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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How to prove that the following function is convex?

I want to prove convexity of the following function: $$f(x) = log_x \left(1 + \frac{(x^a-1)(x^b - 1)}{x-1}\right)$$ for any fixed $a, b \in (0, 1)$ and: $x\in(0,1)$ $x\in(1, \infty)$ I'm trying ...
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1answer
24 views

Subtracting a constant from log-concave function preserves log-concavity, if the difference is positive

I am trying to work out a question from 'Convex Optimization - Boyd' . Specifically, exercise 3.48: Show that if $f : \mathbb R^n \to \mathbb R$ is log-concave and $a > 0$, then the function $g ...
2
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1answer
31 views

Affine to linear like conversion of a concave function

Is the following true: $$\log \left( \frac{1}{f(x)+K}\right)\mathrm{is\;concave}\Longleftrightarrow \log \left( \frac{1}{f(x)}\right)\mathrm{is\;concave},$$ where $K\in\mathbb{R} $ and ...
2
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1answer
60 views

Point-wise converging convex functions on $[0,1]$

Suppose we have a sequence of continuous convex functions $\{f_n\}$ defined on $[0,1]$ which converge point-wise to a limit $f$ on $[0,1]$, i.e. for all $x \in [0,1]$ $$\lim_n f_n(x) = f(x).$$ Let $G ...
0
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1answer
553 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
0
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1answer
38 views

On the decomposition of Stochastic matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
0
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1answer
212 views

Checking convexity

I know that the function $(\mathbf{a}-\mathbf{b})'(\mathbf{a}-\mathbf{b})$ is convex in $\mathbf{a}$ ($\mathbf{a}$ and $\mathbf{b}$ are vectors, not scalars). Would ...
0
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1answer
17 views

Do full-rank linear transformations preserve strong convexity?

Consider a strongly convex function $g$, that is, for all $x,y$ in the domain and $t\in[0,1]$ we have $$ g(t x + (1-t)y) \le tg(x)+(1-t)g(y) - \frac{1}{2}mt(1-t)||x-y||_2^2 $$ for some $m>0$. Also, ...
6
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3answers
222 views

Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
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0answers
18 views

Convex Inequality describing Functions inside specific area

Let us assume that we have two functions $f_1$, $f_2:[0,1] \rightarrow \mathbb{R}^{2}$, which describe each a point trajectory on the plane. Let us further assume that we parametrize those functions ...
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0answers
18 views

Intuitive affirmation on convex sets

Let $D_1, D_2$ two open, bounded and convex domain in $R^n$. Suppose that $D_2 \supset \overline{D_1}$, and the boundaries of these sets are of class $C^1$. Fix $x \in \partial D_1$ and suppose that ...
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1answer
36 views

Prove that $f(x) > g(x)$ where both functions are convex and have the same value and slope at $0$

Let $f: [-a,a] \to \mathbb{R}$ and $g: [-a,a] \to \mathbb{R}$ be two non-negative, convex and smooth functions. We further know $f(0) = g(0)=0$ and $f'(0) = g'(0)=0$. I'd like to show $$f(x) \ge ...
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0answers
26 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
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2answers
141 views

The distribution of barycentric coordinates

Let $\mathcal{X} = \{x_1,\dots,x_n\} \subset \mathbb{R}^d$ and let $Z$ be a random variable uniformly distributed over convex hull of $\mathcal{X}$, denoted as $\text{conv}(\mathcal{X})$. Assuming ...
0
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1answer
22 views

Is strict convexity necessary and sufficient for non-degeneracy of the Hessian?

A function $f$ is called strictly convex if for $\lambda\in(0,1)$, $x\neq y,$ $$f(\lambda x + (1-\lambda)y) < \lambda f(x) + (1-\lambda)f(y)$$ If $f:\mathbb{R}^n\to\mathbb{R}$ is a twice ...
0
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1answer
10 views

How to derive the support function for this set?

