The convex-analysis tag has no usage guidance.
2
votes
1answer
47 views
Given the following two assumptions, How to conclude $\lim_{k \rightarrow \infty}|\nabla g_r(y^{k})|=0 \ $?
Please refer attached 6-page short paper for details.
Let $M(y)=y+2r\nabla g_r(y)$.
Given $\ g_r(M(y^k))>g_r(y^k)+r|\nabla g_r(y^k)|^2, \forall k. \ $ (equation 36 in the paper)
and $ \ \lim_{k ...
1
vote
0answers
33 views
Convexity on large scales
Has the following concept ever been studied/have a standard name?
Let $f:\mathbb{R}\to \mathbb{R}$ be continuous. We say $f$ is (mid-point) convex on large scales provided
$$f\left( \frac{x+y}{2}\...
1
vote
0answers
53 views
Solve simple stochastic variational inequality
Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\...
0
votes
0answers
19 views
Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
1
vote
1answer
101 views
Characterization of state spaces of Boolean algebras
A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces ...
0
votes
0answers
28 views
Minkowski Sum of duals
I'm really struggling to prove the following statement:
Let $\mathbb{E}$ be an Euclidean space, let $K,K_p,S\subseteq\mathbb{E}$ be a proper cone, a polyhedral cone and a subspace, respectively. If ...
1
vote
1answer
63 views
A property on some unbounded metric spaces
Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property:
$\forall x, \forall y \in X, \exists \{z_n\}$ such that $\lim_{n\to +\infty } d(x,z_n)=+\infty$ and $\...
1
vote
0answers
22 views
Reference Request: Differentiability of Moreau Envelope
I've recently come across many results discussing the differentiation of the Moreau envelope defined by
$$
e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z)
,
$$
where $f$ is a convex functional on a ...
4
votes
1answer
96 views
a compact set with nonempty convex sections
Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space.
For every $x \in X$ and every coordinate $i=1,2,\ldots,d$ denote by $x_{-i} := (x_j)_{j \neq i}$.
Given a set $Y \subseteq X$ ...
2
votes
1answer
76 views
Why the convexity condition on the definition of a face of a convex set?
A face of a closed convex set $X\subseteq\mathbb{R}^n$ is defined to be a set $F\subseteq X$ such that:
$F$ is convex.
Every line segment from $X$ whose interior meets $F$ is contained in $F$.
Is ...
2
votes
1answer
92 views
small perturbation of BV function
consider an interesting real analysis question:
define average operator on $[0,1]$:
$A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $
( may clarify ...
4
votes
1answer
132 views
Is the preimage of a face under an affine map a face?
Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\pi{:}\ X\to\mathbb{R}^m$ be a linear map, with $m<n$ (for example, a projection). Let $\pi^{-1}(y)=\{x\in X\mid\pi(x)=y\}$ denote the inverse of $...
4
votes
2answers
123 views
A kind of exponential concavity for polynomials?
Is there $C > 0$ such that the inequality
$$
\prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right)
$$
holds for all finitely supported sequences $(a_n)$ with $a_n\geq ...
9
votes
1answer
318 views
Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension
By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
3
votes
1answer
186 views
concavity of $\log [ (1+\frac{x_0}{1+x_0+x_1/2+x_2/4}) (1+\frac{x_1}{1+x_0/4+x_1+x_2/2}) (1+\frac{x_2}{1+x_0/2+x_1/4+x_2}) ]$
Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be
$$
f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \...
2
votes
0answers
48 views
Star-convex curve and Fourier series
Let x(t) be a periodic function on [0, 2$\pi$].
I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
4
votes
3answers
141 views
Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$
I want to prove that function
\begin{equation}
f(x)=\frac{1}{x+1} \int\limits_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt
\end{equation}
is quasi-concave. One approach is to obtain the closed form ...
1
vote
1answer
77 views
Every closed and convex subset of a uniformly convex metric space is Chebyshev?
I came across the statement ``Every closed and convex subset of a uniformly convex b-metric space is Chebyshev'' in [1]. Here, the term `convex' is in the sense of Takahashi. I tried looking up for ...
1
vote
0answers
49 views
Are the sets whose convex hull surface admits multiple representations a shy set of sets?
