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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An inequality of a multivariate function: $f(x_1,…,x_n) \geq \frac{1}{n}\sum_i^n f(x_i,x_i,…,x_i) $

Let us assume we have a non linear function $f : \Bbb R^{n+} \to \Bbb R ^+$, and let $x = \{x_1, x_2 , ..., x_n\}$, $x_i \in \Bbb{R}^+$, further define $\bar{x} = \frac{1}{n}\sum_i^n x_i$. My ...
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A necessary and sufficient condition for $f(x,y) = \phi(x²+y²)$ be a convex function

Let $f(x,y) = \phi(x²+y²) , \phi \in C^2$ and $\phi$ non-decreasing. Proof that $f$ is convex in the disk $x²+y² \leq a² \iff 2u \phi''(u) + \phi'(u) \geq 0 $ $\forall u \in [0,a]$ Here is my ...
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34 views

Show non-convexity of a function with vector input

How does one go about proving non-convexity of the function d? $$ d(v) = 1/2*||F(v)- p||^2 $$ $$ F(v)=\sum_{i=1}^n l_i*\begin{pmatrix} cos(\sum_{j=1}^i v_j) \\ sin(\sum_{j=1}^i v_j \end{pmatrix} $$ ...
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Subdifferential optimality conditions

I need help with subdifferential optimality. Let $f(x_1, x_2)=x_1^2 + x_2^2 + |x_1 -x_2 - y|$. Find: \begin{align} \min_{x_1, x_2} f(x_1, x_2) \end{align} This is convex, so must have unique ...
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19 views

minimal representation of convex hull

Here is a question about the convex hull. Let $S$ be a set and $\bar{S}$ be the closed convex hull of $S$, i.e., $\bar{S}$ is the smallest convex set that contains $S$. Then is the following claim ...
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1answer
108 views

Solution for $\min_{x^Tx=1} x^TAx-c^Tx$

$\min_{x^Tx=1} x^TAx-c^Tx$ looks like a simple QPQC problem. If $A$ is positive semi-definite, can I get the solution by first getting $x=A^{-1}c$ and then projecting $x:=\dfrac{x}{||x||}$ to make ...
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23 views

Sum of convex and concave functions when one is greater than the other

Given two $C^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ such that $f(x)>g(x)\text{ }\forall x\in\mathbb{R}$. Moreover, we know that $f(x)$ is convex while $g(x)$ is concave. Now, let's define ...
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16 views

Comparison between secant and derivative in a convex function

Imagine that we have a function $f:\mathbb{R}\to\mathbb{R}$ which is convex, that is $f''>0$. We also know that $f'''<0$, that is its first derivative function is concave. Now, we can define its ...
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27 views

Is this function really not concave or convex in any range?

Consider the function $f(x,y)=\frac{y}{1+e^x}$ where $0<y<1$ and $x \in \mathbb{R}$. If you plot this function, it looks like this: Also note that for a given value of $y$ the function $f$ ...
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1answer
33 views

Properties of convex function with Lipshitz continuous gradient (Prof. Nesterov's textbook)

I am reading the Prof. Nesterov's textbook: Introductory lectures on convex optimization - a basic course p.57 I have problem in the following: My ...
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1answer
62 views

interior of convex hull relatively open

Consider $k+1$ affinely independent vectors $\left\{p_0,p_1, \dots, p_k \right \}$ in $n$-dimensional euclidean vector space $n>k$ and consider their convex hull. It is known that each point $x$ of ...
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26 views

If $f$ is log-convex then $f$ is convex

Here's my attempt: $f$ is log-convex. Then: $\log f(\lambda x + (1-\lambda)y )\leq \lambda \log f(x) + (1-\lambda) \log f(y)$ As $e^x$ is increasing, we can apply it to the inequation without ...
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23 views

A function is convex if and only if its gradient is monotonous.

Let a convex $ U \subset_{op} \mathbb{R^n} , n \geq 2$, with the usual inner product. A function $F: U \rightarrow \mathbb{R^n} $ is monotonous if $ \langle F(x) - F(y), x-y \rangle \geq 0, \forall ...
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1answer
12 views

Quasiconvexity (in the sense of Morrey) implies Rank-One convexity

I am trying to understand why Quasiconvexity implies Rank-One convexity. In a standard proof of this fact a sequence of functions is constructed, which converges weakly to zero in $W^{1,p}.$ in ...
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17 views

Composition of convex and concave functions

I had a homework question: "Show that the function f(x,u,v) = -log(uv-xTx) is convex on domain {(x,u,v)| uv-xTx,u,v > 0}". EDIT: x,u,v are Real No.s One pdf I found online says: "We can express f as ...
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1answer
24 views

