Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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Triangle Inequality Like Equation [closed]

If we are in $R^2$ and define $d(a,b)$ as the set of points between $a$ and $b$ we can create an equation like this: $$d(x,z) \subseteq d(x,y) \times d(y,z)$$ where the $\subseteq$ is the subset ...
2
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1answer
96 views

About the Affine hull and Span.

I'm learning Linear Algebra and Convex Optimization simultaneously, I notice that the affine hull is, to some extent, analogous to the span, but when I read the lines "We define the affine dimension ...
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0answers
55 views

Show that if A and B are strictly convex, then A + B is strictly convex or provide a counter example.

We have: If A is open: $\exists x,y \in A,$ $x \neq y$ such that $\lambda x+(1-\lambda y)\in \dot A $ (the interior) and $\exists u,v \in B,$ $x \neq y$ such that $\lambda u+(1-\lambda v)\in \...
2
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2answers
186 views

Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
0
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1answer
35 views

Discontinuous semiconcave functions

A function $u: \mathbb{R}^n \to \mathbb{R}$ is defined to be semiconcave if there is a positive constant $c$ such that for all $x,z$ $$ u(x-z) + u(x+z) - 2u(x) \leq c |z|^2. $$ Alternatively, one ...
0
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1answer
41 views

Check if ray intersects internals of $D$-facet

Given a ray $\overrightarrow{r_0} + \overrightarrow{v} \cdot t, t \in [0;+\infty)$ and a $(D - 1)$-simplex, defined by $D$-tuple of its vertices $p_i = (p_i^1, p_i^2, \dots, p_i^D), i \in \{1, 2, \...
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0answers
25 views

Discrete concavity of a log function

I want to prove that the function $f_i(P)=f_i(P_1,..P_i,..,P_K)=log(1+(\frac{a_iP_i}{\eta+\sum\limits_{i'\neq i}a_{i'}P_{i'}}))$ is discetely concave, which means that I should prove: $\forall \lambda ...
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0answers
39 views

Convergence of backtracking and gradient descent.

I am thinking a bit about the following exercise: Let $f(x) = x_1^2 + x_2^2$ with dom $f = \{ (x_1,x_2):x_1 > 0 \}$. The optimal value of this problem is $p^* =1$, but it is never attained since $(...
1
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1answer
109 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all $x,...
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1answer
73 views

Are unions, sums (…) of quasiconvex functions again quasiconvex?

for a project I need to prove quasiconvexity of several general functions. Can I argue that the union (or sum, or difference...) of quasiconvex functions is again quasiconvex? I do know that the sum ...
0
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1answer
43 views

Prove that $|\int_C f(z)dz| \le M |z_2 - z_1|$ where $M \gt 0$ such that $|f(z)|\le M; \ \forall \ z \in \Omega$

Let $z_1$ and $z_2$ be any two points in $\Omega$ and let $C$ be any oriented contour in $\Omega$ from $z_1$ to $z_2$. Also, assume that $f:\Omega \to \Bbb{C}$ is analytic on an open convex set $\...
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0answers
44 views

Can a non-convex set be partitioned into a set of nearly convex subsets? [closed]

Consider a non-convex bounded subset $S \subseteq \mathbb{R}^{n}$. Is it always possible to partition this set into a finite set of disjoint subsets \begin{equation} S = \bigcup_{i=1}^{n}s_i, \quad ...
-1
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1answer
268 views

Is a distance function on $\mathbb{R}^n$ convex?

