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Is there a notion of "best" convex relaxation of a particular function in a normed vector space? For example, the $\ell^0$ pseudo-norm can be relaxed into the $\ell^1$ norm, and that allows us to solve sparse recovery problems. Is there a general recipe to construct such relaxations given an arbitrary function?

Specifically, I have a function of the form $f(x) = \|x\|_2\|x\|_1$, which is non convex. How would I go about relaxing this? Any resources/links I should look into?

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is there any other constraints? otherwise, ofcourse the solution is zero – dineshdileep Jan 4 '13 at 10:07
Actually, are you sure $\|x\|_2\|x\|_1$ is nonconvex? – Rahul Jul 31 '15 at 18:59

A function is convex if and only if its epigraph $\big\{(\mathbf x,z):z\ge f(\mathbf x)\big\}$ is a convex set. So if your function is nonconvex, take the function corresponding to the convex hull of its epigraph. Other equivalent ways of describing this:

  • Let $\hat f$ be the largest convex function dominated by $f$. $$\hat f(\mathbf x) = \max\big\{g(\mathbf x):\text{$g$ is convex and $g(\mathbf y)\le f(\mathbf y)$ for all $\mathbf y$}\big\}.$$
  • Let $\hat f$ be the maximum of all affine functions dominated by $f$. $$\hat f(\mathbf x) = \max\big\{\mathbf a\cdot\mathbf x + b:\text{$\mathbf a\cdot\mathbf y+b\le f(\mathbf y)$ for all $\mathbf y$}\big\}.$$
  • Let $\hat f(\mathbf x)$ be the smallest $\sum\alpha_i f(\mathbf x_i)$ over all possible convex combinations $\sum\alpha_i\mathbf x_i=\mathbf x$. $$\hat f(\mathbf x) = \textstyle\min\big\{\sum\alpha_i f(\mathbf x_i):\alpha_i\ge0,\sum\alpha_i=1,\sum\alpha_i\mathbf x_i = \mathbf x\big\}.$$

(I'm being sloppy and assuming that the $\max$/$\min$ are attained and we don't need $\sup$/$\inf$ instead; I might be wrong.)

I haven't given any thought to your specific case of $f(\mathbf x)=\lVert\mathbf x\rVert_2\lVert\mathbf x\rVert_1$ yet. If I think of something, I'll edit it into this answer. Surely the symmetry of the function should help a lot here.

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$\min_{z,x}~z^2,~s.t.~||x||_2\leq z,~||x||_1\leq z$, is this problem convex? – dineshdileep Jan 4 '13 at 10:09
@dinesh: I guess so. The minimum is at $x=0, z=0$. But why do you ask? – Rahul Jan 4 '13 at 10:11
adding other constraints which the OP might have, do you think it might be an convex relaxation for the problem he asked for? For instance, $\min_{z_1,z_2,x}~z_1z_2,~s.t.~||x||_2\leq z_1,~||x||_1\leq z_2$, this should exactly solve OP's question along with other constraints. I just put another restriction $z_1=z_2$, thus it should give an upper bound. – dineshdileep Jan 4 '13 at 10:15
But the problem is not to minimize $f$. The problem is to find a convex function that best approximates $f$. – Rahul Jan 4 '13 at 10:22

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