ZDT

ZDT, named after its authors Zitzler, Deb and Thiele, is a benchmark suite of six bi-objective test problems used in evolutionary multi-objective optimization. Introduced around 2000, the ZDT suite is designed to evaluate how well optimization algorithms balance convergence toward a Pareto front with diversity along that front, across a range of front shapes and problem characteristics. The problems share a common structure: a decision vector x = (x1, x2, ..., xn) with most variables constrained to the unit interval, and two objective functions f1 and f2 that are defined through a coupling function g(x) and a helper function h that relates f1 to g. In many formulations, the Pareto front is obtained when g equals 1, yielding a trade-off curve between f1 and f2.

The suite comprises six problems, commonly referred to as ZDT1 through ZDT6. ZDT1 and ZDT2 feature smooth, single-peak fronts with different mathematical forms for f2. ZDT3 is designed to produce a Pareto front consisting of multiple disconnected regions, increasing the challenge for maintaining diversity. ZDT4 introduces a high-dimensional search space with many local optima, creating a deceptive landscape for convergence. ZDT5 and ZDT6 broaden the variety by incorporating different variable types and front geometries, testing robustness to discrete and mixed-variable settings and producing fronts that are non-uniform or non-convex.

ZDT benchmarks are widely used to compare evolutionary algorithms such as NSGA-II, SPEA2, and MOEA/D. They provide standardized reference problems for assessing convergence speed, diversity preservation, and scalability with respect to the number of decision variables. References typically cite the original ZDT 2000 paper by Zitzler, Deb, and Thiele.