Questions tagged [convex-analysis]
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504
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A standard name of a strongly extremal point of a convex set
I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
2
votes
0
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51
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Conjugate of composition in Bochner spaces
Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
0
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4
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444
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Confining a polytope to one side of an affine hyperplane
Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem.
This answer on math.stackexchange.com claims the ...
0
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0
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44
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Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?
Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
1
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0
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78
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Proximity operator of lower semi-continuous and convex functions pre-composed with norm
Let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a some-where finite, lower semi-continuous, increasing, and convex function. It is easy to verify that the function $\Phi:\mathbb{R}^n\...
0
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0
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68
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Can convex functions on product space be approximated by product of convex functions?
I am working on a problem where I need the following property that I guess should be true but I am not able to prove it.
I have a bounded convex function $F(x, y)$ on $X\times Y$ (Think of $X=Y=\...
0
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1
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247
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When does strict inclusion holds for the domain of subdifferential?
Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$
Its effective domain is,
$$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$
The subdifferential ...
1
vote
0
answers
88
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Differential of the gradient of a strictly convex function
For $n\geq 2$, we consider $\mathbb{R}^n$ endowed with the usual scalar product. Let $f\in\mathcal{C}^2(\mathbb{R}^n,\mathbb{R})$ be a striclty convex function such that $\nabla f$ is nowhere ...
23
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2
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Are such functions differentiable?
In my recent researches, I encountered functions $f$ satisfying the following functional inequality:
$$
(*)\; f(x)\geq f(y)(1+x-y) \; ; \; x,y\in \mathbb{R}.
$$
Since $f$ is convex (because $\...
2
votes
1
answer
329
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Normal cones and KKT conditions
I'm trying to understand a statement from the book "Perturbation Analysis of Optimization Problems", by Bonnans and Shapiro. Let me start by providing some context. In page 148, the authors ...
4
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2
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178
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Existence of a fundamental domain for the convex hull of group action on a rational polytope
Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}...
0
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1
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525
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A concave function as supremum of upper semi continuous is upper semi continuous
We define a affine(concave), upper semi continuous function and bounded function $f:X \to \mathbb{R}$, where $X \subset \mathbb{R}^{k}$ is compact and convex set. Assume that $T$ is an affine and ...
0
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1
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One question about how to get $\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0)$?
In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents: they claim that follows a standard argument in Ekeland ...
8
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Probability of a deviation when Jensen’s inequality is almost tight
This is a cross-post to a yet unanswered question in Math StackExchange
https://math.stackexchange.com/questions/3906767/probability-of-a-deviation-when-jensen-s-inequality-is-almost-tight
Let $X>0$...
5
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2
answers
942
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Distance to a closed set. Is this result known?
Given a closed set $\varnothing\neq E\subset\mathbb{R}^n$, let $\operatorname{Unp}(E)$ be the set if points $x\in\mathbb{R}^n$ for which there is a unique point $y\in E$ nearest to $x$.
Clearly $E\...
3
votes
1
answer
361
views
Concavity near the boundary of the distance function
I was reading the paper
Quelques remarques sur les problemes elliptiques quasilineaires du
second ordre, P. L. Lions, Journal d’Analyse Mathématique volume 45,
pages 234–254(1985)
and on page 251 he ...
13
votes
2
answers
654
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Regularity of convex sets in $\mathbb{R}^n$
The following result is Proposition 2.4.3 in [1]:
Theorem. Let $K\subset\mathbb{R}^n$ be a bounded convex set with the non-empty interior. Then $\partial K\in C^{1,1}$ if and only if
there is $r>0$...
0
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1
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Is it true that every uniformly continuous strictly convex function on convex compact subset of a finite-dim normed vector space has unique minimizer? [closed]
Let $C$ be a convex compact subset of a finite-dimensional normed vector space and let $f:C \to \mathbb R$ be strictly convex and uniformly continuous function (e.g it is sufficient that $f$ be ...
