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For any matrix, $X$, $\lambda>0$ and $j$ vectors, $v_1$, $v_2$, ..., $v_k$. We want to solve the following optimization problem. $$\textrm{max}_vv^TX^TXv\textrm{ s.t. }\|v\|_2\leq1, \|v\|_1\leq \lambda\textrm{ and }v\perp v_1, ..., v_k$$ Some Thoughts
Optimization of PCA can be converted into a convex programming problem by tricks introduced in this link. Therefore, we can view it as a convex programming with absolute values in contraints.
Relevant Work
If the optimization were the following, $$\textrm{min}_vv^TX^TXv+\lambda\|v\|_1\textrm{ s.t. }v\perp v_1, ..., v_k$$ we could solve it by LASSO with linear constraint. The reference work is here

Yanqi Huang
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  • What's your question? – Sycorax Sep 04 '20 at 02:47
  • I want to solve the above optimization problem. I think it can be categorized as a convex programming problems but my knowledge is limited. I did some research and found that if we were to minimize the objective and remove the norm constraint. It can be converted to a LASSO problem with linear constraint. – Yanqi Huang Sep 04 '20 at 02:52

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