Fix $z \in \mathbb{R}^n$. Let $||\cdot||$ be a norm on $\mathbb{R}^n$, and define the distance function $f(x)=||z-x||$ for $x\in \mathbb{R}^n$ Then, is it true that $f(x)$ is convex?
2
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1answer
103 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
1
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1answer
33 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid \sum_{...
1
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1answer
43 views

Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if $T:\...
2
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2answers
92 views

Uniformly convex and strictly convex

I have the following definitions of uniformly convex and strongly convex Let $f:R^n \to R$ be smooth. (1) $f$ is uniformly convex if there exists $\theta > 0 $ such that $$\Sigma_{i,j}f_{x_i x_j} ...
0
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1answer
120 views

Supporting hyperplane of convex function

Below is the appendix B of Evan's PDE book on supporting hyperplanes of convex functions. In the remark (1), he says that the mapping $y\to f(x)+r\cdot(y-x)$ determines the supporting hyperplane to ...
2
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1answer
79 views

Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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1answer
56 views

Showing that a given function is convex.

I am trying to show that the function $f(x,\vec{y})=\alpha\ln(1+\exp(x+\vec{y}\cdot \vec{z}))+(1-\alpha)\ln(1+\exp(-x-\vec{y}\cdot\vec{z}))$ is a convex function of $(x,\vec{y})$ (where $x\in\mathbb{...
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1answer
43 views

Parametrizing the Boundary of a Convex Set

Let $K$ be a compact convex set in $\mathbb{R}^2$. In the proof of a proposition in a paper I am reading, they are concerned with parameterizing $\partial K$ in the following way: If $K$ is ...
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52 views

Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
2
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1answer
108 views

Strong convergence of convex combinations of a weakly convergent sequence

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
2
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1answer
53 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed (weak-* sense) set of probability measures. $m$ is a probability ...
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0answers
16 views

does convexity implies $g(u)>cu$?

So I have been doing some self study and I was wondering if my results are true, or if I am misreading something. Say we have a function $g$ which is concave on values of $u \in \mathbb{R}$. Then we ...
0
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1answer
35 views

convexity and first derivative

Let $\phi$ be a differentiable function on an interval $(a,b)\subset R^1$. If $\phi '$ is non-decreasing, then $\phi$ is convex. But, is the converse true? Does the convexity of $\phi$ necessarily ...
2
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43 views

Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) d(u,...
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1answer
102 views

Show the Gini Coefficient is Quasiconvex

The Gini-coefficient is defined as $$ G(x) = \sum_{i = 1}^n \frac{i}{n} - \sum_{j=1}^{i} \frac{x_{(j)}}{\mathbb{1}^{T}x}, $$ where $x_{i} $ is nonnegative numbers with positive sum. $x_{(j)}$ denotes ...
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2answers
32 views

To show $f(x)$ has ONLY one Max in $x\in[0,1]$

I have function $$f(x)=\left(\frac{1-x}{2-x}\right)x^{p-1}~\text{where}~p>1;~x\in[0,1]$$ I want to show that $f(x)$ has ONLY one maximum in $x\in[0,1]$. I get the second derivative as $$f"(x)=\...
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1answer
56 views

Show that f(x) is convex

Show that f(x) = inf{g(x1)+h(x2)} is convex subject to x1+x2=x and where g(.) and h(.) are convex functions. Can I just go about this by using the regular definition of a convex function or which ...
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1answer
48 views

Convexity/concavity of a strictly increasing and continuous function

Consider a continuous, strictly increasing function $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ with $f(0)=0$, and $x>f(x)$ for all $x>0$. Is this enough to conclude anything about convexity/...
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1answer
49 views

Are these two definitions of an affine subspace equivalent?

I've seen the notion of an affine subspace defined differently as follows: $S \subset \mathbb R^3$, non-empty, is an affine subspace if $(1-t)u + tv \in S$ whenever $u,v \in S$. $S$ is an affine ...
0
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2answers
54 views

compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
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1answer
24 views

Extending a convex function

Suppose $f:(a,b) \to \mathbb R$ is twice differentiable with the property that $c_1 \leq f''(x) \leq c_2$ for every $x \in (a,b)$, where $c_1, c_2$ are positive constants. Is it possible to extend $f$ ...
2
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1answer
60 views

Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
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4answers
335 views

Euclidean norm second derivative

I really need Your help. I need to prove that Euclidean norm is strictly convex. I know that a function is strictly convex if $f ''(x)>0$. Can I use it for Euclidean norm and how? $||x||''=\frac{||...
0
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1answer
110 views

Log-convexity preserved by sum?