Minkowski sum and Polygons

The problem:.. Given two convex polygons $A$ and $B$, we can define Minkowski sum as A + B = {a + b: a $\in$ A, b $\in$ B}, where $a + b$ vector sum. Prove that: every vertex $p \in A + B$ is a ...
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1answer
17 views

How to prove a two variable set is convex

$X=\{(x,y)\in R^2\ :\ 3\le 2x+3y\le 8\}$ i tried to solve it as: Let set $X$ is convex for $x_2,y_2\in X$ such that $\alpha x_1+(1-\alpha)x_2$,$\alpha y_1+(1-\alpha)y_2\in X$ Now, $3\le 2x+3y\le 8$ ...
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1answer
26 views

Proximal-type support function properties - nonnegative & strongly convex (proof)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems The following part confuses me: $\\$ $\\$ ...
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69 views

LASSO constrained and penalised forms: first - order conditions?

The LASSO problem can be formulated in two ways: 1) Constrained formulation: $$ \|Xw-y\|^2\to\min_{w}\\ \text{s.t. } \sum_{i}|w_i|\leq{t}. $$ 2)Penalised formulation: $$ ...
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1answer
33 views

Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems I am confused about the green part of the following: $\\$ ...
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32 views

How to prove this function is quasi-concave? [closed]

Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?
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Generalization of Brouwer’s fixed-point theorem

Perhaps the most widely known version of Brouwer’s famous fixed-point theorem reads as follows: For any $n\in\mathbb N$, let $A\subseteq\mathbb R^n$ be a compact (with respect to the Euclidean ...
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35 views

rconvex image under nonlinear function

Let $X\subset R^3$ be a compact and convex set, and let $f: X\rightarrow R^3$ be a nonlinear function, with $f\in C^k$. What are the tools to investigate if the image $K=f(X)$ is also convex, in the ...
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17 views

Probability that a point lies in an uncertain convex hull

Given $n+1$ independent random vectors $X_i \sim N(\mu_i,\Sigma_i)$, where each $\mu_i \in \mathbb{R}^n,$ let $C$ denote the random region formed by taking the convex hull of a realization of the set ...
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1answer
32 views

A convexity argument

Let $(\alpha_n)$ be a sequence of positive real numbers s.t. $\sum \alpha_n=1.$ Consider a sequence of complex numbers $(\beta_n)$ s.t $|\beta_n|=const$ for all $ n \in \mathbb{N}.$ Suppose that ...
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121 views

Which is bigger $e^{(a+b)}$ vs $e^a + e^b$?

I understand that exponential function is a convex function so for any convex function $\theta(a+b) > \theta(a) + \theta(b)$, but can someone provide a more formal proofs ?
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28 views

Can we solve minimax in this way?

I am working to use proximal operators for solving a minimax optimization problem. It is known that if you use alternative optimization, the algorithm cycles, see an answer to this question ...
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1answer
55 views

Minkowski sum and vectors

Problem: Given two convex polygons A, B, we can define Minkowski sum, as A + B = {a + b: a $\in$ A, b $\in$ B}, where a + b vector sum. Prove that: for every external perpendicular u to an edge of ...
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53 views

convexity of a nonlinear function

I tried to search similar answers to my problem here, but unfortunately I'm a little bit lost on this subject, originally from a physical problem, but can be stated as: Let $A\subset R^3$ be a ...
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1answer
38 views

How to prove conv(M)+u=conv(M+u) and Aff(M)+u=Aff(M+u)

I don't know how to prove that if $ M \subseteq R^n, \forall u \in R^n $ then conv(M)+u=conv(M+u) and Aff(M)+u=Aff(M+u). conv is convex hull and Aff is affine hull. Yes it is a homework question, ...
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1answer
54 views

Required conditions for product of two submodular functions to be submodular?

3.32 of Boyd's Convex Optimization book says: Products and ratios of convex functions. In general the product or ratio of two convex functions is not convex. However, there are some results that ...
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3answers
38 views

Convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant 1$

I would like to ask the convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant1$. We know that $y\geqslant\frac{1}{x}$ is convex for $x>0$. But if we transform the inequality into ...
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If we know the convex conjugate of $f(x,y)$, what can we say about the conjugate of $f$ in $x$?

