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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Definition of a function being unimodal

For a function $f: \mathbb{R}^n \to \mathbb{R}$, I am looking for the definition of $f$ to be unimodal. From Wikipedia $f$ is unimodal if there is a one to one differentiable mapping $X = G(Z)$ ...
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How to prove that this function is convex

My problem is that: The domain is $\mathbb{R} ^n _{++}$ . I need to prove that $f(x_1,...,x_n)=\sum_{i=1}^{n}x_i\cdot ln(x_i) -(\sum_{i=1}^{n}x_i)\cdot ln(\sum_{i=1}^{n}x_i) $ is convex. I tried to ...
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Jensen's inequality and $L^p$ norms

Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
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277 views

Prove that the spaces have the same homotopy type

This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima. "Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...
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51 views

Relation about Gateaux differentiable and differentiable

Assume there is a multivariable function $f:\mathbb{R}^n\to\mathbb{R}$. I want to know the relation between Gateaux differentiable and differentiable. That is, if $f$ is differentiable, is it Gateaux ...
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Special type of convexity

Does anyone know if there is a name for a function $f:W\rightarrow\mathbb{R}$ ($W$ is a vector space) that satisfies $$ f\left(\theta x + \left(1-\theta\right) y\right) \leq \theta^\alpha ...
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Lipschitz functions in $\mathbb{R}^n, \ \ \mathbb{R}^m$, extension

I've found the following lemma : Let $\{x_1, . . . , x_k\}$ be a finite collection of points in $\mathbb{R}^n$ , and let $\{y_1, . . . , y_k\}$ be a collection of points in $\mathbb{R}^m$, such that ...
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53 views

Does the Border (Boundary) Points of a convex shape in the positive quadrant make a convex function?

Let $\mathbb{S}$ be a convex body in 2-D with some non-zero intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis such that $(c,y)\in ...
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Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
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Convexity of $\log \det(I+(P-\text{tr}(X))\cdot X)$

where $P-\text{tr}(X)>0$ and $X$ is a diagonal matrix with diagonal elements are $(x_1,x_2,\dots x_m)$
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Solution of a Quadratic Optimization Problem

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem, \begin{align} ...
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Given some points in the Euclidean space, find a plane satisfying some restrictions

In a 3-D Cartesian coordinates, suppose we are given $n$ points and their coordinate values in the form $(x,y,z)$. Obviously there are uncountable planes which divide those points into two groups, ...
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42 views

Robust feasibility with halfspace?

Consider a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, such that for all $x \in \mathbb{R}^n$ we have $$ a_1^\top x + b_1 \leq f(x) \leq a_2^\top x + b_2 $$ for some given $a_1, a_2 ...
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128 views

About Balanced-Convex Hull of a Set

Definition. $\newcommand{\bco}{\operatorname{bco}}$Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, deonoted $\bco A$, is the intersection of all balanced-convex ...
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117 views

How is the concave closure operation defined?

I learned the in a vector space over an ordered field, the convex closure operation of a subset is defined as the smallest convex set that contains the subset. I was wondering how the concave closure ...
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Explain about convexity in geometry and in optimization.

My question is 'what is a difference between convexity in geometry and optimization?'
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Minkowski Inequality for $p \le 1$

I've been trying to prove the concavity of a particular function which I reduced to proving the reverse Minkowski Inequality for $p \le 1$, $p \ne 0$ for arguments in $\mathbb{R}^{n}_{+}$. That is, ...
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61 views

How to prove that is a cone

I have to prove that the set $$K=\lbrace u\in C[0,1]\mid u(t)\geq a(t)||u|| \text{ on } [0,1]\rbrace,$$ where $a(t)=\displaystyle\frac{t(2p-t)}{p^2}$ with $\frac12<p<1$, and ...
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307 views

Show that the maximum of a set of convex functions is again convex

Let $f_1(x), f_2(x), \ldots, f_n(x)$ be a set of convex functions. We define $f(x)$ as $$ f(x) = \underset{i}{\text{max}} \left\{ f_i(x) \right\}. $$ How do I show that $f(x)$ is also convex, and ...
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On the convexity of element-wise norm 1 of the inverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$ \|A\|_1= \sum_{i,j} |A_{i,j}|. $$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
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78 views

Proving function is convex

How do you show that $c + max(0,1-x)^{2}$ is convex where $c$ is a constant? I can graph it and observe that the function is below any line segment between any two points but I am not sure how to ...
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203 views

Epigraph of a function f: D $\rightarrow$ R is convex iff epif(f) is a subset of D*R which is a convex set

As in the topic, how to show that $epi(f)$ is convex iff $epi(f)$ belongs to D*R which is a convex set.
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Does $\inf_{(a,b)\in A} ax+by=\inf_{(a,b)\in B} ax+by$ imply $A=B$?

