Within the framework of complex systems design, such as launch vehicles, numerical optimization is an essential tool as it allows to reduce the design process time and costs. The inclusion of discrete variables in the design optimization process allows to extend the applicability of numerical optimization methods to a broader number of systems and sub-systems. In this paper, a recently proposed adaptation of the Efficient Global Optimization method for constrained mixed-variable problems is applied to the design optimization of a launch vehicle thrust frame, which depends on both continuous sizing parameters and discrete variables characterizing the number of structural reinforcements. The Efficient Global Optimization adaptation that is considered is based on a redefinition of the Gaussian Process kernel as a product between a standard continuous kernel and a second kernel representing the covariance between the discrete variable values. From the results obtained on an analytical test-case as well as on the launch vehicle thrust frame design optimization, it is shown that the use of the mixed variable Efficient Global Optimization algorithm allows to converge towards the neigh-borhoods of the problems optima with fewer function evaluations when compared to reference optimization algorithms.