1
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0
answers
99
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On the differentiability of max-type functions
If $f(x, y)$ is strongly concave in $y$, then from Danskin's theorem we know that $\max_{y\in Y} f(x, y)$ is differentiable if $Y$ is convex and bounded. What if $Y$ is not bounded? Is $\max_{y\in Y} ...
0
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1
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105
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If $C_1\subseteq C_2$ are two closed convex cones that are pointed with $\partial C_1\subseteq \partial C_2$ then is $C_1=C_2$?
Let $C_1$ and $C_2$ be two proper full dimensional closed convex cones in $\mathbb{R}^n$ that are pointed. Suppose that $C_1\subseteq C_2$ and that the boundary of $C_1$ is contained in the boundary ...
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4
answers
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Convexity and Lipschitz continuity
It is probably an easy question, but somehow I am stuck.
Question Is the following statement true? If yes, how to prove it?
Suppose that $f\in C^1(\mathbb{R}^n)$ is convex and
$$
\langle\nabla f(x)-\...
3
votes
3
answers
215
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Example of a (strictly) proper scoring rule on a general measurable space?
Most of the literature on scoring rules that I know of deals with discrete measurable spaces, but in this paper by Gneiting and Raferty a very general definition of a scoring rule is given. I don't ...
4
votes
0
answers
83
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Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms
Let $F$ be the set of all convex functions $f\colon[0,\infty)\to[0,\infty)$ with $f(0)=0=f'_+(0)$ and $f_+(\infty-)=\infty$, where $f'_+$ is the right derivative of $f$. For any function $f\in F$, its ...
2
votes
0
answers
98
views
A strong duality for convex functional optimization that admits Lipschitz continuity constraints?
Problem Statement
I am looking for formal proof---hopefully textbook material---of two items:
an analogue to Slater's condition [1] that obtains strong duality for optimization of convex functionals; ...
1
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1
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136
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Find this reference or an alternative where I can find this result
I need this reference, but I couldn't find it online as a PDF. Any help please?
J. Sun, X, Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear ...
0
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1
answer
605
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Linear combination of convex functions is constant
Let $\Phi : \mathbb{R}_{++}\to \mathbb{R}$ be a convex function (not necessarily differentiable). Fix an $\alpha \in (0,1)$ and define
$$g(t) = \alpha t .\Phi\left(\frac{1}{t} + 1\right) + 2(1-\alpha)...
4
votes
0
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252
views
Can this function be minimized?
Let $X$ be a locally convex TVS, and let $A$ and $B$ be convex and compact subsets of $X$ with $A \subset B$.
Let $f: A \times B \to [0,\infty]$ have the following properties:
(1) For all $b \in B$, $...
1
vote
0
answers
68
views
Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
1
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0
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78
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Conditions on triangle inequality for integral kernel
Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$.
Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$
A(t,v)=\int_0^{1/v}L(1/t,s)ds,
$$
which is decreasing with $v$ and ...
1
vote
1
answer
188
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Is this $(\Bbb R^{n \times n})^n \to \Bbb R$ function convex?
Let $W := (W_1, W_2,\dots, W_n)$, where $W_i \in \Bbb R^{n \times n}$. Let $x$ be a constant vector. Is the following function convex?
$$f(W) := x^TW_1^TW_2^T \cdots W_n^TW_n \cdots W_2W_1x $$
1
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0
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240
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Möbius function and polynomials
Let $\mu$ be the Möbius function. It is well known that $\sum_{n|k} \mu(n) = 0$ for $k>1$. What could be said about the polynomials $R_k = \sum_{n|k} \mu(n) x^n$ for $x \in [0,1]$? There does not ...
0
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1
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204
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the subdifferential at points of differentiability in infinite dimensional space
Let $ f: X\to (-\infty,+\infty]$ that $ X$ is an infinite dimensional space.
What are the conditions for $f$ and space $X$ to have the following equality correct?
$$\partial f(x)=\{\nabla f(x)\}$$ for ...
7
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2
answers
547
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Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?
This is a cross-post.
Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a strict local minimum point of $f$.
Let $df^k(x):(\mathbb R^n)^...
