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 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
11 views

Conic hull of a proper function

Suppose $f$ is a proper function pn $\mathbb{R}^{n}$with $f(0)>0$.Now consider $$ g(x) = \text{inf}\{t: (t,x) \in \text{cl(cone(epi(}f)))\} $$ Can I always say that $\exists y \in \mathbb{R}^{n} : ...
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20 views

What is a “symmetric convex function”?

I was reading a document (on doubly stochastic matrices) and they write : "a symmetric convex function $f \, : \, \mathbb{R}^{n} \, \longrightarrow \, \mathbb{R}$". I don't understand what "symmetric" ...
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1answer
42 views

Intersection of strictly monotone, convex functions on interval with two given intersections

Given two functions $f,g:[0,1]\rightarrow[0,1]$ that are smooth, strictly convex, strictly monotone s.t. $f(1)=g(1)=0$, $g(0)=f(0)=1$ and $f(x_0)>g(x_0)$ holds for some point $x_0$, does it follow ...
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4answers
113 views

Why is the projection of a closed polytope closed?

In general, projection of a closed set into a subspace does not result in a closed set. However, I was able to prove that in $\mathbb{R}^n$, the projection of a closed polytope (intersection of ...
3
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1answer
68 views

Algebraic Proof that a Disk is Convex

After searching on Google for a while, I cannot seem to find an algebraic proof that a disk is a convex set. Intuitively, this seems obvious: if you take any two points $x, y$ in a disk, then the line ...
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1answer
14 views

Problem with convex function

In Papadimitriou book I found a problem. If I know that function $f$ is a convex function, and I have values $x_2,...,x_n$, is function $g(x_1) = f(x_1,x_2,...,x_n)$ also a convex function? I know ...
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184 views

How to formally prove that f(x,y) is jointly convex if f(x,y)=h(g(x,y))?

I know that this function should be concave, I am working on the Hessian proof but I would rather use this property. I know that h(a) is convex and decreasing in a, and g(x,y) is linear, ...
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1answer
55 views

Invariant convex subset

Let X be a banach space and $T$ a bounded operator on X with $||T||$ less than or equal to 1. If $T$ is an isometry and $r(T)$<1 show that there is a closed convex non-zero proper subset $C$ of the ...
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3answers
337 views

How to prove that $e^x$ is convex? [closed]

I need a help with proving of $e^x$ function. We know $e^x$ is a convex function. $$f(c(x')+(1-c)(x'')) ≤ c\cdot f(x')+(1-c)\cdot f(x''), \quad 0 ≤ c ≤ 1$$ Thanks.
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Why the cone generated by a closed convex set containing the origin may not be closed?

I encounter this problem in the proof of Theorem $1.28$, page 21, Skiadas' Asset Pricing Theory. $X$ is a constrained market which is defined to be a closed convex subset of $\Bbb R^{K+1}$, $C = \{kx ...
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0answers
64 views

KKT conditions for nonsmooth convex problems

What are the KKT conditions for a non-smooth convex function? Is the vanishing gradient of Lagrangian, replaced by $0$ in sub-differential of the Lagrangian, and all other things remain the same? I ...
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1answer
45 views

Sets defined by distance to a convex set

Let $Y \subset \mathbb R^n$ be a bounded convex set, let $R>0$, and let $$Z := \left\{z \in \mathbb R^n : d(z,Y) > \dfrac12R \right\}$$ where $$ d(z,Y) = \inf_{y\in Y}|y-z|. $$ If you like, ...
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28 views

Convex Function defined on a closed convex set, Semi-Definiteness of the Hessian Matrix

I have a real-valued function that is defined on the subset of the non-negative (possibly zero) n-dimensional real numbers that add to one, i.e., a closed and convex set. Moreover, the Hessian matrix ...
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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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273 views

