zdt2

ZDT2 is a bi-objective benchmark problem used in multi-objective optimization to evaluate the performance of evolutionary algorithms. It is part of the ZDT suite introduced by Zitzler, Deb, and Thiele to provide a simple yet informative set of test problems with known Pareto-optimal fronts.

The decision vector x = (x1, x2, ..., xn) consists of n decision variables with each xi in the interval [0, 1]. A common choice is n = 30, though any n ≥ 2 is allowed. The objective functions are defined as follows:

f1(x) = x1

g(x) = 1 + 9 * (sum_{i=2}^n xi) / (n - 1)

f2(x) = g(x) * (1 - (f1(x) / g(x))^2)

The Pareto-optimal front of ZDT2 is obtained when g(x) = 1, which occurs when all xi for i ≥ 2 are zero. In that case, the front is the curve f2 = 1 − f1^2 for f1 in [0, 1]. This yields a smooth, curved front in the objective space.

Characteristics and purpose: ZDT2 is designed to test an algorithm’s ability to handle non-linear, non-convex front shapes and to balance convergence toward the front with diversity along a curved front. The g function introduces a separation between the decision variables and the Pareto front, allowing researchers to study how search strategies manage deception and scaling as the influence of the non-first variables grows.

Usage: ZDT2 is widely used in benchmarking MOEAs (multi-objective evolutionary algorithms) such as NSGA-II, SPEA2, and related methods. It provides a standard reference point for comparing convergence speed and spread of the obtained solutions.