I want to ask how to derive the support function of the convex set (in $\mathbb{R}^2$) that is described as the intersection of $x_1\leq \frac{3}{4}$, $x_2\leq \frac{3}{4}$, $x_1+x_2\leq 1$, and ...
0
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1answer
413 views

Convex conjugate of $\ell_1$ and $\ell_2$ norm

Given a function $f$, we can define a function $f^{*}$, called the convex conjugate (also known as the Fenchel conjugate) of $f$ as follows: $$ f^*(\vec{z})= \sup_{\vec x \in \mathbb ...
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1answer
299 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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2answers
51 views

Prove that $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex

At this link there is a demonstration that for $f$ continuously differentiable on $C \subseteq \mathbb{R}^n$ convex, $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex. This ...
3
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1answer
38 views

Generalization of log-convexity (log-concavity): log-log-convexity (log-log-concavity)?

$\underline{\mathrm{Background\; on\; function\; Convexity}}$ A function, $f$, is convex if: $$f( x\theta+y(1-\theta) ) \leq \theta f(x) + (1-\theta)f(y).$$ $f$ is concave if $-f$ is convex, [1]. If ...
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1answer
25 views

Difference quotients are increasing for $f$ convex

Problem 11A.9(c) in Spivak's Calculus (4th edition) asks the following (I'm paraphrasing): Suppose $f$ is convex. Show that $f'(a)$ exists iff $f_+'(x)$ is continuous at $a$. ($f_+'(x)$ is the ...
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263 views

Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any ...
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1answer
35 views

Prove that a Hilbert space is convex of power type $2$

Let $X$ be a Banach space. For $\epsilon \in (0,2]$, define: $$\delta_X(\epsilon) = \inf_{x,y \in X}\{1 - \|\frac{1}{2}(x + y)\| : \|x\| = \|y\| = 1, \|x-y\| \ge \epsilon\}.$$ Then we say that $X$ ...
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1answer
44 views

Are strictly convex functions with positive second derivatives on compact domains strongly convex?

Claim: Let $\chi$ be a compact set. If $f''(x)>0$ for all $x\in\chi$, then $f$ is strongly convex. This seems to be true, intuitively, as I can't think of a counterexample. All of the examples ...
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1answer
32 views

On a Banach space $X$, is the functional $x \mapsto \frac{1}{p}\|x\|^p$ convex?

Let $X$ be a Banach space. Let $p > 1$ and, consider the functional $X \to \mathbb{C}$ given by: $$x \mapsto \frac{1}{p}\|x\|^p$$ I would like the know if the above functional is convex. That ...
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0answers
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A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
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0answers
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How to convexify (relax) this L0 eigenvalue optimization problem?

Let $C_1,\dots,C_L$ be $N\times N$ hermitian matrices. Let $d<0$ be a given negative constant. Then consider the optimization problem \begin{align} \max_{r\in \mathcal{R}^{L\times 1}} &\mid\mid ...
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3answers
100 views

Is every such function convex or concave?

I was pondering convex functions today, and the following questions naturally posed themselves. Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ $2$-limited iff for all $m,c \in \mathbb{R},$ ...
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1answer
103 views

Knuth's Sandwich Theorem: requesting proof clarification

The question is about F6 of Section 8 ("Elementary facts about cones") in Donald Knuth's Sandwich Theorem (http://arxiv.org/pdf/math/9312214.pdf). He claims to prove $(A \cap B)^* = A^* + B^*$ when ...
2
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1answer
73 views

$f$ is convex function iff Hessian matrix is nonnegative-definite.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le ...
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0answers
42 views

Strictly Convex Functions

I am trying to show the equivalence of two definitions of strictly convex functions. Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth function. The function $f$ is strictly convex if for each ...
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6answers
227 views

Convex Sets in Functional Analysis?

Why did Bourbaki choose to study convex sets, convex functions and locally convex sets as part of the theory of topological vector spaces, and what is so important about these concepts? I'd like to ...
2
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1answer
63 views

How can the sum of two closed cones be not closed?

