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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Non-empty intersection of convex sets

Assume that $X_1,\ldots,X_n$ are open, convex subsets of $\Bbb R^d$ such that for any $i,j,k$ with $1\le i,j,k\le n$, we have $X_i\cap X_j\cap X_k\neq\emptyset$. Is it possible for $\bigcap_{i=1}^n ...
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1answer
83 views

Maximal intersection of slabs in $\mathbb{R}^n$ with a compact convex centrally symmetric set

Let $K \in \mathbb{R}^n$ be a compact convex set containing the origin and symmetric with respect to the origin. Let $S_i(t_i)$ be a finite set of slabs of various widths and orientations, translated ...
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Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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1answer
49 views

Minimizing a non-convex rational function of two variables

I need to minimize the following function $$f(x,y)= \frac{a}{x}+\frac{bx}{y}+\frac{cy}{x}+dy+\frac{e}{y}$$ where $a,b,c,d$, and $e$ are positive constants, and $x$ and $y$ are both strictly positive. ...
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Write the convex hull of the points $(0,0), (1,1), (2,0) \in \mathbb R^2$ as the set of solutions to a finite number of inequalities.

Write the convex hull of the points $(0,0), (1,1), (2,0) \in \mathbb R^2$ as the set of solutions to a finite number of inequalities. The convex hull of $x_1, \ldots, x_n \in \mathbb R^n$ is defined ...
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1answer
115 views

Strictly increasing and strictly convex function that does not go to negative infinity

Let $f : \mathbb{R} \to \mathbb{R}$ be continuous, strictly increasing, and strictly convex for all $x$. What are necessary and sufficient conditions for $f$ to be bounded below? Strong convexity is ...
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25 views

Some good reference on convex functions

I want some good reference on convex functions i.e. all of their properties along with their proofs.
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87 views

how to check an optimization function is convex or not

This is the sparse coding optimization function: $\operatorname*{argmin}_{B, \alpha} \sum_j \| \bf{x}_j - B\bf{\alpha}_j \|_2^2 + \lambda\sum_j |\bf{\alpha}_j|_1$ I read in the literature that this ...
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24 views

Does $\mu\coth(\mu)=A\mu^{2}+B$ have at most two positive solutions $\mu$?

Is it true that $$ \frac{\mu\cosh(\mu)}{\sinh(\mu)} = A\mu^{2} + B $$ has at most two solutions $\mu > 0$ for any choice of $A$, $B$? I believe this is true; it looks true when I ...
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Convexity of sigmoid-based squared error loss

Assume $w, a_1,a_2 \in \mathbb{R}^d$ and $\sigma = \dfrac{1}{1+e^{-x}}$ the sigmoid function. Is the following squared difference a convex function? $$J(w)= (\sigma(w^Ta_1)\times \sigma(w^T ...
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Directions of sublevel sets ofa convex function

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a convex (continuous ) function. Let's assume that some sublevel set $L_a:=\{x\in \mathbb R^n : f(x) \leq a\}$, where $a \in \mathbb R$, contains a ...
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30 views

Existence criterion for solution in quadratic programming

I have the problem $$ \begin{align*}\min \quad&f(x)= c^Tx + x^TQx \\ &x\in D \end{align*}$$ with $D=\{ x \in \mathbb{R}^n \mid Ax \leq b\}$, $A,Q\in \mathbb{R}^{n\times n}$ and $b,c \in ...
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107 views

Shrinkage operator for matrices

Here http://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf, on page 188, you can see the derivation of the soft thresholding operator or shrinkage operator for the case of vectors using Moreau ...
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38 views

what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
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Distributed Convex Optimization Algorithm

Consider the convex optimization problem $$ \min_{x_1, \cdots, x_N, y} \sum_{i=1}^{N} f_i(x_i,y) $$ $$ \text{subject to: } x_i \in X_i \ \ \forall i, \ \ y \in Y, \ \ y = \sum_{i=1}^{N} x_i $$ ...
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Convex interpolation between two points with given derivatives

Let's say I have two real values $x_1$ and $x_2$, to each of which I associate $y_i$ and $y'_i$ satisfying $$ (y_2-y_1)(y'_2 - y'_1) \geq 0. \tag{1} $$ I would like to find a polynomial ...
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1answer
101 views

Firm non-expansiveness in the context of proximal operators

$\newcommand{\prox}{\operatorname{prox}}$ Probably the most remarkable property of the proximal operator is the fixed point property: The point $x^*$ minimizes $f$ if and only if $x^* = \prox_f(x^*) ...
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21 views

What is the nature of this one dimensional function?

Let $\mathcal{S}$ be a 2-D convex set whose elements can be represented as $(x,y)\in\mathcal{S}$. Let $p_L$ and $p_U$ be two real constants such that $p_L\leq p_U$. For $p\in[p_L,p_U]$, I define the ...
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Convex Combination of 3 point in R2 and Triangle

I am new to convex combination, and I am quite amazed by some easy result. I know that convex combination of 2 points($P_1P_2$) in $R^2$ is all points in the line segment $P_1P_2$. And then I see a ...
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1answer
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Interesting and intuitive affirmation involving convex sets

Let $\Omega_1$ and $\Omega_2$ two open, bounded and convex domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ and $0 \in \Omega_2.$ Suppose that for each $x_0 \in \partial (\Omega_1 ...
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Common subdifferentials of convex function

Let $f: \mathbb R^n \rightarrow \mathbb R$ be a convex function. By a subdifference of $f$ in $x\in \mathbb R^n$ we mean an $h\in \mathbb R^n$ such that $f(x) \geq f(p)+<x-p,h>$ for all ...
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Dual norm of the matrix L1 norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
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Suitable composition of concave and convex functions is convex?

Let $f:[0,1]\to[0,1]$ be a strictly increasing continuous concave function with $f(0)=0$ and $f(1)=1$. Let $g$ be the inverse of $f$. Then $g$ is strictly increasing and convex. It seems that the ...
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Basic question about tangent cone

The following is from Prof. Jahn's book " Intro. to the theory of nonlinear optimization" about tangent cone: After definition, he gave an example: My question is: Does the tangent cone include ...
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190 views

Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
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104 views

Convolution of Strictly Convex Function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^2$, strictly convex function, and $\theta_\epsilon$ the standard approximation to identity ...
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Is the distance between disjoint closed convex subsets of a Hilbert space positive? Is it attained?

Let $H$ be an infinite dimensional and separable Hilbert space. Let $A,B$ be infinite, closed and convex subsets of $H$. If $A$ and $B$ are disjoint and if at least one of them is bounded, is the ...
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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 ...
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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 ...
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1answer
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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, ...
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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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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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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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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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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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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 ...
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1answer
28 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 ...
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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 ...
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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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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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1answer
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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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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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On the decomposition of stochastic matrices as convex combinations of zero-one 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 & ...
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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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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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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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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
134 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 ...
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127 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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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 ...