Elastic Net like problem

[raypenper]

Can QSQP solve the Elastic Net like problem? It has the form $$min_{x}||Ax - b||^2_2+\lambda||x||_1+\mu||x^2-y||^2_2$$ where y is a given vector. This problem is basically a lasso problem with L2 regularization, but it has a non-linear term $x^2$ and a reference solution $y$. Anyone know how to solve the problem? Thanks a lot.

[imciner2]

Hi @raypenper, just so I understand better. $y$ is a constant that is given before solving the problem and not an optimization variable? Making this regularization term trying to minimize the difference between $x^2$ and $y$?

[raypenper]

Yes, that's correct. I think the last term is not convex and maybe impossible to solve. But the problem has practical application.