Can there be two closed cones $K_1$ and $K_2$ in $\mathbb{R}^3$ such that $K_1+K_2$ need not be closed?
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1answer
40 views

Optimization problems on the circle

Consider the optimization problem $$ \min_{x \in \mathbb{R}^2} x^{\top} P x + q^{\top} x$$ subject to: $$ A x = b, \ x \in X, \ x_1^2 + x_2^2 = 1$$ where $X$ is compact and convex. Then ...
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2answers
32 views

Convexity of functions

I'm trying to read up on convex/concave functions (is that the same as concave up, concave down?) If I were asked to prove a convexity of a function, what are the general steps to follow? (So far I ...
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1answer
12 views

Quasi-Concavity and Quasi-Convexity

My book states that: $f$ is a quasiconcave function on $U$ if for all $x,y \in U $ and $t \in [0,1]$: $f(x) \geq f(y) \implies f(tx + (1 - t)y) \geq f(y)$ $f$ is a quasiconvex function on $U$ ...
0
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1answer
47 views

$F(x) = f(x) + g(x) + h(x)$, where h(x) is strictly convex , is also strictly convex

$\newcommand{\prox}{\operatorname{prox}}$ $\newcommand{\argmin}{\operatorname{argmin}}$ $\newcommand{\Tr}{\operatorname{Tr}}$ Suppose $g: \mathbb{R}^n \rightarrow \mathbb{R}$ is a continuous convex ...
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1answer
38 views

Two Strictly Convex Functions with Contact of Order 1

Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f\geq g$. Suppose that for some $x_0\in \mathbb{R}$: ...
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1answer
87 views

Gradient descent (with line search) for convex functions viewed as alternation

I have fundamental confusion about gradient descent (with line search) and the reason it works. I try to explain my view here, and please tell me where it goes wrong. Let $f: \mathbb{R}^n \to ...
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1answer
17 views

Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
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1answer
30 views

Strictly Convex and Differentiable Implies

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be strictly convex and differentiable. Is $f$ strongly convex when restricted to a closed and bounded interval $[a,b]$? This is true if $f$ is smooth but am ...
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1answer
30 views

Lipschitz in $\mathbb R^1$ implies Lipschitz along any line in $\mathbb R^k$ (for convex functions)

I'm trying to solve a homework problem (baby rudin course) and realized that Lipschitz continuity really makes the problem easier for me. However, since it is not defined in Rudin I have to prove the ...
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2answers
254 views

How to compute the image of a polyhedron under a linear transformation

Suppose $P$ is a polyhedron whose representation as a system of linear inequalities is given to us: $$ P = \{ x ~|~ Ax \leq b\}$$ Define $P'$ be the image of $P$ under the linear transformation which ...
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0answers
19 views

convex set but not convex cone

Let S be a subset of R^n. If we say S is convex cone, it means S is convex and it is a cone. Obviously, a convex set contains convex cone. So can we take an example to say that it is a convex set ...
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0answers
17 views

separating hyperplane theorem proof

I need help understanding the last part of this proof, the lemma they are refering too is just a lemma about convex sets that shows us that p is unique: I do not see how H* separates C and z. The ...
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1answer
100 views

Convex Functions are Continuous

In W. Rudin's Principles of Mathematical Analysis, we read in Chapter 4 that real-valued functions defined on an open interval $(a,b)\subseteq\Bbb R$ are continuous (specifically, Exercise 23). I am ...
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1answer
67 views

Convex polygon is connected by broken lines with each chain parallel to one of two vectors

Given is convex polygon $P$ on the plane. Settle if there are two vectors $\vec{a}$, $\vec{b}$ such that any two points belonging to the polygon $P$ you can connect a broken line contained in the ...
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2answers
100 views

On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
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2answers
32 views

Prove that the set $C = \{x \in\Bbb R : ax\le b\}$ is convex

Prove that if a and b are real numbers, then the set $C = \{x \in\Bbb R : ax\le b\}$ is a convex set. My solution so far: To show that a set $C$ is convex it needs to be shown that for for every ...
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0answers
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Prove $T(e) = \int_{e}^{\infty} (y-e) \mathrm{d}F(y)$ is convex

Prove that $T(e) = \int_{e}^{\infty} (y-e) \mathrm{d}F(y)$ is a convex function, where $F(y)$ is cumulative distribution function of some real random variable $Y$. This is an exercise from ...