I'm currently reading Boyd's book on Convex Optimization (free book which can be found here : http://stanford.edu/~boyd/cvxbook/) , and there is one proof I do not understand : p. 105, the book says ...
2
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1answer
66 views

Are the coefficients of a Taylor series bounded when the function is?

Say that I have three real functions $f(x)$, $g(x)$, and $h(x)$ such that $f(x)\le g(x)\le h(x)$ for all real $x$. Additionally, $f(x)$ and $h(x)$ are logarithmically convex. Can I make any definite ...
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1answer
39 views

Is this function of matrix F convex?

A scalar function $f(F)=vec(F)^{H}\Big(\Omega^{H}\big(M\otimes F(I+F^{H}F)^{-1}F^{H}\big)\Omega\Big)^{-1}vec(F)$ convex? $\Omega$ is arbitrary matrix and $M$ is positive definite. F can be reviewed ...
3
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1answer
90 views

Relationships between affine closures and convex closures

Let $V$ denote a vector space. Then the following concepts make sense: affine subset of $V$ affine closure (affine "hull") of a subset of $V$ Suppose $V$ is in fact a real vector space. Then the ...
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2answers
106 views

Can $\sin (x)$ be represented as difference of two convex functions?

In my optimization homework, I am supposed to prove that every differentiable function with Lipschitz continuous gradients can be represented as difference of two convex functions. I think I have come ...
2
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1answer
82 views

Pointwise convergence implies uniform convergence under concavity?

Suppose that $X$ is a subset of a Euclidean space and $f_n:X\to\mathbb{R}$ converges pointwise to $f:X\to\mathbb{R}$. A book I'm reading seems to say that: if $X$ is convex and compact and each $f_n$ ...
2
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1answer
68 views

Hint on how to proof that $x^2$ is convex

Note: I can't differentiate 2 times and prove that $f''(x) > 0$ The exercise requires me to prove that the function $f(x) = x^2$ is convex by using the following Theorem: $f(x) \ge f(x^*) + \...
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1answer
128 views

Convex optimization problem: linear equality and inequality constraints

When linear equality constraints can be converted in an inequality constraints for a strongly convex optimization problem? I mean, I got the same solution for both the following problem: 1) $\min_x ...
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1answer
53 views

Continuity of subdifferential mapping

I'm reading Cedric Villani's book topics in optimal transportation, and I have a problem on page 53: If $\varphi$ lower semi-continuous, then the subdifferential mapping $\partial\varphi$ is always ...
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1answer
51 views

Let S ⊂ R 2 . Show that if cl(S) is convex, then S is convex.

Let $S ⊂ R^2$. Show that if $cl(S)$ is convex, then S is convex. This seems intuitive but, I am having trouble thinking of a proof or counter example.
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1answer
57 views

How to prove convexity of a given set

I have the set $$ C_c = \{(x,y,z) \epsilon \mathbb{R}^3 : (2x-x^2+y)(2y-3z)(5x-z) > 1, |x| < 1, y > 3, z < 2\} $$ and I need to prove whether it's convex or not. I know that the ...
2
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1answer
52 views

Existence of complicated convex functions

In Stochastic Finance: An Introduction In Discrete Time (by Follmer, Schied), page 400, I found the following proposition: Proposition A.4. Let $I\subseteq\mathbb R$ be an open interval and $f:I\to\...
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1answer
19 views

Set of points at which a function coincides with its convexification is compact?

Let $f:[0,1]\rightarrow\ \mathbb{\bar{R}}$, and let $\tilde{f}$ be the convexification of $f.$ (i.e., $\tilde{f}$ is the pointwise supremum of all affine functions that lie everywhere below $f$.) Let $...
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0answers
103 views

Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal ...