Generalized semi-infinite optimization problems are optimization problems having infinitely many constraints. In addition, the infinite index set depends on the decision variable of optimization. In this study, as an application of generalized semi-infinite optimization problems a type of design centering problems is considered. In a general design centering problem some measure of a parametrized body is maximized under the constraint that parametrized body is inscribed in a fixed body. In this study, diamond cutting problem is considered as a type of design centering problems. Here the aim is to maximize the volume of a round cut diamond from functional approximation of a rough irregularly shaped gemstone. First of all, the problem is converted to a generalized semi-infinite optimization problem, then corresponding first order optimality conditions are obtained. Several numerical examples are presented by solving reformulated Karush-Kuhn-Tucker optimality conditions with semismooth Newton method. The advantage of the method is that a linear system of equations has to be solved in each iteration.