Say $f^*(x,y)$ is the convex conjugate of $f(x,y)$. Now take $g_{y_0}(x) := f(x, y_0)$. Is there any relationship between $g^*_{y_0}(x)$ and $f^*(x, y_0)$?
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Subset relation between convex cone and its dual

Let $C$ be a convex cone and $C^*$ its dual cone. It seems for me that either $C\subseteq C^*$ or $C^* \subseteq C$ at least in 2 dimension. Is it correct? if so, is it the case also for higher ...
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1answer
41 views

Open ball around a point in the hyperplane in a locally convex space

Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$. It is my intuition that $V_z$ should ...
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Convex dense subset of $\Bbb{R}^n$ is the entire space

Say we have a convex dense set $X\subset\Bbb{R}^n$, does it follow that $X=\Bbb{R}^n$ ? For $n=1$ it's true because convex set of real numbers are intervals, and if it's dense then it's $\Bbb{R}$. ...
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1answer
17 views

Optimality condition

I was looking at a few results of convex optimization and I'm stuck with a part of a proof. Consider the following minimization problem: \begin{align} \text{minimize} \quad &\Phi(x) \\ ...
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linear inequalities and reduction

Let $X\subset \mathbb R^m$ be a convex compact subset. Let $E=\{-1,0,1\}^m$. What would be the necessary or sufficient conditions on $X$ such that for any $f\in [-1,1]^m$, there exists at least ...
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Can someone confirm this sketch of a proof for the existence of smooth partitions of unity on an unbounded convex open euclidean set?

EDIT!! The problem originally described (see below) has been reduced to the correctness of a simple extension of an argument from Rudin's PMA. Feel free to skip to the proposed solution, below. As ...
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Convexity of distance function

For any $n \in \mathbb{N}$, $a \in \mathbb{R}$ with $a > 1$ and $k_i > 0$ for $i = 1,\ldots,n$ define the following function: $$f: \mathbb{R}_{>0}^n \to \mathbb{R}, x \mapsto \sum_{i=1}^n ...
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Find maximum value of ${a}_{0}=A$ ${a}_{n}=B$

We have numbers $A$ and $B$, sequence ${a}_{0},\cdots,{a}_{n}$ such that: ${a}_{0}=A$ ${a}_{n}=B$. All greater than 0. Find maximum value of $\prod_{i=1}^{n}\frac{{a}_{i}}{{a}_{i-1}+{a}_{i}}$ Hint: ...
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1answer
32 views

Convexity, Hessian matrix, and positive semidefinite matrix

I would like to ask whether my understanding of convexity, Hessian matrix, and positive semidefinite matrix is correct. For a twice differentiable function $f$, it is convex iff its Hessian $H$ is ...
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49 views

general definition of concavity, mean-preserving spread and concavity

The usual definition of concavity is: for any $x_1$ and $x_2$ and any $t\in[0,1]$, $$f(tx_1+(1-t)x_2)\geqslant tf(x_1)+(1-t)f(x_2).$$ I am wondering how to generalize this definition to more than 2 ...
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Taylor's series question on bounds

$f(x) = f(y) + f'(y)(x-y) + \frac{f"(y) (x-y)^2}{2} + ....$ is the taylor's series. I am aware of bounds which specify the error in approximating $f$ with a polynomial. I want sufficient conditions ...
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1answer
33 views

Convexity of $f(x,y)=\frac{x}{y^2}$

I would like to ask the convexity of function $$f(x,y)=\frac{x}{y^2}$$ where $x\geqslant0, y>0$. Since $f(x,y)$ is differentiable but not twice differentiable, I used the first order condition ...
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21 views

How to perform a quasiconvex optimization

I have a quasiconvex objective function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ which I would like to minimize over a simplex $S\subseteq \mathbf{R}^n$. I have looked pretty hard but have been unable ...
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1answer
36 views

Functions mapping convex sets on convex sets

A function $f:\Bbb R^n\to\Bbb R^m$ is said that maps convex sets on convex sets if it satisfies: (1) the image $f(A)$ of any convex set $A$ is a convex set; (2) the preimage $f^{-1}(B)$ of any convex ...
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21 views

function that is higher at constant sequence than random sequence is concave

I have a T-sequence $x=(x_1,x_2,...,x_T)$ where each $x_i$ random, but they have same expectation $c$, i.e., $E[x_i]=c$, for all $i$. Another T-sequence that is constant, i.e., $x_i=c$, for all $i$. ...
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23 views

Is the subdifferential always convex and closed set?

Two properties of the subdifferential set are stated as follows: Given a function $f : \mathbb{R}^n → \mathbb{R}$, (i) the subdifferential set $\partial f(x)$ is always convex and closed, even if ...
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10 views

Whether it is jointly convex or not?

I know that $f(x, Y)=x^HY^{-1}x$ is a convex function on $x$ and $Y$ jointly, where Y is positive definite. Now, if $Y=\Sigma_{n=1}^{N}A_nSA_n^H+I$, where $S$ is positive semidefinite and $I$ is the ...