Questions tagged [convex-analysis]
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507
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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?
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$\arg\max$ 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 ...
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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 ...
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improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
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Concavity of a function along a path
Suppose that $f(x,y)$ is a continuously differentiable function
and $g(x,y) =xy-f(x,y)$. I know that $g$ is concave
if and only if $(-f_{xx})(-f_{yy}) -(1-f_{xy}) ^{2}>0$ and $f_{xx}>0$.
Now ...
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When is the log-permanent concave?
Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...
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Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?
I guess my question is a follow up question of this one: usul, Existence of a strictly convex function interpolating given gradients and values, version: 2019-04-13.
In usul's question, the answer ...
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Implicit function theorem for subdifferentiable convex functions
I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
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Finding the conjugate of a function
I know that the Fenchel conjugate of a function is
$$f^*(x^*) = \sup_x\{\langle x, x^*\rangle - f(x)\}.$$
However, how do I find the Fenchel conjugate of the function
$$f(x) = \frac{1}{p}\sum\limits_{...
4
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Convergence of semi convex functions
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
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When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?
Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality:
$$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$
For ...
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Generalization of minimal selection theorem
Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection
\begin{equation*}
m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\},
\end{...
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Subgradient chain rule
Suppose $$F:\mathbb{R}^n \to \mathbb{R},\; F(x)=\mathrm{max}_\mathrm{eig}(C-\mbox{diag}(x)).$$
I am trying to find a subgradient of $F$ at $x_0$. A subgradient of $\mathrm{max}_\mathrm{eig}$ is given ...
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Variational forms of non-convex functions
I am trying to understand what kind of variational forms exist for non-convex functions. Alternatively, are there conjugate forms which attain strong duality? For a non-convex function $f$, I am ...
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Aleksandrov's proof of the second order differentiability of convex functions
Aleksandrov [A], proved a remarkable property of convex functions.
Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
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Second order differentiability of convex functions
Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...
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Inequalities for upper semi-continuous affine functions on compact sets by using extreme points
Suppose $f_1\colon K\to [0,\infty)$ and $f_2\colon K\to[0,\infty)$ are two upper semi-continuous affine functions,
$$
f_i(\lambda x+(1-\lambda)y)=\lambda f_i(x)+(1-\lambda)f_i(y)\ \mbox{ for all }\ x,...
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Extreme points of an intersection of convex set with countably many linear spaces
Let $V$ be some `nice' vector space and let $T: V\to \mathbb{R}$ be a linear functional over $V$.
Define
\begin{align}
M= K \cap \bigcup_{i \in \mathbb{N} } \{ v \in V: T(v)=c_i \}
\end{align}
...
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Log-concavity of function
Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
My goal is to show that
$$ G(y):=\frac{(f_2*f_0)(y)}{(f_0*f_0)(y)}- \left(\frac{(f_1*f_0)(y) }{(f_0*f_0)(y)}\right)^2$$
is log-concave.
Let us ...
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Monotonicity of maximum of convex combination of two scaled concave functions
Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...
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Subgradient in a predual under weak* continuity
Let $X$ be a Banach space. Suppose $f:X^*\to\mathbb R\cup\{\infty\}$ is convex, has weak*-compact effective domain, and is weak*-continuous on its effective domain. In particular, $f$ is weak*-lower ...
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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_{...
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Prove the equicontinuity of a maximizing sequence
Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...
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Does midpoint-convex imply rationally convex?
Say that a function $f$ defined on $\mathbb{Q}^n$ is midpoint convex if $f((x+y)/2) \le (f(x) + f(y))/2$. Say that it is rationally convex if, for $\lambda \in \mathbb{Q} \cap [0,1]$ and $\bar \lambda ...
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Most general form of Jensen's inequality
What is the most general form of Jensen's inequality?
Wikipedia gives for example this more general form, which holds in every topological vector space.
Are there even more general forms, for ...
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Generic shadows of convex bodies
If $K \subset \mathbb{R}^m$ is a convex body (i.e., a compact convex set) and $T \mathbin\colon \mathbb{R}^m \to \mathbb{R}^n$ is a linear map then $TK \subset \mathbb{R}^n$ is a convex body as well. ...
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What is a non-trivial example of an unbounded subdifferential?
Let $f: X \to [ -\infty, \infty]$ be some function,
Can someone provide a non-trivial example where the subdifferential evaluated at a point $x$,
$$\partial f(x)$$ is "unbounded"? (trivial examples ...
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Expectations, double integrals and Jensen's inequality
$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and
$c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and
$v$ be $[x,y]$....
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Sufficient condition for convex conjugate to be second-order differentiable
Let $f:\mathbb{R}^n\to (-\infty,\infty]$ be a convex lower-semicontinuous function, we then define its conjugate by
$$
f^*(y)=\sup_{x\in \mathbb{R}^n}\{x^Ty-f(x)\}.
$$
Then there exist well-known ...
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A strange variant of the Gaussian log-Sobolev inequality
Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, ...
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On the area-perimeter ratio of a convex limited set
(Previously asked on MSE)
Let $C\subset \mathbb{R}^2$ be a convex limited set. We define the average radius of $C$ as
$$a_C=\frac{\int_{v\in C}d(v,C)dxdy}{A(C)}$$
Where $d(v,C)$ is the distance ...
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Goldowsky-Tonelli theorem for upper semi continuous function
Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a convex and continuous function. We know that $\partial^{e} f(. )$(either right or left derivative) is non-decreasing and upper semi-continuous function. ...
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How does one translate from convex hull to a set of facets (inequalities)? [duplicate]
Suppose I have defined a convex set as the convex hull of a set of points. (I know that all these points are "extremal points" of the convex set.) I know want to translate this description of the ...
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1
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Separation in $l^1$ (Kreps-YanTheorem)
I have a question about the hypotheses of the Kreps-Yan Separation Theorem. I use the notation $l^p_+$ for the subspace of vectors all of whose coordinates are non-negative and define $l^p_- = -l^p_+$....
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Maximization of an integral functional over a closed convex set
I want to maximize $$F(w):=\sum_{1\le i,\:j\le2}\int\lambda^{\otimes2}({\rm d}(x,y))\left(w_i(x)f_j(x,y)\wedge w_j(y)f_i(y,x)\right)g_{ij}(x,y)$$ over the closed convex set $$S:=\left\{w\in{\mathcal L^...
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Gradient formula for Clarke's generalized gradient on a general Banach space
In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula:
($\operatorname{co}$ deotes the convex hull).
Is there an ...
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How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?
Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
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Intersection of Subharmonic and Harmonic Functions (also: Extension of harmonic functions)
Let $\Omega \subset \mathbb{R}^n$ be open, convex and bounded with smooth boundary. Define
$$\mathcal{A}(\Omega) = \left\{ C \subset \Omega \ \big\vert \begin{array}{l} \text{for any subharmonic ...
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When do convexity and lower semicontinuity imply continuity?
Let $X$ be a nonempty compact convex subset of a locally convex space. Say $X$ has the convex function property if every convex, lower semicontinuous $f:X\to\mathbb R$ is also continuous.
Question: ...
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Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$
Let $f: \mathbb R \rightarrow (-\infty,+\infty]$ be a lower-semicontinuous convex function.
Question
Under what futher conditions does there exists a convex decreasing function $\phi: \mathbb R \...
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Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?
It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual ...
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Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article
To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question.
We consider the ...
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A property of convex cones in Euclidean spaces
EDIT: Let $K$ be a closed convex cone in a Euclidean space of finite dimension. Assume it is non-zero and not the whole space.
Does there exist a non-zero point $x\in K$ such that
$$(x,y)\geq 0 ...
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730
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Monotone function which is separately convex but not convex
I am looking for a function $f\,:\,\mathbb{R}^n\to\mathbb{R}$ which is separately convex and increasing but not convex.
That is to say, the function is convex and increasing in each coordinate while ...
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Why is it difficult to solve the Monge problem directly?
I'm trying to understand something about the Monge problem. The Monge problem is:
Let $c(x,y): \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $$\mathcal{T}(\mu_1,\mu_2) = \{ T: \...
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Inheriting quasiconvexity from convex function after re-parametrisation of space into the Stiefel manifold
Consider a function $f(S): \mathcal{S} \to \mathbb{R}$, where $\mathcal{S}$ is the convex cone of all positive semidefinite complex $M\times M$ Hermitian matrices.
The function $f(\cdot )$ is concave ...
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"Moduli space" of isotropic convex bodies?
A lot of questions in convex geometry revolve around the geometry of isotropic convex bodies in $\mathbb{R^n}$.
To my knowledge there is no, or very little study of a space such as :
$$C_n = \{...
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John's ellipsoid of a polytope
Suppose that $X$ is $\mathbb R^n$ with some polyhedral norm, that is, the unit ball of $X$ is an $n$-dimensional polytope. Assume that the John ellipsoid of $X$ is an Euclidean ball that touches every ...
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Upper bound on number of vertices in intersection (and union) of simplices
Let $S_1, \dots, S_k \subset \mathbb{R}^n$ be a set of (non-regular) simplices. Let $m_i$ indicate the number of vertices of simplex $S_i$ (we do not assume it is equal to $n-1$).
Is there a simple ...
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Lipschitz min implies Lipschitzian argmin?
Let $X$ be a Hilbert space, and suppose that $f:X^2\rightarrow \mathbb{R}$ is a Lipschitz, supercoercive, convex function such that (for every $y \in X$) the set
$$
\operatorname*{argmin}_{x\in X} f(x,...
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