4
votes
1
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289
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Variance of random variable decreasing in parameter
I did quite a few numerical computations and think the following is true, but I cannot prove it:
Let $\varphi(x):=\sum_{i=1}^n \varphi_i(x_i)$ where $x=(x_1,...,x_n) \in \mathbb{R}^n$ and $\varphi_i \...
4
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5
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616
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Elementary inequality generalizing convexity of a function on a segment
I am looking for a proof of the following statement which is known to be true as far as I heard.
Let $g\colon [a,b]\to \mathbb{R}$ be a smooth function. Assume that
$$b-a< \pi.$$
Assume also $$g(a)\...
1
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1
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445
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Convexity at a point and Jensen inequality
I am looking for a reference for the following claim:
Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed.
Suppose that "$\phi$ is convex at $c$". ...
1
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0
answers
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Minimax theorems in nonconvex setting
Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$).
Question. ...
2
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1
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334
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Lower bound on $L^2$ norm of a strongly convex function
Let $f\colon[0, 1] \to \mathbb R$ be an $m$-strongly convex function and $\mu$ be a probability measure on $[0,1].$ For any $t<1$, the goal is to find a lower bound on $\int_{0}^t f^2(x) d\mu(x)$ ...
5
votes
1
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362
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Proving equivalence of two definitions of a convex-type Hamming distance
Update: If somebody can answer my question there, then I will be able to fully answer my question here.
Consider $n\in\mathbb N$ and a non-empty set $M\subset\{0,1\}^n$. I have the following ...
2
votes
1
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356
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Concavity of entropy difference
Suppose that $\mathrm{A}$ is a $n\times n$ random matrix with a given distribution. Suppose that $\mathrm{U}$ is a diagonal unitary random matrix, defined as
\begin{align*}
\begin{bmatrix}
\exp(i\...
5
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0
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133
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Conv A = Dual B
I have two cones $A$ and $B$ in a Euclidean space.
I want to show that $\mathrm{Conv}\,A=\mathrm{Dual}\,B$; that is, $a\in\mathrm{Conv}\,A$ if and only if $\langle a,b\rangle\ge 0$ for any $b\in B$.
...
1
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1
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74
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How to compare the minimums of two discrete convex functions?
I have a question that troubled me for a long time.
If I have two convex discrete function $f(·)$ and $g(·)$ such that $f(·) \ge g(·)$. (may be not necessary?)
Let $x_1 = \text{argmin } f(·)$, ...
11
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1
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502
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A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
1
vote
1
answer
184
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On examples of subspaces of $C(X)$ for which state spaces are Choquet Simplices
Let $C(X)$ be the Banach space of all Real valued continuous functions on a compact Hausdorff space $X$. What are examples of uniformly closed subspace $\mathcal{A}$ of $C(X)$ such that $\mathcal{A}$ ...
-1
votes
1
answer
182
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Dense linear span implies closed convex hull has non-empty interior
Let $X$ be a Fréchet space and let $Y\subseteq X$ such that $\overline{\operatorname{span}(Y)}=X$. It seems intuitive to me that $\operatorname{int}\big(\overline{\operatorname{co}(Y)}\big)$ is a non-...
1
vote
0
answers
92
views
strict convexity of the Legendre-Fenchel transform
Let $d$ be a positive integer.
Let $L:\mathbb{R}^d\to\mathbb{R}$ be a differentiable function with continuous derivatives.
Assume that the Legendre-Fenchel transform of $L$ exists everywhere, is ...
4
votes
1
answer
231
views
Is there a non-convex function with non-decreasing average rate of change?
$\newcommand{\R}{\mathbb R}$
Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...
0
votes
0
answers
52
views
Separability of Minkowski Sum of well-behaved sets
Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
2
votes
1
answer
969
views
$\arg\max$ in the dual norm of the nuclear norm
Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by
$$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$
whereas the nuclear norm is ...
3
votes
0
answers
46
views
A complete metric space with some convex-type property
Let $(X,d)$ be a complete metric space with this property:
for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$.
I want to know if the family ...