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A Literature Survey of Benchmark Functions For Global Optimization Problems
Momin Jamil, Xin-She Yang
TL;DR
Because no agreed benchmark set exists, the paper reviews and compiles unconstrained test functions with diverse landscape properties for algorithm validation and comparison. It presents 175 benchmark problems and concludes that this comprehensive collection can support future testing of optimization algorithms, while acknowledging possible omissions.
Problem
The literature lacks an agreed, diverse, and unbiased benchmark set for validating and comparing optimization algorithms.
Method
The paper reviews available literature and compiles unconstrained benchmark functions classified by properties including modality, separability, and valley-related landscape features.
Results
175 unconstrained optimization test problems are presented as a comprehensive collection for validating optimization algorithms.
Takeaways & Limitations
The collection is intended to provide a more complete view of algorithm performance in future validation and comparison studies.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper argues that optimization algorithms require broad, unbiased benchmark suites because small or specialized test sets cannot characterize where algorithms perform well. It therefore reviews scattered literature and collections to compile a comprehensive set of unconstrained test problems with diverse landscapes.
- Common studies use only a few to about two dozen test functions, despite the need for diverse and unbiased evaluation.
- Specialized problem sets cannot reliably characterize which problem types favor one optimization algorithm over another.
- Benchmark functions are dispersed across textbooks, research articles, websites, and existing online collections, making comprehensive retrieval cumbersome.
- The survey focuses on modality and separability because multimodality, flat surfaces, narrow optima, and scaling differences create distinct search difficulties.
2 Characteristics of Test Functions
The paper characterizes benchmark functions through optimization objectives and landscape properties that affect search difficulty. It emphasizes modality, basins, valleys, separability, and dimensionality as key dimensions for understanding algorithm behavior.
- Global optimization seeks the best solution in a search space according to objective functions, commonly formulated as minimization problems.
- Optimization problems may have multiple or infinitely many optimal solutions and can vary across continuity, convexity, modality, and separability.
- Benchmark functions can be classified by modality, basins, valleys, separability, and dimensionality to relate landscape features to search difficulty.
- Multiple ambiguous peaks can trap algorithms away from the true optimum, while basins and valleys can attract searches and slow progress.
- The paper treats partially separable functions as separable functions.
- Separable functions are generally easier because independent variables can be optimized separately, whereas non-separable functions contain interrelated variables.
- Higher dimensionality expands the search space exponentially and can become a major barrier for highly nonlinear problems.
3 Benchmark Test Functions for Global Optimization
The paper presents 175 unconstrained optimization test problems for validating optimization algorithms, spanning varied structural properties and difficulty levels. The collection includes functions differing in modality, separability, scalability, differentiability, domains, and known optima.
- Collection scope: 175 unconstrained optimization test problems are presented for validating optimization algorithms.The collection specifies dimensions, variable bounds, and optimal solutions, although some optima are known only numerically.
- Problem specifications: The benchmarks provide diverse domains and optimum specifications, including bounded search regions and analytically or numerically reported global minima.Examples report domains such as −10 ≤xi ≤10 and optima including the origin, multiple points, or approximate objective values.
- Benchmark characteristics: The listed functions vary in continuity, differentiability, scalability, modality, and separability.Examples include scalable and non-scalable functions, differentiable and non-differentiable functions, and both unimodal and multimodal benchmarks.
- Landscape properties: The benchmarks cover unimodal and multimodal landscapes, including functions with multiple local or global minima.Multimodal functions are used to test whether algorithms can escape local minima; Bukin functions are described as especially difficult because of fine seesaw edges near minima.
4 Conclusions
The paper argues that benchmark test functions are important for validating and comparing optimization algorithms, especially newly developed methods. It offers a comprehensive literature-based list while acknowledging that some functions may have been missed.
- Benchmark test functions are important for validating and comparing optimization algorithms, especially newly developed methods.
- The paper aims to provide the most comprehensive list of known benchmark or test functions available from the authors’ surveyed literature.
- The authors expect that using the full collection can provide a more complete view of algorithm performance.