OMADS 2601.0

Optimization using Mesh Adaptive Direct Search (MADS)

OMADS

OMADS: A Python implementation for the optimization using the mesh adaptive direct search (MADS) method; Optimization using MADS (OMADS). OMADS can solve both single- and multi-objective optimization problems.

For technical and code documentation, please visit OMADS Webpage.

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Version 2.1.0

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License & copyright

© Ahmed H. Bayoumy 2022-2026

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License either version 3 of the License, or (at your option) any later version.

See the LICENSE file for the full license text.

Citation

If you use this code, please cite it as follows.

   @software{OMADS_AB,
   author       = {Bayoumy, A.},
   title        = {OMADS},
   year         = 2022,
   publisher    = {Github},
   version      = {2407},
   url          = {https://github.com/Ahmed-Bayoumy/OMADS}
   }

How to use the OMADS package

After installing the OMADS package from PYPI website, the functions and classes of OMADS basic module can be imported directly to the Python script as follows:

from OMADS import *

How to run OMADS from the terminal

After installing OMADS, SEARCH, POLL, and MADS modules can be called directly from a terminal window under the src directory. The path of the JSON template, which contains the problem input parameters, should be entered as an input argument to the aforementioned modules call.

python ./OMADS/POLL.py ../../tests/unconstrained/rosenbrock.json
python ./OMADS/SEARCH.py ../../tests/unconstrained/rosenbrock.json
python ./OMADS/MADS.py ../../tests/unconstrained/rosenbrock.json

Input parameters

Input parameters are serialized in a JSON template using predefined attributes (keywords) under four dictionaries; evaluator, param, options, and search. Here is a brief description of each dictionary and its key attributes.

• evaluator: in this dictionary, we define the blackbox location and the name of input and output files (if exist)
  • blackbox: blackbox executable file name, or the function name if this is an internal function defined within the BM_suite
  • internal: the name of the testing category that holds your internal/external test function or blackbox evaluator
    • con: internal constrained single-objective function
    • uncon: internal unconstrained single-objective function
    • exe: external executable blackbox evaluator
  • input: the name of the input file (considered if an external executable was defined)
  • output: the name of the output file (considered if an external executable was defined)

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• param: problem setup
  • baseline: this is the initial starting point (initial design vector)
  • lb: lower bounds vector
  • ub: upper bounds vector
  • var_names: list of design variable names
  • var_types: list of variable types
    • R: real variable
    • I: integer variable
    • C_<set name>: categorical variable. A set name from the sets dict should be added after the underscore that follows C
    • D_<set name>: discrete variable. A set name from the sets dict should be added after the underscore that follows D
  • Sets: a dictionary where its keys refer to the set name and their value should be assigned to a list of values (the values can be of heterogeneous type)
  • scaling: scaling factor
  • post_dir: the location of the post directory where the results file shall be saved if requested
  • constraints_type: list of the constraints barrier type, i.e., progressive barrier (PB) and extreme barrier (EB)
  • LAMBDA: list of the initial Lagrangian multipliers assigned to the constraints
  • RHO: list of the initial penalty parameter
  • hmax: the maximum feasibility threshold
  • nobj: number of objectives
  • isPareto: boolean for running a Pareto study
  • meshType: specify whether to use a granular or an orthogonal mesh
    • GMESH: granular mesh
    • OMESH: orthogonal mesh
  • name: problem name (that name will be used for naming the output files)

