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I am wondering if I minimize a non-smooth convex function, which solver should I choose. I think I should choose a fastest one with a big convergence rate. Subgradient descent is always on the textbook. But it is slow. I am seeking a state of the art optimization algorithm for non-smooth case.

I actually study a general case. E.g., the regression problem $$\|Kx-y\|+|x|$$, where $$x\in\mathbb{R}^d,K\in\mathbb{R}^{n\times d}, y\in \mathbb{R}^n$$. I mean I not only try to solve the regression problem, but also others. So I cannot make it too specific. Got a good studff,http://napsu.karmitsa.fi/nsosoftware/. There are several algorithms for nonsmooth case. Which should I use?

Vivian
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    It depends on a number of things. First of all if your problem is linear or nonlinear. I am not sure what the best algorithm is out there, but if you want to do some research a great person to look at is Carsten Gräser. He has done convex minimisation problems in the linear and nonlinear case. His PhD thesis would probably have some excellent references – Keeran Brabazon Feb 14 '14 at 17:54
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    There is no one right answer here; it depends considerably on the specific function. Perhaps if you offered more details about the functions you are considering, we can offer more specific advice. – Michael Grant Feb 14 '14 at 18:35
  • @KeeranBrabazon, Thank you. I will check it. – Vivian Feb 14 '14 at 18:53
  • @MichaelC.Grant, Thanks. I consider a general case here. Happy weekend. – Vivian Feb 14 '14 at 18:54
  • This can be expressed as a linear program for which there are many good solvers from simplex to interior point... – copper.hat Feb 14 '14 at 19:19
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    Yep. And that's why it's good to get as specific as possible. This is a relatively trivial problem to solve. (I assume you meant $|\cdot|$ instead of $|\cdot|$ above?) – Michael Grant Feb 14 '14 at 19:48
  • @MichaelC.Grant, yup, I correct it. – Vivian Feb 14 '14 at 22:02

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