Projection onto an ellipsoid

[Nghĩa Nguyễn Trung]

Dear group,

I'm not sure whether my discussion suits our group, but if it's not, please let me know so I can remove this.

I'm trying to calculate the orthogonal projection of a given point onto an ellipsoid of the form E = {x \in \mathbb R^n: x^TQx <= 1} where Q is a symmetric positive definite matrix. At first, I tried some fast algorithms such as Lin-Han's alg. or Dai's Alg. (https://doi.org/10.1016/j.cam.2017.01.008). They work well with single numerical experiments, but when I embedded the function into my code, the projecting algorithms failed to converge.

While trying to fix this issue, I also want to implement this calculation as a QCQP (quadratically constrained quadratic programming). Specifically, I want to solve the problem: min 0.5*|| x - a ||^2, subject to: x^TQx <= 1. While I can use QP solvers in CVXOPT, I have no idea about QCQP. Could anyone help me with my problem?

Thank you so much, I appreciate your consideration.