A beautiful inequality for convex functions

Let $f\in \mathcal{C}([0,1],\mathbb R_+)$ increasing. Prove that there exist $g,h\in \mathcal{C}([0,1],\mathbb R)$, convexs, such that $g\leqslant f \leqslant h$ and : $$\displaystyle ...
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1answer
76 views

Level set of convex functions

Let $f:\mathbb R^n \to\mathbb R \cup\{+\infty\}$ be a proper convex function, assume that there exists $c\in\mathbb R$ such that the $c$-level set $L_{\leq c}=\{x\in R^n: f(x)\leq c\}$ is nonempty and ...
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27 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
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1answer
76 views

Explain the convexity by looking the hessian matrix of a function

The hessian matrix of a function is given by, $$ H = \begin{bmatrix} a & b & c \\[0.3em] b & b & 0\\[0.3em] c & 0 & c \end{bmatrix} $$ where, ...
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3answers
206 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
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1answer
37 views

A characterization of convexity for functions with vectors as domain.

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a continuously differentiable function. By $df(w)$ I denote the Frechet derivative of $f$ at $w$ Prove that $$f \:\text{is convex} \Leftrightarrow ...
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58 views

Monotonically increasing maximum eigenvalue

Let a matrix $A \in \mathbb{R}^{n \times n}$ be the convex combination of two matrices as $A = qB + (1-q)C$. Define $B$ as unit anti-diagonal. Define $C_{i,j} = \delta_{i,i+1}$. Consider $A$ for ...
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1answer
31 views

Computation of convex conjugate

I am learning convex analysis by myself and I need help. How to show that if $X=U=\mathbb{R}$ and $f\left(x\right)=\frac{|x|^{p}}{p}$ then the convex conjugate ...
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32 views

How to handle concave-concave constraints?

there. I have an optimization problem, which takes the following form ...
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1answer
138 views

Proof of the Moreau decomposition property of proximal operators?

Given the prox operator i.e. $ prox_h (x) = arg min_u (h(u) + 1/2 ||u-x||^2_2) $ the Moreau decomposition property says that $ x = prox_h (x) + prox_{h^*} (x) $ where $h^*$ is the conjugate of ...
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36 views

Express a function as difference of convex functions (DC)

is there a way to express the function $$1-\exp \Big( \frac{-\max(0,x)^2}{\alpha} \Big)$$ as the difference of two convex functions (DC)? Thanks
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204 views

convexity of inverse function

I have a question on the reverse of a convex function. Let $f(x)$ be a convex function. Is the reverse function, say $g(x)=f(x)^{-1}$, is necessarily a concave function ? Considering that such ...
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26 views

Proving that $f(x) = (1+x)^{-\theta}$ is convex using only definition

How can I prove that the function $f(x) = (1+x)^{-\theta}$ (where $\theta>0$ is a parameter) is convex using only definition, i.e. without using derivatives? I actually want to prove convexity of ...
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111 views

Inequality related with concave property

Assume that $f>0,f'<0$ and $f$ is logconcave(the log of $f$ is concave) and twice differentiable. Can we prove, or give a counter example to the following claim: there exists $\bar x>0$ such ...
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Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
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2 questions - Convexity of a function

I have 2 questions about Convex functions Let f(x) be a convex function. Is the reverse function, say g(x), is necessarily a concave function ? (Considering that such function g does exist (f is ...
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Uniqueness of a convex linear decomposition

Let $X$ be composed of $d$ different vectors of $\mathbb{R}^n$ : $X=\{x_1,\ldots,x_d\}$ and $H$ be the convex hull of $X$. Each vector $y\in H$ can be expressed as $$y=\sum_{i=1}^d a_i x_i,$$ with ...
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Relax equality into inequality in convex problem

Let $\mathbf{x}, \mathbf{z}, \underline{\mathbf{x}}, \overline{\mathbf{x}} \in \mathbb{R}^{I}$, where the first two are variables and the last two are given data. I have the following problem: ...
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35 views