Let $A,B\subseteq\Bbb [0,1]\times [0,1]$, and for all $(x,y)\in \Bbb R^2$ with $x^2+y^2=1$, $$\inf_{(a,b)\in A} ax+by=\inf_{(a,b)\in B} ax+by$$ Can we deduce $\overline A=\overline B$.
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Coding Distributions as a Convex Constraint

In convex optimization, how can we impose a constraint that a variable has certain distribution? e.g. elements of vector $v$ have power law distribution?
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37 views

Find a point in a polytope that always cuts off a constant fraction of the polytope.

I have $k$ linear inequalities in $\mathbb{R}^n$, which we can express as $Ax \ge c$ (where $A \in \mathbb{R}^{k \times n}$ and $c \in \mathbb{R}^k$). Assume that the set of $x \in \mathbb{R}^n$ that ...
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93 views

Is permutation-invariance of an objective a problem in convex optimization?

I have difficulty understanding how permutation-invariance and convexity are related in an optimization problem. Let $f(.)$ be a convex function defined on $d$-dimensional vectors. Suppose, also that ...
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127 views

Trace Constraints and rank-one positive semi-definite matrices.

Let $C_1$,$C_2$...$C_N$ be $M\times M$ hermitian matrices and $c$ be a given positive constant. Let $W$ be a positive (possibly semi) definite matrix such that \begin{align} \text{trace}\{WC_1\}\geq ...
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Pointed Convex cone: one-to-one correspondence extreme rays - extreme points

Hoi, let $V$ be a finite dimensional real vector space with inner product $\left\langle .\right\rangle$. Let $\Gamma\subset V$ and $\Gamma \neq \left\{0\right\}$ a pointed convex cone. (Pointed means ...
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Are these convex optimization problems equivalent?

Consider the optimization problem $$ \mathcal{P}_1: \qquad \min_{x \in \mathbb{R}^n} c^\top x \quad \text{sub. to } \ g(x,y_i) \leq 0 \ \ \forall i = 1,2,...,M$$ where $c \in \mathbb{R}^n$, and ...
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Prove that if $A, B $ are convex , $B$ is closed, $C$ is bounded and $A+C \subset B+C$ then $A\subset B$

Given the sets $A,B,C \in \mathbb{R}^n$ such that: $$A+C \subset B+C$$ Show that if $A,B$ are convex, $B$ is closed and $C$ is bounded then $A\subset B$. I kind of understand the geometrical ...
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A property of the minima of a sum of convex functions, take 2

This is a follow-up to my previous question. Let $g_1(x), \ldots, g_k(x)$ be convex functions from $\mathbb{R}^n$ to $\mathbb{R}$, and lets assume that global minimum of each $g_i$ is unique and is ...
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A property of the minimum of a sum of convex functions

Let $g_1(x), \ldots, g_k(x)$ be convex functions from $\mathbb{R}^n$ to $\mathbb{R}$, and lets assume that global minimum of each $g_i$ is unique and is achieved, denoting $$x_i = \arg \min_{x \in ...
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convexity of the product of two entropy-like functions

Consider the functions $T_p(q)= \sum_i q_i^p$, where p>1 and q is a finite-dimensional vector satisfying $\sum_i q_i = 1, q_i >0$ (ie, a probability mass function). In information-theoretic terms, ...
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Hessian of a function that takes matrix arguments

I have a function that that takes a matrix and returns a scalar, $f : \mathbb{R}^{m\times n} \rightarrow \mathbb{R}$. I know how to calculate the derivative of this function with respect to the matrix ...
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128 views

Convexity of product of elements from two convex set

Given two convex set $X\subseteq \mathbb{R}^N$,$Y\subseteq \mathbb{R}^{N\times N}$ Given a $x\in X$, is the set $\{z|z=yx,\forall y \in Y\}$ convex? If no, by adding what can force it to be convex? ...
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Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property $$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$ So it's like convexity ...
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Prove the concavity of $F(x,y) =\ln(x)+y$ by arguing the definition of concavity.

