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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determining whether a set is convex

I came across this exercise question from a course on optimisation. It only discussed basic aspects of convex functions. The question asks: if the solutionn set of the following inequalities convex. ...
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146 views

How many hyperplanes does it take to separate $n$ points in $\mathbb{R}^m$?

By 'separate', I mean that each point lies in its own little region/cell. For instance, it takes a minimum of $P = 4$ lines to separate $n = 7$ points in $\mathbb{R}^2$ ($m=2$), assuming that no 3 ...
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189 views

What is the expected convex depth of a set of $m$ randomly chosen points in the unit square?

Definition. Let $X$ be a set of points in $\mathbb{R}^2$. Then the vertex sequence of $X$ is defined by $X_{0}=X$, and $X_{i+1}=\left\{x\in \operatorname{Conv}(X_{i}): x \notin ...
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1answer
18 views

Let $Ax = b$ be a system of hyper-planes that form a bounded convex $D$. Can $D$ be partitioned into union of adjacent simplices?

Let $Ax = b$ be linear system that forms $q$ bounded region $D$. If the columns of $A$ are independent, can $D$ be written as a union of adjacent simplices?
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1answer
117 views

Composition of a convex function

If $f:[a,b]\rightarrow R$ is convex function and $f'(x)\geq 0$ for all $x\in [a,b]$ and $g:U\rightarrow [a,b]$ is convex function, how to show that $f(g(u)), u\in U$ is convex function?
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1answer
41 views

supremum of an array of a convex functions

If $\{J_n\}$ is an array of a convex functions on a convex set $U$ and $G(u)=\sup J_i(u), u\in U$, how to show that $G(u)$ is convex too? I've done this, but I am not sure about properties of a ...
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1answer
120 views

separation theorem and convex cone

I have just a small question in a proof in my functional analysis script. I have a set $A\subset L^p$, where the latter is the usual $L^p$ over a space with finite measure $\mu$. The set $A$ is also ...
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Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$

Can you help me to prove that $$(x+y)^n\leq 2^{n-1}(x^n+y^n)$$ for $n\ge1$ and $x,y\ge0$. I tried by induction, but I didn't get a result.
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1answer
44 views

Convexity of a function depending on value of parameters

Check out convexity of a function $J(u)=cu^r$, $J:[a,b]\rightarrow R$, $0<a<b<\infty$, depending on values of parameters $c,r\in R$. I know a definition of a convexity: "Function ...
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1answer
803 views

Strictly convex sets

If $S\subseteq \mathbb{R} ^2$ is closed and convex, we say $S$ is strictly convex if for any $x,y\in Bd(S)$ we have that the segment $\overline{xy} \not\subseteq Bd(S)$. Show that if $S$ is compact, ...
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How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
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1answer
72 views

Show that an affine function on a convex and compact set $\Omega \subset \mathbb{R}^d$ is convex?

Please check this out Prove the supremum of the set of affine functions is convex The answer generalizes without proof that ''every affine function $f_i$ is convex'' on $\Omega \subset \mathbb{R}^d$. ...
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1answer
52 views

Convexity of expected value

I am trying to understand if the expected value of a variable is convex in that variable or not. I know that expectation is a linear operator, so must be convex. But I do not see why it does not ...
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1answer
79 views

Convexity of a bi-variate function when its second derivatives are discontinuous?

A piecewise function $F(x_1,x_2)$ is continuous, so are its 1st partial derivatives. However, its 2nd partial derivatives are discontinuous, that is , $F$ is not of class $C^2$. But its Hessian ...
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1answer
272 views

Convex homogeneous function

Prove (or disprove) that any CONVEX function $f$, with the property that $\forall \alpha\ge 0, f(\alpha x) \le \alpha f(x)$, is positively homogeneous; i.e. $\forall \alpha\ge 0, f(\alpha x) = \alpha ...
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234 views

Show that any convex function is locally bounded

Show that a convex function $f:\mathbb{R}^n \rightarrow \overline{\mathbb{R}}$ is bounded in a neighborhood of $x\in \text{ri}(\text{dom}(f))$. Showing that it has an upper bound is not difficult ...
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78 views

Does convexity of all projections imply convexity in higher dimensions?

If I have $n$ 2-dimensional convex "regions" that are the projections along $n$ (independent) dimensions of a n-dimensional compact subspace: Does that imply that the latter is convex? Is the ...
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1answer
153 views

Determining the convex hull of the union of two polyhedra

I'm doing an introductory course to linear programming and I'm working through some exercises to prepare for the final exam, I'm stuck on an exercise and I would really appreciate a hint: Let ...
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1answer
635 views

Pointwise supremum of a convex function collection

In Hoang Tuy, Convex Analysis and Global Optimization, Kluwer, pag. 46, I read: "A positive combination of finitely many proper convex functions on $R^n$ is convex. The upper envelope (pointwise ...
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672 views

maximum and minimum singular values

I was studying the Stephen Boyd's textbook on convex optimization and have a question. The book says the following: The singular values of $A$, $\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n$ are ...
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4answers
288 views

Is second derivative of a convex function convex?

