Module mixed 

Test functions of continuous and integer variables together: those of R. Hamano, S. Saito, M. Nomura and S. Shirakawa, “CMA-ES with Margin: Lower-Bounding Marginal Probability for Mixed-Integer Black-Box Optimization”, GECCO 2022, https://arxiv.org/abs/2205.13482, §5.1 (p. 7). The first ⌊n/2⌋ variables are continuous and the rest integer (Variable::integer); each function is given the values the optimizer encodes, whole numbers in the integer variables.

Function	Integer variables	Minimum
sphere_int	from −10 to 10	0 at x = 0
ellipsoid_int	from −10 to 10, with the largest coefficients	0 at x = 0
sphere_one_max	0 or 1	0 at continuous 0, integer 1

Functions

ellipsoid_int

EllipsoidInt, f(x) = Σ (1000^(i/(n − 1)) xᵢ)²: ellipsoid, whose coefficients are the same, so the integer variables, the last, have the largest.

sphere_int

SphereInt, f(x) = Σ xᵢ² over every variable: sphere.

sphere_one_max

SphereOneMax, f(x) = Σ xᵢ² + n_b − Σ x_k, the first sum over the first ⌊n/2⌋ variables (continuous) and the second over the n_b others (0 or 1).