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
19 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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1answer
150 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
128 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
262 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
23 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 ...
2
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1answer
746 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
61 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
1k 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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1answer
168 views

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
108 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
64 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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1answer
44 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 ...
13
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1answer
319 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
125 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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0answers
35 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
96 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, ...
4
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3answers
235 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
46 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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0answers
70 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
39 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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57 views

How to handle concave-concave constraints?

there. I have an optimization problem, which takes the following form ...
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1answer
342 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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39 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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3answers
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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 ...
5
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1answer
121 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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0answers
48 views

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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0answers
27 views

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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93 views

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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56 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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1answer
46 views

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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1answer
37 views

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
178 views

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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2answers
380 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
133 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
40 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
91 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
68 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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2answers
71 views

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
60 views

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
78 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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0answers
36 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
189 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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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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112 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 $a(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, ...
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3answers
329 views

Concave function divided by a convex function. What is the result?

Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...
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0answers
56 views

An extreme point that is not strongly exposed

I want to construct an example that a point is extreme but is not exposed. This example can be in the following : A compact convex subset $K‎\subset‎\mathbb{R}^{2}$ and a point $u\in K$ such that ...
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2answers
52 views

Convex set for a set of points in 2d plane

There are set of five points $A(0,0) ,B(1,1) ,C(2,0) ,D(2,2).E(0,2) F(1.5,1.5)$ $S=\{A,B,C,D,E,F\}$ Please tell me whether my understanding is correct or not! The points $A,C,D,E$ forms a convex ...
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1answer
84 views

On convex hulls and intersections of chains of compact sets

Let $V$ be a topological vector space, let $\{ C_i \}_{i \in I}$ be a set of compact subsets of $V$ which forms a chain with respect to inclusion. For now, assume the following stronger properties: ...
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2answers
76 views

computation of subdifferential

This question mainly deals with subdifferential of a convex function with respect to the cost function $c(x,y)=\frac{|x-y|^2}{2}$ I want to compute the cost-subdifferential $\partial^{c}\phi$ of the ...