I need help with this problem. Prove the concavity of $F(x,y) = \ln(x) + y$ by arguing the definition of concavity. A function $f$ is concave is for any $x_0, x_1 \in \mathbb{R}^2$ and $t \in ...
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What is the name for functions with this property similar to concavity?

What's the name of the class of functions satisfying the following property: Given some convex space $D\subseteq \mathbb{R}^k$, a function $f:D \to \mathbb{R}$ is called $\mathbf{(?)}$ if for all ...
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How to prove that the space created by pointwise Bernoulli random variables are compact

I have a function $$\delta:\mathbb{R}\rightarrow [0,1].$$ We obtain this funtion pointwise as follows: For each point $y\in\mathbb{R}$, $\delta(y)$ is a real number in $[0,1]$. More explicitely, ...
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$\sup\{t_{1}f_{n}(x_{1})+t_{2}f_{n}(x_{2})\mid n\geq n_{0}\}=\sup\{t_{1}f_{n}(x_{1})\mid n\geq n_{0}\}+\sup\{t_{2}f_{n}(x_{2})\mid n\geq n_{0}\} $

I was thinking, if this is correct: Let $f_n$ is a series of convex, limited function $I \rightarrow \mathbb{R}$ $t_1, t_2 \in \mathbb{R} \ \ \ \ \ t_1 + t_2 = 1$ Is that a true statement : ...
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Why is the negative entropy Lipschitz with respect to the $1$-norm (Over)?

Let $\left\|x \right\| = \sum_{i=1}^{i=n}\left|x^i\right|$ and $d\left(x\right)=\sum_{i=1}^{i=n}x^i\ln x^i$ where $x\in R^n $ and $ \sum_{i=1}^{i=n}x^i=1$ How to prove: For all $x, x'$, $$\left| ...
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343 views

Some questions on Minkowski's functional

I'm reading Wikipedia's article on Minkowski's functional. They state that if the set K used in defining Minkowski's functional pK is convex then pK is sub-additive. They argue as follows: Suppose ...
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Is the correlation function convex or not?

Suppose the function for statistical correlation is a non linear constraint in a non linear programming model: $$ \frac{\sum_{t=1}^T (p_t - \bar{p})(R_t - \bar{R})}{\sqrt{\sum_{t=1}^T (p_t - ...
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Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
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61 views

Distributed Robust Optimization

Consider the following constrained optimization problem $\mathcal{P}$. $$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$ ...
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223 views

Strictly Convex Function and Well-Separated Minimum

Suppose $\Theta \subset \mathbb{R}^d$ is a convex set, and $f:\Theta \rightarrow \mathbb{R}$ is a strictly convex function that has a minimum at $\theta_0\in\Theta$. Is it true then that $\forall ...
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400 views

Is this sum of convex and concave functions a convex function?

Is this a convex function in $X$, where all the entries are real and $Y,\beta$ are constants where $X,Y$ are rectangular matrices and $\beta$ is a constant vector and $A,B$ are constant p.s.d ...
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55 views

Achieving equality in the definition of the support function

Suppose that $B$ is a convex body (compact closed with nonempty interior), and let $$h_B(u) = \sup_{x \in B} \langle u,x\rangle$$ be its support function. Is there a nice description of the set $E ...
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162 views

Convex and concave functions

This maybe a silly question... So mercy me. Let $m,v:[0,S]\to \mathbb{R}$ be two Lebesgue integrable, monotone functions, say $m$ decreasing and $v$ increasing and set: $$M(s):=\int_0^s m(\sigma)\ ...
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90 views

Convex hull area of projected points are convex respect to rotations?

Let $A$ be a finite list of points in $\mathbb{R}^3$ and $c$ the centroid of $A$. Let $P$ be an orthographic projection onto a plane in $\mathbb{R}^2$ and $h$ be the convex hull of $P(A)$. Let further ...