μ_s^j and σ_s^j are the surrogate prediction and uncertainty for constraint j. A solution is Class I (feasible/near-boundary) if min_j Z_s^j ≥ e (default e = -0.5). Otherwise it is Class II (largely infeasible).

Survival Selection Procedure (Algorithm 1)

• Step 2 (|Class I| ≥ N): Select N solutions from Class I using the MFS metric per reference vector. For each selected solution, evaluate constraints j with e ≤ Z_s^j ≤ -e (near boundary) and the objective m with the highest ρ_s^m.
• Step 3 (0 < |Class I| < N): Use Step 2 for all Class I solutions, fill remaining slots with best Class II solutions by PG_s (no HF evaluation for infeasible solutions).
• Step 4 (|Class I| = 0): Select the solution with the highest PG_s and evaluate its least-violated constraint (argmax_j Z_s^j).

Overall Constraint Satisfaction Probability — Eq. (3)

$$PG_s = \prod_{j=1}^{J} P(g_j(s) \leq 0) = \prod_{j=1}^{J} \Phi(Z_s^j)$$

Alpha Update — Eq. (4)

$$\alpha = \frac{(T - \tau) - (T_{\max} - T)}{T_{\max} - \tau}$$

where T is elapsed time, τ = N_DoE is the initial DoE budget, and T_max is the total budget. This shifts emphasis from cheap to expensive objectives over time.

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Installation

Clone the repository:

git clone https://github.com/santoshbalija/HET-NSGA-III.git

If you want to create a new conda environment (recommended), use the following commands:

conda create -n hetemo python=3.11
conda activate hetemo

Then install the necessary libraries:

cd HET-NSGA-III
pip install -r requirements.txt

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Usage

MFE-NSGA-III (constrained, non-block)

python evaluator_data_args.py \
    --algorithm HE_Cons_f \
    --problem "MW3(n_var=10)" \
    --n_obj 2 \
    --ET_f1 0.7 --ET_f2 0.1 \
    --ET_g1 0.15 --ET_g2 0.05 \
    --termination 300 \
    --n_doe 100 \
    --pop_size 50 \
    --surr_gen 30 \
    --seed 1 \
    --sigma_factor 20 \
    --cons_sigma_level -0.5

Key arguments:

Argument	Description	default
--algorithm	HE_Cons_f (MFE-NSGA-III) or base (NSGA-III) or batch (SA-NSGA-III)	—
--problem	Any pymoo problem, e.g. MW3(n_var=10), C2DTLZ2(n_var=12, n_obj=3)	—
--ET_f1..ET_fM	Relative evaluation time for each objective (must sum with constraints to 1.0)	varies
--ET_g1..ET_gJ	Relative evaluation time for each constraint	varies
--termination	Total time budget T_max	300 (2-3 obj), 400 (5-8 obj)
--n_doe	Initial DoE size N_DoE	100 (2-3 obj), 150 (5-8 obj)
--pop_size	Population size N (= n_infills = n_ref)	50
--surr_gen	Surrogate generations t_S per iteration	30
--sigma_factor	Surrogate uncertainty exponent η in Eq. (2)	20
--cons_sigma_level	Constraint boundary threshold e in Step 1 (must be negative)	-0.5
--seed	Random seed	varies

Results are saved to Data/two_obj_two_const_cons_sigma/ as .pkl files per seed.

MFE-NSGA-III with Block Evaluation (Section 4, Paper 2)

Block evaluation applies when multiple objectives and/or constraints are always evaluated together by the same third-party software call.

python evaluator_data_args_block.py \
    --algorithm HE_Cons_f_block \
    --problem "Carside()" \
    --n_obj 3 \
    --ET_f1 0.05 --ET_f2 0.95 \
    --ET_g1 0.95 --ET_g2 0.95 \
    --block_obj_cons "[[1, 2, 3]]" \
    --termination 300 \
    --pop_size 50 \
    --seed 1

The --block_obj_cons argument specifies which objectives and constraints are evaluated together. Indices follow the convention: objectives are 0..M-1, constraints are M..M+J-1. For example [[1, 3]] means objective index 1 (f2) and constraint index 3 (g2, i.e. M+1=3 for M=2) are always evaluated together. Multiple blocks: [[0, 2, 4], [1, 3]].

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Experimental Setup (from papers)

All results in both papers use:

• Population size N = 50, reference directions = 50 (Riesz energy method)
• T_max = 300 for 2–3 objective problems, T_max = 400 for 5–8 objective problems
• N_DoE = 100 for 2–3 objective problems, N_DoE = 150 for 5–8 objective problems
• t_S = 30 surrogate generations per iteration
• η = 20, e = -0.5
• 15 independent runs per problem scenario, IGD+ as performance metric
• Kriging with six model configurations (constant/linear/quadratic × ARD/no-ARD), best selected by 3-fold cross-validation MAE

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IGD+ is computed on the feasible non-dominated front of the true population. Both the reference Pareto front and the obtanied front are normalized using the same min/max bounds of the reference Pareto front before IGD+ computation.

Test Problems

The following problems from the papers are supported out of the box via pymoo and AddProblems/:

Constrained multi-objective: C2DTLZ2, TNK, MW3, MW7, MW14
Engineering design: Welded beam (M=2, J=4), Carside impact (M=3, J=10), Disc brake (M=2, J=5)
Unconstrained (degenerate case): ZDT1, ZDT2, ZDT3, DTLZ2
Single-objective constrained (degenerate case): G1, G4, G10

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Citation

If you use this code, please cite both papers:

@article{santoshdebtevc2025,
  author={Santoshkumar, Balija and Deb, Kalyanmoy},
  journal={IEEE Transactions on Evolutionary Computation}, 
  title={Handling Objectives With Heterogeneous Evaluation Times in Surrogate-Assisted Evolutionary Multi-Objective Optimization}, 
  year={2025},
  volume={},
  number={},
  pages={1-1},
  keywords={Optimization;Linear programming;Computational modeling;Measurement;Predictive models;Evolutionary computation;Probabilistic logic;Computational efficiency;Vectors;Uncertainty;Heterogeneous Evaluation Times;Heterogeneous Objectives;Non-uniform Latency;Surrogate-assisted Optimization;Multi-objective Optimization;Machine Learning;Reference Vector Guided Probabilistic Dominance},
  doi={10.1109/TEVC.2025.3560922}}


@article{santoshdeb2026,
  title={Handling of Objectives and Constraints with Heterogeneous Evaluation Times for Surrogate-Assisted Evolutionary Multi- and Many-Objective Optimization},
  author={Santoshkumar, Balija and Deb, Kalyanmoy},
  journal={Swarm and Evolutionary Computation},
  volume={100},
  pages={102260},
  year={2026},
  publisher={Elsevier},
  doi={10.1016/j.swevo.2025.102260}
}

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Contact

Balija Santoshkumar — balijasa@msu.edu and santhoshkumar.balija@gmail.com

COIN Lab, Michigan State University — https://www.coin-lab.org/