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• options: algorithmic options
  • seed: the random generator seed that ensures reproducibility of results. This should be an integer value
  • budget: the evaluation budget; the maximum number of evaluations for the blackbox defined
  • tol: the minimum poll size tolerance; the algorithm terminates once the poll size falls below this value
  • psize_init: initial poll size
  • display: a boolean for displaying verbose outputs per iteration in the terminal window
  • opportunistic: a boolean for enabling opportunistic search
  • check_cache: a boolean for checking if the current point is a duplicate by checking its hashed address (integer signature)
  • store_cache: a boolean for saving evaluated designs in the cache memory
  • collect_y: currently inactive (to be used when the code is integrated with the PyADMM MDO module)
  • rich_direction: a boolean that enables capturing a rich set of directions in a generalized pattern
  • precision: a string character input that controls the dtype decimal resolution used by the numerical library numpy
    • high: float128 1e-18
    • medium: float64 1e-15
    • low: float32 1e-8
  • save_results: a boolean for generating a MADS.csv file for the output results under the post directory
  • save_coordinates: saving poll coordinates (spinners) of each iteration in a JASON dictionary template that can be used for visualization
  • save_all_best: a boolean for saving only incumbent solutions
  • parallel_mode: a boolean for parallel computation of the poll set

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• search: the search step options
  • type: search type can take one of the following values
    • VNS: variable neighbor search
    • sampling: sampling search
    • BO: Bayesian optimization (TODO: not published yet, as it is still in the testing and BM phase)
    • NM: Nelder-Mead (TODO: not published ye,t as it is still in the testing and BM phase)
    • PSO: particle swarm optimization (TODO: not published yet as it is in the testing phase)
    • CMA-ES: covariance matrix adaptation evolution strategy (TODO: not published yet, as it is in the testing phase)
  • s_method: can take one of the following values
    • ACTIVE: Online active sampling using kernel density estimators (KDE)
    • LH: Latin Hypercube sampling\
    • RS: random sampling
    • HALTON: Halton sampling
  • ns: number of samples

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Multiobjective MADS

The poll, search, and MADS algorithms are capable of solving both unconstrained and constrained multiobjective problems by conducting a Pareto study. The approximated Pareto front (non-dominated solutions) are saved in a separate CSV file named with the suffix Pareto "{problem_name}_Pareto.out". The figures below display the results of test functions chosen from the literature.

Constrained test functions

Unconstrained test functions

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Benchmarking

To benchmark OMADS, per se, you need to install the non-linear optimization benchmarking project NOBM (will be installed automatically when you install OMADS) from PYPI. Two benchmarking suites are provided under the BMDFO package -- BMDFO stands for benchmarking derivative-free optimization algorithms. The benchmarking suites have different constrained and unconstrained optimization problems with various characteristics. The BMDFO modules can be imported directly to the Python script as shown below:

from BMDFO import toy

For more details about the NOBM project and its use, check this link. After running the benchmarking suite using various seed values, which are used to initialize the random number generator, a BM_report.csv file will be created in the post directory under the examples folder.

Example

import OMADS
import numpy as np

def rosen(x, *argv):
    x = np.asarray(x)
    y = [np.sum(100.0 * (x[1:] - x[:-1] ** 2.0) ** 2.0 + (1 - x[:-1]) ** 2.0,
                axis=0), [0]]
    return y

eval = {"blackbox": rosen}
param = {"baseline": [-2.0, -2.0],
            "lb": [-5, -5],
            "ub": [10, 10],
            "var_names": ["x1", "x2"],
            "scaling": 10.0,
            "post_dir": "./post"}
options = {"seed": 0, "budget": 100000, "tol": 1e-12, "display": True}

data = {"evaluator": eval, "param": param, "options":options}

out = {}
# out is a dictionary that will hold output data of the final solution. The out dictionary has three keys: "xmin", "fmin" and "hmin"

out = OMADS.main(data)

Results

 --- Run Summary ---
 Run completed in 0.0303 seconds
 Random numbers generator's seed 0
 xmin = [1.0, 1.0]
 hmin = 1e-30
 fmin = 0.0
 #bb_eval = 185
 #iteration = 46
 nb_success = 4
 psize = 9.094947017729282e-13
 psize_success = 1.0
 psize_max = 2.0

https://github.com/Ahmed-Bayoumy/OMADS/assets/22842095/5dc72c34-4722-4d93-8c84-d17f5595556d