Consider a compact subset $A$ of $R^n$. Let me call $A$ special if any point $x$ that is on the boundary of $\textsf{conv.hull}(A)$ admits a unique representation as a convex combination of points in $...
0
votes
0answers
70 views
Volume of parametric integral of convex set
Given $t,\mu>0$. I am interested in computing the volume of the $n$-dimesnional set
$$\int_{0}^{t}e^{A(t-\tau)}\begin{pmatrix}0\\ 0\\ \vdots\\ 0\\1\end{pmatrix}U\:{\rm{d}}\tau,$$
where the set $U$ ...
2
votes
1answer
110 views
Quasi-concavity of $f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$ on $[0~1000]$
I want to prove that function $f:[0~1000]\rightarrow R$, $$f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$$ is quasi-concave. Any idea how to do the proof? I already tried to prove that any super-level set is ...
4
votes
2answers
133 views
Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix
Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that
$\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that
$$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...
0
votes
0answers
45 views
Optimization of generalized Fisher information over space of probability measures
Assume we already have $p(\theta)$ apriori desnsity function about unknown parameter $\theta$, and we consider an optimization problem as follows:
\begin{equation} \max_{P \in \mathcal{M}} \int_{\...
0
votes
1answer
87 views
1D functional equation: solve for function with given expected value w.r.t normal density
Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation
$$
\begin{split}
\mathbb ...
0
votes
0answers
96 views
Convexity of the Frobenius norm of the product of matrices
I have a question similar to Convexity of the Frobenius norm of the product of two matrices. I am not able to comment on that question as I don't have enough reputation, and that is why I have asked a ...
-1
votes
1answer
79 views
subgradient calculus [closed]
I have to calculate a subgradient of the following function
$$
p(x) = \max\{e(Cy-Cx)\ :\ Cy \geq Cx, y \in X\},
$$
where $C$ is a $p\times n$ matrix and $X$ is a convex polyhedral set.
It is a non-...
5
votes
1answer
215 views
On faces of convex sets of positive semidefinite matrices
A face $F$ of a convex set $C$ is a set such that, if $x \in F$ is a convex combination of other elements in $C$, then they must also be in $F$. I will denote by $F(C,x)$ the smallest face of $C$ ...
1
vote
0answers
40 views
In convex optimization we know that the optimum solution is on which hyper plane
We have a standard linear program, I mean a set of inequalities $c_i^Tx\leq b_i$ where $i\in \{1,\ldots ,k\}$ and we want to find $max\{c^Ty| y\in \{\cap \{x|c_i^Tx\leq b_i\}\}$. I put some condition ...
-1
votes
1answer
95 views
Does a half plane contain intersection of some other half planes? [closed]
I'm doing research in Optimization and I have found this obstacle in the way.
If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
1
vote
0answers
32 views
How is the minimax oracle used to find the oracle complexity of projected subgradient?
I am going through a set of blog posts on the complexity of projected gradient method.
https://blogs.princeton.edu/imabandit/2013/03/15/orf523-oracle-complexity-large-scale-optimization/ defines the ...
3
votes
0answers
104 views
Weak to weak$^*$ continuity of the duality mapping
Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
1
vote
1answer
68 views
Norm of solution of quadratic program
In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}_{m\times n}$ and $q\in\mathbb{R}^n$. Define
$$x^* ...
4
votes
2answers
245 views
Is this function always bounded below?
Is there any global constant $C$ such that $$C<\frac{\sum_{i=1}^{n}x_{i}\log x_{i}-x_{i}+(1-x_{i})\log(1-x_{i})}{\sum_{i=1}^{n}x_{i}}+\log(\sum_{i=1}^{n}x_{i})-\log(\sum_{i=1}^{n}x_{i}^{2})$$ for ...
6
votes
2answers
138 views
Inequality on permutation polytope
Let $a = (a_1, \cdots, a_n), b = (b_1, \cdots, b_n), c = (c_1, \cdots, c_n) \in \mathbb{R}^n$ with $a_1 \geq \cdots \geq a_n, b_1 \geq \cdots \geq b_n, 0 < c_1 \leq \cdots \leq c_n$.
In addition, ...
