Benchmarking for Metaheuristic Black-Box Optimization: Perspectives and Open Challenges

Research on new optimization algorithms is often funded based on the motivation that such algorithms might improve the capabilities to deal with real-world and industrially relevant optimization challenges. Besides a huge variety of different evolutionary and metaheuristic optimization algorithms, also a large number of test problems and benchmark suites have been developed and used for comparative assessments of algorithms, in the context of global, continuous, and black-box optimization. For many of the commonly used synthetic benchmark problems or artificial fitness landscapes, there are however, no methods available, to relate the resulting algorithm performance assessments to technologically relevant real-world optimization problems, or vice versa. Also, from a theoretical perspective, many of the commonly used benchmark problems and approaches have little to no generalization value. Based on a mini-review of publications with critical comments, advice, and new approaches, this communication aims to give a constructive perspective on several open challenges and prospective research directions related to systematic and generalizable benchmarking for black-box optimization.

Global optimization test problems based on random field composition

The development and identification of effective optimization algorithms for non-convex real-world problems is a challenge in global optimization. Because theoretical performance analysis is difficult, and problems based on models of real-world systems are often computationally expensive, several artificial performance test problems and test function generators have been proposed for empirical comparative assessment and analysis of metaheuristic optimization algorithms. These test problems however often lack the complex function structures and forthcoming difficulties that can appear in real-world problems. This communication presents a method to systematically build test problems with various types and degrees of difficulty. By weighted composition of parameterized random fields, challenging test functions with tunable function features such as, variance contribution distribution, interaction order, and nonlinearity can be constructed. The method is described, and its applicability to optimization performance analysis is described by means of a few basic examples. The method aims to set a step forward in the systematic generation of global optimization test problems, which could lead to a better understanding of the performance of optimization algorithms on problem types with particular characteristics. On request an introductive MATLAB implementation of a test function generator based on the presented method is available.

SQG-Differential Evolution for difficult optimization problems under a tight function evaluation budget

In the context of industrial engineering, it is important to integrate efficient computational optimization methods in the product development process. Some of the most challenging simulation-based engineering design optimization problems are characterized by: a large number of design variables, the absence of analytical gradients, highly non-linear objectives and a limited function evaluation budget. Although a huge variety of different optimization algorithms is available, the development and selection of efficient algorithms for problems with these industrial relevant characteristics, remains a challenge. In this communication, a hybrid variant of Differential Evolution (DE) is introduced which combines aspects of Stochastic Quasi-Gradient (SQG) methods within the framework of DE, in order to improve optimization efficiency on problems with the previously mentioned characteristics. The performance of the resulting derivative-free algorithm is compared with other state-of-the-art DE variants on 25 commonly used benchmark functions, under tight function evaluation budget constraints of 1000 evaluations. The experimental results indicate that the new algorithm performs excellent on the 'difficult' (high dimensional, multi-modal, inseparable) test functions. The operations used in the proposed mutation scheme, are computationally inexpensive, and can be easily implemented in existing differential evolution variants or other population-based optimization algorithms by a few lines of program code as an non-invasive optional setting. Besides the applicability of the presented algorithm by itself, the described concepts can serve as a useful and interesting addition to the algorithmic operators in the frameworks of heuristics and evolutionary optimization and computing.