Convexity of support function

Let $C$ be a closed non-empty set, but not necessarily convex. The support function of $C$ is given by $$S(z) = \sup_{c \in C} \langle z,c\rangle. $$ Prove that this is a convex function. ...
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Minimizing difference and individual variables in convex problem

Let's say I have the following optimization problem: $$ \begin{align*} \min_{\mathbf{x},\mathbf{y}} & \sum_i x_i-y_i \\ \mathrm{s.t.} & \{\mathbf{x},\mathbf{y}\} \in ...
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Prove that the intersection of convex sets is convex using the following three points…

I want to prove each point, then, use points (1) and (2) to prove (3). $C_{1} = \lbrace x \in \mathbb{R}^{n} \mid h(x) = 0 \rbrace $ is convex iff $h(x)$ is affine in $C_{1}$ $C_{2} = \lbrace x ...
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1answer
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Matlab - Generate square convex function with positive definite Hessian Matrix

So, I have to generate a square convex function in Matlab and it's Hessian Matrix must be positive definite but I can't find any function that can help me do that. Is there anything I should search ...
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163 views

Why is this weighted least squares cost function a function of weights?

Here is a picture from my book regarding weighted least squares: Totally lost here, so I extracted the main nested issues confusing me: First Question: I know that in any LSE we want to minimize ...
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1answer
77 views

Largest eigenvalue of symmetric matrix

I am trying to understand why the $\lambda_{\max}$ function is convex given an $n\,x\,n$ symmetric matrix, let's call it $A$. I know from elementary property of eigenvalues that all the eigenvalues of ...
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1answer
22 views

Polytope - Convex Hull

After doing some reading on the V-representation of a convex polytope (finite set of extreme points, also the convex hull?), it's often simply stated that the convex hull is compact. Can anyone show ...
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1answer
56 views

“Support function of a set” and supremum question.

I have already learned about what a supremum means from wikipedia and from another answer here. However I am not quite sure what 'supremum over a set of functions' means exactly. As an example, my ...
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1answer
50 views

Caratheodorys lemma proof

I have to proof caratheodorys lemma for my oral exam. The proof is given here. I dont get the last part. "This process can be repeated until x is represented as a convex combination of at most d + 1 ...
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Inequality involving max is confusing me.

I am trying to understand one line in a derivation here. Simply put, the statement is that: $$ max\{\theta \ f_1(x) + (1-\theta) \ f_1(y) \ , \ \theta \ f_2(x) + (1-\theta) \ f_2(y)\} \leq \theta \ ...
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1answer
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Examples of affine functions and convex sets

I'm just learning about convexity and affineness, and I've read over some similar questions asked here, but those were more about general properties. I need some help applying those properties to a ...
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1answer
40 views

Hausdorff metric and convex hull

Let $X$ be a normed linear space and $A, B \in P(X)$. We define $\overline{co}(A) =$ the closure of the convex hull of $A$. Let $h$ denote the usual Hausdorff metric. We need to show that: $$h( ...
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35 views

The equivalent definition of denting point

How i can prove that If $K$ is a subspace of Banach space $X$, $x$ is denting point of $K$,when for every $\varepsilon>0$,there is a unit vector $x^{*}\in X^{*}$ and $\delta>0$ such that ...
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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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Calculation of the set for the polar tangent cone?

I have the following theorem in my book. Assume that $\tilde{x}$ is a local minimum from a minimization problem and that f(.) is differentible at $\tilde{x}$ Let $T_X(\tilde{x})$ be the tangent cone ...
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The convexity of $f(x)/x$ if $f(x)$ is concave/convex?

I was wondering if you know a theorem that states that the function $f(x)/x$ is convex in $x$ if $f(x)$ is concave or convex in $x$. $f(x)$ is convex and increasing in $x$. when $\lambda>0, ...
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73 views

Is the geometric-to-arithmetic function convex or concave?

Consider a vector $\mathbf{x} \in \mathbb{R}_{++}^N$. Also consider two functions, $g(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, and $a(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, ...