All Questions
12 questions with no upvoted or accepted answers
9
votes
0
answers
978
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
- 91
7
votes
0
answers
249
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
4
votes
0
answers
481
views
Generalized Jensen's inequality for positively homogeneous functions
The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
- 519
3
votes
0
answers
63
views
Is the collection of Schur convex functions sequentially compact?
We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of ...
- 185
1
vote
0
answers
79
views
Conditions on triangle inequality for integral kernel
Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$.
Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$
A(t,v)=\int_0^{1/v}L(1/t,s)ds,
$$
which is decreasing with $v$ and ...
- 343
1
vote
0
answers
303
views
Continuity of the Legendre transform of a Lipschitz function
Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
- 31
1
vote
0
answers
78
views
Potential for a Monotone Operator
[Cross-posted from math.stackexchange]
I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The ...
- 291
1
vote
0
answers
389
views
The perturbation of a convex function can also be convex?
$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
- 263
1
vote
0
answers
123
views
Generalization of concave envelope
Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
- 1,835
1
vote
0
answers
145
views
convergence of supergradient
Let $\{g_n\}$ be a sequence of concave functions defined on $\mathbb{R}$ and set
$$\lambda_n(x)=\lim_{\Delta x\to 0+}\frac{g_n(x+\Delta x)-g_n(x)}{\Delta x}$$
Assume there exists a concave function ...
- 1,835
1
vote
0
answers
217
views
convergence of concave envelope
Let $\{f_n\}$ be a sequence of uniformly upper bounded functions defined on $\mathbb{R}$ s.t. for every $x\in\mathbb{R}$
$$f_n(x)\to f(x),~ n\to\infty$$
Define $g_n$ and $g$ as the concave envelope ...
- 1,835
0
votes
0
answers
80
views
Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
- 85