A Literature Survey of Benchmark Functions For Global Optimization Problems (1308.4008v1)

Abstract: Test functions are important to validate and compare the performance of optimization algorithms. There have been many test or benchmark functions reported in the literature; however, there is no standard list or set of benchmark functions. Ideally, test functions should have diverse properties so that can be truly useful to test new algorithms in an unbiased way. For this purpose, we have reviewed and compiled a rich set of 175 benchmark functions for unconstrained optimization problems with diverse properties in terms of modality, separability, and valley landscape. This is by far the most complete set of functions so far in the literature, and tt can be expected this complete set of functions can be used for validation of new optimization in the future.

• The paper systematically compiles 175 benchmark functions, providing an extensive resource for testing optimization algorithms.
• The paper analyzes functions with varied properties such as modality and separability to ensure rigorous evaluation.
• The paper bridges theoretical and practical challenges by contrasting synthetic benchmarks with real-world problem tests.

Overview of Benchmark Functions for Global Optimization

In the paper titled "A Literature Survey of Benchmark Functions for Global Optimization Problems" by Momin Jamil and Xin-She Yang, a comprehensive set of 175 benchmark functions for unconstrained optimization problems is meticulously compiled and analyzed. This extensive repertoire is invaluable for the robust validation and comparison of global optimization algorithms. These benchmark functions, crucial to the optimization community, present diverse characteristics including modality, separability, and landscape complexity, aimed at evaluating new optimization algorithms in varied and challenging scenarios.

Key Aspects and Contributions

1. Comprehensive Compilation:
   • The authors have systematically gathered and documented 175 benchmark functions, providing the most extensive collection known to date. This compilation spans functions with varying landscape properties from unimodal to multimodal, separable to non-separable, and low-dimensional to high-dimensional.
2. Diverse Properties for Fair Evaluation:
   • The benchmarks encompass a wide range of properties to ensure unbiased evaluation of optimization algorithms. They include:
     • Modality: Functions classified as unimodal or multimodal to test an algorithm's ability to escape local minima.
     • Separable and Non-separable: Tests the independence of variables and interactions among variables.
     • Valley and Basin Landscapes: Aimed to evaluate how algorithms handle narrow valleys and broad basins where search processes can be misdirected or slowed.
3. Real-world Problems versus Test Functions:
   • The paper distinguishes between synthetic test problems and real-world problems, emphasizing that while the former are essential for structural testing and controlled manipulation, the latter are indispensable for practical applicability. The authors highlight the necessity of addressing both for a holistic assessment of an algorithm’s performance.
4. Historical and Online Sources:
   • The survey aggregates functions from a plethora of sources including textbooks, research articles, and online repositories such as the GLOBAL library and CUTE collection. This breadth ensures the collection’s thoroughness and accessibility.
5. Illustrative Examples:
   • The paper provides detailed descriptions and mathematical formulations for each function. For instance, the Ackley function is defined as:

     \( f(x) = -20 \exp\left(-0.2 \sqrt{\frac{1}{D} \sum_{i=1}^D x_i^2}\right) - \exp\left(\frac{1}{D} \sum_{i=1}^D \cos(2 \pi x_i)\right) + 20 + e \)

     subject to $-35 \leq x_i \leq 35$.

Implications and Future Directions

The implications of this work are both practical and theoretical:

• Practical Implications:
  • This extensive collection provides a fundamental resource for developers and researchers of new optimization algorithms, ensuring that these algorithms undergo rigorous and varied testing. Researchers can benchmark their methods against a wider range of scenarios, aiding in the identification of strengths and weaknesses.
• Theoretical Implications:
  • The diverse properties of the collected functions facilitate the theoretical analysis of algorithms with respect to different optimization landscapes. Understanding an algorithm’s performance over these varied test cases can lead to the development of more robust and adaptive optimization strategies.

Potential Developments in AI and Optimization

Looking forward, several developments can be anticipated:

1. Automated Algorithm Design:
   • With a comprehensive suite of benchmark functions, automated or semi-automated approaches could be developed to fine-tune optimization algorithms for specific problem landscapes, leading to custom-built and highly efficient optimization strategies.
2. Hybrid Optimization Methods:
   • The performance data collected from these benchmarks could inform the development of hybrid optimization methods that combine the strengths of various algorithms to handle different types of functions effectively.
3. New Test Function Proposals:
   • As new optimization challenges emerge, especially from fields like machine learning and deep learning, there will be a continuous need for the introduction of new test functions. This collection lays the groundwork for such developments, providing a template for the systematic evaluation of future functions.
4. Comprehensive Databases and Community Contributions:
   • Establishing online, continually updated databases where researchers can contribute new benchmark functions and share performance results can foster collaboration and accelerate advancements in global optimization research.

By consolidating this extensive set of benchmark functions, the paper stands as a critical resource in the ongoing pursuit of advancing optimization algorithms. This work ensures that future algorithms are subjected to rigorous, diverse, and comprehensive evaluation, thus fostering the development of truly robust optimization methodologies.