3
votes
2answers
183 views
Mixtures of log-convex functions are log-convex: a reference
A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
2
votes
1answer
194 views
example of convexity
Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $(S_1,S_2)\in \mathcal{B}(F)^2$. We define
$$W(S_1,S_2)=\{(\langle S_1 y\; ,\;y\rangle,\langle S_2 ...
4
votes
0answers
126 views
Sharp constant for inequality with convex functions
This is a follow up to this question, where the optimal constant was left open.
Let $P \subset \mathbb{R}^n$ be bounded, convex, and open. Let
\begin{equation}
\mathcal{H} := \{f : P \rightarrow \...
2
votes
0answers
32 views
Do internal stable sets contain big manifolds?
Given two strictly concave functions $u_{i}$ with continuous derivatives in $\mathbb{R}^{k}$. We define their upper levels at a point $x$ of these functions as the set of points y such that $u_i(y)>...
0
votes
0answers
14 views
characterize a family of convex function parametrized by maximal monotone sets
To start, a set $G\in \mathbb R^2$ is called maximal monotone if it is not a strict subset of a monotone set. For each maximal monotone set $G$, we can define a function
$$u_G(x_1,x_2)=\inf_{y\in G} \{...
3
votes
0answers
49 views
Behavior of vector field that is the arg min of a convex function
Say $f(x,y)$ is a real valued function that is strongly convex and twice differentiable in both $x$ and $y$. Here $x$ and $y$ are real vectors of different dimensions. We define $y^\star(x)= \arg \...
13
votes
1answer
411 views
Average measure of intersection of a convex region with its translate
Let $\lambda$ denote the Lebesgue-measure on $\mathbb{R}^n$, and let $C\subset\mathbb{R}^n$ be a convex region.
My question is about
$$f(C):=\int_{C} \lambda(C \cap (x + C) ) \mathrm{d} x.$$
How ...
6
votes
1answer
294 views
Fenchel-Rockafellar Duality in Villani's Book
Villiani writes (some notation changed) in Topics in Optimal Mass Transportation:
Theorem 1.9. Let $E$ be a normed VS, $E^*$ it topological dual. $\Theta$ and $\Psi$ are two convex functions on $E$ ...
1
vote
1answer
48 views
Calculate $k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$ for a special $\ell$ function
We consider the function $\ell:\mathbb{R}^{m}\rightarrow \mathbb{R}$ given by
$$\ell(\xi):=-\max\left\{-\left\langle x,\xi\right\rangle+10 \tau, -51\left\langle x,\xi\right\rangle -40\tau \right\}$$
...
1
vote
2answers
119 views
Let $X\subseteq\mathbb{R}^n$ and let $F$ be a face of $\mathop{\rm conv} X$. Then $F=\mathop{\rm conv}(X\cap F)$
This theorem is obviously true if the set $X$ is finite (so that $\mathop{\rm conv} X$ is a convex polytope). I believe it is true for any set $X\subseteq\mathbb{R}^n$ but I cannot prove it. Can ...
2
votes
0answers
88 views
Covering a space by cones
Let $X\subset\mathbb{R}^n$ be some connected and bounded $n$-dimensional manifold, e.g. a space homeomorphic to an open/closed ball with possibly some parts of it being removed.
I am interested in ...
1
vote
1answer
55 views
Properties of Relative Entropies
I'm looking at a paper by Arnold et al. (CPDE, 2001), in which they make use of convex functions in the context of relative entropies. There, they assume that on $(0,\infty)$, their entropy function ...
2
votes
0answers
35 views
Mean width of intersection of two elipsoid
My question is regarding mean widths. For a set $\mathcal{T}$ define the mean width
\begin{align*}
\omega(T)=\mathbb{E}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I})}\bigg[\underset{\mathbf{u}\in\mathcal{...
3
votes
1answer
120 views
closure of a separating set of pure states
Let $A$ be a unital C*-algebra, and let $\mathcal R$ be a separating family of irreducible representations of $A$. Each vector state of a representation in $\mathcal R$ is a pure state, and the span ...
1
vote
0answers
18 views
Find conditions over $U$ such that optimization over a convex cone generated by $U$ is equal to optimization over $U$ [duplicate]
Reading several pappers to prepare my thesis I found the following problem:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \max\limits_{x\in\mathcal{C}} & f(x) \\[...
8
votes
1answer
521 views
property of convex functions
I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
