Search Results
Search type | Search syntax |
---|---|
Tags | [tag] |
Exact | "words here" |
Author |
user:1234 user:me (yours) |
Score |
score:3 (3+) score:0 (none) |
Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
Views | views:250 |
Code | code:"if (foo != bar)" |
Sections |
title:apples body:"apples oranges" |
URL | url:"*.example.com" |
Saves | in:saves |
Status |
closed:yes duplicate:no migrated:no wiki:no |
Types |
is:question is:answer |
Exclude |
-[tag] -apples |
For more details on advanced search visit our help page |
1
vote
Envelopes of functions with respect to some convex cone $\mathcal{F}$
If you switch from your perspective of functions to their epigraph, I think that you end up with a "closure operator", see Wikipedia.
I am not sure about other examples, but if $\mathcal{F}$ consists …
1
vote
Accepted
Directional derivates and unique subgradients
You can take $X = \ell^\infty$ (actually $\ell^p$ with $p > 2$ should work) and define
$$
f(x) = \frac12 \| x \|_{\ell^2}^2.
$$
Note that $f$ equals $\infty$ on $\ell^\infty \setminus \ell^2$. One can …
2
votes
Accepted
When are infimal convolutions contractions?
Here are just some thoughts. I think it is a matter of curvature, so let us assume that $\varphi$ and $\psi$ are smooth. Then, $y(x)$ solves the optimality condition
$$
\psi'(y(x)) = \varphi'(x - y(x) …
0
votes
Optimality condition for strongly convex function under sparsity constraint
I don't think that this is true. Let us take $p = 2$, $s = 1$ and $f(x) = \frac12 \|x - (1,1)\|^2$. Then, $\theta_0 = (1,0)$ is a minimizer, but with $\theta = (0,1)$ we get
$$
\nabla f(\theta_0)^\top …
2
votes
Accepted
Lipschitz smooth convex extension
Such an extension is not always possible, a counterexample can be found in Section 2 of https://arxiv.org/abs/1812.02419v3.
1
vote
Accepted
Is this notion of being "fully" convex closed under set addition?
Have a look at the example from https://mathoverflow.net/a/37683/32507. This gives two convex, closed sets which cannot be separated by linear functionals (even not by discontinuous ones). If I unders …
2
votes
Accepted
A converse question about the polyhedrality under linear mapping
I think we can argue as in https://mathoverflow.net/a/423284/32507 to answer the question in the affirmative.
Let $\mathcal R_K(x)$ be the radial cone of $K$ at $x$ (as defined in the other answer). F …
0
votes
On faces of polytopes
The set $K_A$ is essentially a polar of $A$. Indeed, we have
$$
A = \{ x \in \mathbb R^n \mid l(x) \ge t \; \forall (l,t) \in K_A\} =: B.$$
The inclusion "$\subset$" is clear and in order to check "$\ …
1
vote
Is a Lipschitz continuous gradient equivalent to this condition?
Yes, the converse is also true. This follows from the answer in https://math.stackexchange.com/questions/4227159/characterization-of-lipschitz-derivative.
In fact, your condition yields
$$
| (\nabla f …
3
votes
Accepted
Subgradient in a predual under weak* continuity
Finally, I was able to cook up a counterexample. We choose $X = c_0$ (zero sequences equipped with supremum norm). Thus, the dual spaces are (isometric to) $X^* = \ell^1$ and $X^{**} = \ell^\infty$.
W …
1
vote
Accepted
When is a convex function continuous on its domain?
I don't think that this is true.
Let us take
$$
C := \{ x \in \mathbb R^2 \mid x_1^2 \le x_2 \le 1\}$$
and
$$
f(x) = \frac{x_1^2}{x_2}
$$
for $x \in C \setminus \{0\}$, $f(0,0) = 0$.
This function is …
2
votes
A strange variant of the Gaussian log-Sobolev inequality
Here is a simple proof that
$$\frac1\lambda \mapsto \frac1\lambda \, \log \int \exp(\lambda \, \phi(x)) \, \mathrm d \gamma(x)$$
is convex.
This does not need any assumptions on $\phi$ or $\gamma$.
Ma …
1
vote
Second order necessary and sufficient conditions for convex nonsmooth optimization
For convex optimization problems, you do not need second-order conditions, because already the optimality conditions of first order characterize global optimality.
4
votes
Characterization of convex functions
This has been shown by Dudley, see https://www.jstor.org/stable/24490947.
2
votes
Accepted
Generalization of standard convex problem
Let me sketch a proof that the KKT conditions imply global optimality in the case that the objective $f$ and $S$ is convex. No constraint qualification is needed.
Let us assume that the KKT condition …