In 2019, Yoshida et al. developed tropical Principal Component Analysis (PCA), that is, an analogue of the classical PCA in the setting of tropical geometry and applied it to visualize a set of gene trees over a space of phylogenetic trees which is an union of lower dimensional polyhedral cones in an Euclidean space with its dimension $m(m-1)/2$ where $m$ is the number of leaves. In this paper, we introduce a projected gradient descent method to estimate the tropical principal polytope over the space of phylogenetic trees and we apply it to apicomplexa dataset. With computational experiment against Markov Chain Monte Carlo (MCMC) samplers, we show that our projected gradient descent has a lower sum of tropical distances between observations and their projections on an estimated best-fit tropical polytope compared with the MCMC approach proposed by Page et al.~in 2020.