Mathematical modelling of UMDAc algorithm with tournament selection. Behaviour on linear and quadratic functions

This paper presents a theoretical study of the behaviour of the Univariate Marginal Distribution Algorithm for continuous domains (UMDAc) in dimension n. To this end, the algorithm with tournament selection is modelled mathematically, assuming an infinite number of tournaments. The mathematical model is then used to study the algorithm’s behaviour in the minimization of linear functions L(x) = a0 + ∑ n i=1 aixi and quadratic function Q(x) = ∑ n i=1 x2i , with x = (x1, . . . , xn) ∈ IR n and ai ∈ IR, i = 0, 1, . . . , n. Linear functions are used to model the algorithm when far from the optimum, while quadratic function is used to analyze the algorithm when near the optimum. The analysis shows that the algorithm performs poorly in the linear function L1(x) = ∑ n i=1 xi. In the case of quadratic function Q(x) the algorithm’s behaviour was analyzed for certain particular dimensions. After taking into account some simplifications we can conclude that when the algorithm starts near the optimum, UMDAc is able to reach it. Moreover the speed of convergence to the optimum decreases as the dimension increases.