Design of Test Problems

Test functions help to evaluate multi-objective optimization algorithms. Toolkits and suites support for constructing tunable test multi-objective problems. Generating tunable methods are helpful for controlling a different kind of complexities as in the K. Deb’s approach. We may be faced with two types of difficulties, which concern the convergence process and the diversity of Pareto-optimal solutions. Firstly, the convergence of the iteration process to a Pareto-optimal front may not be achieved. The reasons can be due to the multi-modality of objective functions, to the presence of a deceptive attractor, or to flat areas surrounding the global optimum. Secondly, the non-diversity of the Pareto-optimal solution should be devoted to geometric anomalies of the Pareto-optimal front, such as convexities or non-convexities, discontinuities, and non-uniform distributed solutions. Generating tunable methods are helpful for controlling a different kind of complexities. K. Deb (1999) suggested the construction of tunable two-objective test problems. The objective functions of such test problems are composed of a particular function for which we know the impacts. A nonlinear multivariate ‘distribution function should affect the diversity in the Pareto-optimal front. A multi-modal ‘distance function’ should disturb the convergence to the true Pareto-optimal front. The convexity or discontinuity in the Pareto-optimal front should be affected by choice of a “shape function.” An interactive and controlled document demonstrates the resolution process of the Kursawe's test function.

Keywords: Benchmark problems, DTLZ test suite, Generating tunable methods, Kursawe`s test function, Non-overlapping arguments, Okabe`s test function, Test suites, Toolkit, WFG test suite, ZDT test suite.