GLODS-SI

Scale-Invariant Global-Local Direct Search for Engineering Design Optimization

GLODS-SI is a derivative-free direct-search method for bound-constrained global optimization problems whose decision variables span widely different physical scales. It extends the original GLODS framework (Custódio & Madeira, 2015) by performing all geometric operations — polling, distance computation, point merging — in normalized coordinates y ∈ [0, 1]^n, while objective evaluations remain in the original variable space x ∈ [ℓ, u]. This two-space formulation makes the method's behaviour invariant to the units chosen for each variable and robust to heterogeneous variable scales.

This repository contains the reference MATLAB implementation that accompanies the paper:

J. F. A. Madeira, GLODS-SI: Scale-Invariant Global-Local Direct Search for Engineering Design Optimization, Journal of Computational Design and Engineering, 2026, qwag049. Open Access — DOI: 10.1093/jcde/qwag049.

The numerical results that produced Tables 3 and Figures 3–6 of the paper are provided as final artifacts in results/.

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Repository contents

GLODS_SI/
├── README.md                  (this file)
├── LICENSE                    (GNU LGPL v3)
├── CITATION.cff               (citation metadata)
├── glods_si.m                 (main solver)
├── parameters_glods_si.m      (default parameters; matches paper Section 4)
├── driver_glods_si.m          (example driver)
├── aluffi_pentini_2D.m        (illustrative test problem, 2D)
└── results/                   (numerical results that produced the paper's
                               tables and figures; see results/README.md)

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Installation

Requirements: MATLAB R2018b or later (no toolboxes required).

1. Clone or download this repository.
2. Add the repository folder to your MATLAB path:

   addpath('path/to/GLODS_SI');

That's it.

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Quick start

Run the bundled demo on the 2D Aluffi-Pentini problem:

cd GLODS_SI
driver_glods_si

The driver runs GLODS-SI with the default parameters listed in parameters_glods_si.m and reports:

• objective value reached;
• best point in work-space coordinates;
• total number of function evaluations;
• a convergence profile (best-so-far value vs. function evaluations).

Aluffi-Pentini is a 2D problem with a single global minimum at x* = (-1.0465, 0) and f* = -0.3523, plus two local minima. GLODS-SI typically reaches the global minimum within a few hundred function evaluations.

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Using GLODS-SI on your own problem

GLODS-SI expects:

• a function handle @my_objective that accepts a column vector x ∈ R^n and returns a scalar f;
• bound vectors lb, ub of size n × 1.

The solver is then called as:

[profile, Plist, flist, alfa, radius, fevals] = ...
    glods_si(@my_objective, [], [], lb, ub);

Outputs:

Output	Meaning
profile	Best-so-far objective value vs. function evaluations
Plist	Final list of points returned by the solver
flist	Objective values at the points in Plist
alfa	Final step sizes (one per point in Plist)
radius	Final comparison radii
fevals	Total number of function evaluations performed

For benchmarking against the test suite used in the paper, see the companion repository DFO_Benchmark_Suite, which provides 504 self-contained wrappers (63 instances × 8 scaling strategies) ready to be passed as the first argument to glods_si.

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Default parameters

The default values in parameters_glods_si.m match the convention used in the experimental section of the paper (Section 4):

Parameter	Value	Meaning
alfa_ini	0.1	Initial step size in normalized space
radius_ini	0.2	Initial comparison radius (= 2·alfa_ini)
tol_stop	1e-5	Stopping tolerance
max_fevals	20000	Maximum number of function evaluations
nPini	30	Number of initial sample points
list	6	Initialization: Sobol sequences
suf_decrease	0	Integer-lattice sufficient decrease
cache	1	Cache previously evaluated points

Two changes relative to the original GLODS defaults are worth noting:

• alfa_ini and radius_ini are dimensionless (fractions of each variable range), since geometric operations occur in [0, 1]^n.
• radius_ini = 2 · alfa_ini = 0.2 is the convention used to generate Table 3 of the paper. It satisfies r_0 ≥ d_max · alfa_ini (with d_max = 1 for the positive basis [I -I]), preventing nearby initial points from being merged immediately and favouring exploration of the global structure.

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Numerical results

The results/ folder contains the numerical results that produced the tables and figures of the accompanying paper, organized into three sub-folders:

Main comparison (3-way)

results/GLODS_vs_NOMAD_vs_GLODSSI/ contains the joint 3-algorithm data profiles, including the figures used in the manuscript:

• Figure 3 — baseline scaling (κ = 1)
• Figure 4 — extreme scaling (κ = 10⁸)
• Figure 5 — Halton oscillatory scaling (κ = 10⁶)
• Figure 6 — spatial–thermal scaling (κ ≈ 9 × 10⁴)

Profiles for the four scaling strategies not shown in the paper are also provided, together with the auto-generated LaTeX summary table.

Pairwise comparisons

• results/GLODS_vs_GLODSSI/ — source of the GLODS and GLODS-SI columns of Table 3.
• results/NOMAD_vs_GLODSSI/ — source of the NOMAD column of Table 3.

Each sub-folder contains, for the eight scaling strategies considered in the paper: PDF data profiles, ASCII summary tables, and the auto-generated LaTeX success-rate tables. See results/README.md for the full description.

These files are final artifacts and can be inspected directly without rerunning the experiments.

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License

This software is distributed under the GNU Lesser General Public License version 3 (LGPL-3.0-or-later). This license is inherited from the original GLODS framework (Custódio & Madeira, 2015), on which GLODS-SI is based.

See LICENSE for the full license text.

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Citation

If you use GLODS-SI in academic work, please cite the accompanying paper:

@article{Madeira2026GLODSSI,
  author  = {Madeira, J. F. A.},
  title   = {{GLODS-SI}: Scale-Invariant {Global--Local} Direct Search
             for Engineering Design Optimization},
  journal = {Journal of Computational Design and Engineering},
  year    = {2026},
  doi     = {10.1093/jcde/qwag049},
  note    = {Article qwag049, Open Access}
}

A CITATION.cff file is also provided for tools that consume that metadata format (GitHub, Zenodo, etc.).

Underlying framework

Custódio, A. L., Madeira, J. F. A. (2015). GLODS: Global and Local Optimization using Direct Search. Journal of Global Optimization, 62, 1–28. doi:10.1007/s10898-014-0224-9

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Acknowledgments

This work was supported by Fundação para a Ciência e a Tecnologia (FCT) through LAETA (project UID/50022/2025).

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Contact

J. F. A. Madeira IDMEC, Instituto Superior Técnico, Universidade de Lisboa ISEL, Instituto Politécnico de Lisboa Email: aguilarmadeira@tecnico.ulisboa.pt ORCID: 0000-0001-9523-3808