If $f$ is twice differentiable and convex, is it true that $f''$ is a convex function ?
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1answer
64 views

Convex functions on real vector spaces

So I'm trying to solve the following problem, Suppose that $f$ is a non-zero convex real-valued function on a real vector space $V$ with $f(0) = 0$ Show that there is a linear functional $g$ on $V$ ...
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indirectly convex

Let g:$\mathbb{R^2}$$\rightarrow$$\mathbb{R}$ be defined by g(x)=Max{$x_1$,$x_2$} at each x=($x_1$,$x_2$)$\in$ $\mathbb{R}$. Determine whether or not g is indirectly convex on $\mathbb{R^2}$. ...
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50 views

Example on Correspondences

Giva an example of correspondence $F : \mathbb{R} \rightarrow \mathbb{R}$ such that the closure of $F$ is $ \overline{F}: \mathbb{R} \rightarrow \mathbb{R}$, upper semi continuous on $\mathbb{R}$, ...
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Finding convex conjugate of a bounded function

The convex conjugate of a function $f:\mathcal{X}\mapsto \mathbb{R}$ is formally defined as $$f^\star\left(y\right)=\sup_{x\in\mathcal{X}}\ \left\langle x,y\right\rangle-f\left(x\right).$$ In cases ...
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Linear Difference Equations

Let $g_1, g_2 \in \mathbb{R}^{[0,5]}$ be defined by $g_1(x) = x$ and $g_2(x) = 1-x$ for each $x \in [0,5]$. Find a second-order homogeneous linear difference equation on $[0,3]$ such that {$g_1, g_2 ...
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35 views

Indirect Concavity

Let $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $g(x_1,x_2) = e^{\min{(x_1,x_2)}}$ at each $ x_1, x_2 \in \mathbb{R}^2$. Find whether or not $g$ is indirect concave on $\mathbb{R}$.
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How can i show this inequality?

Let $n>1$ and $a_1,...,a_n \in \mathbb{R}^+$ be such that $\sum a_i=1$. For evey $i$, define $b_i=\sum_{j=1,j\neq i}a_j$. Show that $\sum_{k=1}^n \dfrac{a_k}{1+b_k}\ge \dfrac{n}{2n-1}$ Thanks a ...
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convexity of norm

I want to show that $f(v)=\|v\|^p$ for $1\leq p<\infty$ is strictly convex. In the simplest case when $p=2$, we have: ...
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73 views

Does convex and radially open imply open?

I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace. Here the 'openness' we are talking about is from any normed space. ...
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1answer
38 views

secant method twice on convex decreasing function

I have a continuous, decreasing and convex function $f$. Given an interval $[a, b]$ such that $f (a)>0 $and $f (b)<0$, if I apply the secant method twice, where the outcome point will be ...
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Suggestions for a reference-level text on optimization theory?

I'd be interested in knowing if anybody has suggestions on an advanced but still self-contained reference on optimization theory, centered around linear and convex problems. The key feature of my ...
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1answer
271 views

Maximum of quasi-convex functions

A function $f$ is quasiconvex if all its sub-level sets are convex (i.e., $\{ x: f(x) \le \alpha\}$ is convex for all $\alpha$.) For a convex function $f$, it is true that $f$ acheives its maximum ...
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1answer
139 views

How to prove the property of convex function in higher dimension

Suppose $f:\mathbb{R}^n\mapsto \mathbb{R}$ is convex, could anyone tell me how to prove the following fact? (1) If $f\in C^1$, then for any $u,v$ $$ f(v)\geqslant \langle\nabla f(u),v-u\rangle $$ ...
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On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has coordinate-wise Lipschitz constant $M$ (meaning that for all ...
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117 views

KKT conditions of this convex optimization problem

Consider a continuous function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that $\forall y \in \mathbb{R}^m$ $f(\cdot,y)$ is convex, and $\forall x \in \mathbb{R}^n$ $f(x,\cdot)$ ...
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1answer
559 views

Convex analysis: relative interior in finite and infinite dimension

Let $X$ be a normed space. Given a nonempty convex set $C \subset X$, the relative interior of $C$, denoted by $\text{ri} C$, is the set of the interior points of $C$, considered as a subset of ...
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Convexity of polylogarithms

I want to prove the following proposition: The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$. And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
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Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as $\Omega_f(x) \triangleq ...
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1answer
46 views

Where the gradient of a convex function approaches zero

Does there exists a differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with a unique global minimum which we denote by $x^*$ such that there exists a sequence $x_k$, not ...
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1answer
86 views

Find function with given properties

Find a smooth function $g: \mathbb{R} \to \mathbb{R}$ that domain $g$ is $\mathbb{R}$ range of $g$ is a subset of $\mathbb{R^+}$ $g$ is concave.
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Are there logarithm functions for arbitrary rings?

The logarithm function for $\mathbb{R}$ satisfies $\log xy = \log x + \log y$ whenever both $\log x$ and $\log y$ are defined. Are their conditions for a ring $R$ which guarantee the existence of a ...
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1answer
105 views

Analytic proof that $\log{\Phi(x)}$ is concave?

How can one prove that $\log{\Phi(x)}$ is a concave function in x? I tried taking second derivative, but so far it isn't helpful. I read a hint on my textbook that says it is easy to show its first ...
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1answer
342 views

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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A robust convex optimization problem

Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
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1answer
113 views

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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364 views

Every exposed point is a extreme point

Let $C$ be a non-empty convex subset of $\mathbb{R}^n$. We say that $x\in C$ is a extreme point of $C$ if for every $z,y\in C$ and $t\in [0,1]$ such that $x=ty+(1-t)z$ we have $x=z$ or $x=y$. Or ...
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1answer
48 views

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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1answer
48 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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50 views

Extreme points and positive linear combinations

Let $C$ be a non-empty convex subset of $\mathbb{R}^n$. We say that $x\in C$ is a extreme point of $C$ if for every $z,y\in C$ and $t\in [0,1]$ such that $x=ty+(1-t)z$ we have $x=z$ or $x=y$. Or ...