How to solve a set of $L_1$ quadratic problems with a simple constraint efficiently?

Asked Feb 20 '18 at 13:28

Active Feb 20 '18 at 13:35

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I have a set of problem each in the following form except with different $i$ in the constraint:

$$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$ $$s.t. \quad x_i = a$$

where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive definite, and $\lambda\in\mathbf{R}$ is positive?

How do you suggest to solve this set of problems? It is essentially this problem with just a constraint $x_i = a$.

edited Feb 20 '18 at 13:35

Parcly Taxel

asked Feb 20 '18 at 13:28

user85361

2

Do not post the same question twice (even with minor edits). – Parcly Taxel Feb 20 '18 at 13:29
Oh, sorry. I didn't see that I submitted twice. What should I do? – user85361 Feb 20 '18 at 14:13
Delete the other question! – Parcly Taxel Feb 20 '18 at 14:14
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It is the same as that problem. If you substitute $x_i=a$ into the objective, you can construct a reduced $\tilde{A}$ and $\tilde{b}$. – Michael Grant Feb 20 '18 at 15:52
@MichaelGrant, what about solving all of the problems efficiently? Can't we use the solution of a problem to obtain the solution of others? – user85361 Feb 20 '18 at 16:09
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I doubt there's an easy answer here. You might get some savings by using warm start from one solution to another. – Michael Grant Feb 20 '18 at 17:08
Thank you @MichaelGrant – user85361 Feb 21 '18 at 06:48

0 Answers0