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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Convexity of a complex function

I have a problem in which I need to find the minimum of the function $f:\mathbf{C}\rightarrow\mathbf{R}$ given by $$f(s) = v^*M(s)^*M(s)v$$ with $v \in \mathbf{C}^{n+1}$ and so that $|v_i|$ ...
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1answer
72 views

An inequality regarding the derivative of two concave functions

Let $f:\mathbb{R}^+\to \mathbb{R}^+$ and $g:\mathbb{R}^+\to \mathbb{R}^+$ be two strictly increasing, strictly concave and twice differentiable functions with $f(0)=g(0)=0$ and $f'(0)=g'(0)>0$. We ...
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24 views

Log-concave function changes when scalar is added

A function $f$ is log-concave if $\log(f)$ is concave. Intuitively, one might guess that adding a scalar to a function would not affect properties like concavity, log-concavity, quasi-concavity etc., ...
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23 views

Subgradient of function of two variables

i do not have any experience in convex analysis and I would be most grateful if you would help me with the concept of subgradient. I get the concept of subderivative (one dimension) but it is hard ...
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1answer
28 views

Subgradient inequality for strongly convex functions

I need some help to follow the argument made here which says that $$ f(x_t) - f(x^*) \geq \frac{\alpha}{2}\|x_t-x^*\|^2 $$ if $f$ is $\alpha$ strongly convex and $x^*$ the minimizer of $f$. From the ...
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1answer
41 views

Prove that $tx+(1-t)x \ge x^ty^{1-t}$

Given conditions are $x>0$ $y>0$ and $0 \le t \le 1$ There is a hint given which says $Log$ is a concave increasing function. How do I apply this here? There is also a generalization of this ...
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2answers
19 views

Show $A=\{(x,y) \in \mathbb{R^2} \mid x^2+2y^2<2p\}$ is convex

Show $A=\{(x,y) \in \mathbb{R^2} \mid x^2+2y^2<2p\}$ is convex (i.e. $(a,b) \in A \implies ta+(1-t)b \in A\ \forall\ 0\leq t \leq 1$. I have $x_1^2+2y_1^2 <2p$ and $x_2^2+2y_2^2 <2p$ for $(...
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18 views

If F is convex and lower semicontinuous in norm, then F is weakly lower semicontinuous

I have to prove the following statement: If $F$ is convex and lower semicontinuous in norm, then $F$ is weakly lower semicontinuous.
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39 views

Quasiconvexity of a function on the positive orthant using rays

It is seen in Boyd's book on Convex Optimization book that to show a function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ is quasiconvex, it is enough to show that $f$'s restriction to a line is ...
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33 views

Is a $k$-differentiable convex function $k$-continuously differentiable?

It is known that a differentiable convex function is continuously differentiable. Is a $k$-differentiable convex function $k$-continuously differentiable?
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25 views

convex hull of union of positive definite matrices

Is it true that any element of ${\rm co}\Big\{\bigcup_{x \in [a,b]} S(x) \Big\}$ is in $\mathbb{S}_{> 0}^n$ (cone of positive definite $n \times n$ matrices), given that $S(x) \in \mathbb{S}_{> ...
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1answer
28 views

Convexity of n $4u^4x^{2.5}y^{-5}$ over $x>0, y >0$ and $u>0$

I am trying to find if the function $4u^4x^{2.5}y^{-5}$ is convex over $u>0, y>0$ and $x>0$. The thing which comes to my mind immediately is to check the positive semi-definiteness of the ...
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11 views

Property of a $C^\infty$ convex function

Hey guys I need your help. Let $\Omega$ be a bounded, 2 or 3 dimensional domain with smooth boundary. Let $c\in H^2(\Omega)$ with Neumann boundary conditions. We define $\overline{c}=\frac{1}{|\Omega|...
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15 views

Supporting hyperplane of a polarity of a convex body.

Recently, I am studying in combinatorial convexity and related topics. I use the book "Combinatorial Convexity and Algebraic Geometry" (GTM 168) as my main reference. The book is very good, all the ...
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1answer
33 views

Are these two optimization problems equivalent?

I have two problems as follow. $min_x: ||x-y||_2^2 + \lambda_1 ||x|| \quad \ \ (1)$ and $min_x: ||x-y||_2^2 + \lambda_2 ||x||^2 \quad (2)$ Here $||\cdot||$ could be any norm and $\...
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Explain why for an extreme point some inequalities in the constraints become equalities

Suppose we have a linear program $$ \max_x c^Tx\\ \text{subject to } Ax \leq b $$ where $\leq$ denotes pairwise inequality, i.e. $a^T_i x \leq b_i, i = 1,..., n$. If $y$ is an extreme point, why is ...
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73 views

Is the intersection of 2 convex hulls a convex hull?

Given two finite sets of points, $X$ and $Y$, in $\mathbb R^d$ and assuming that $\text{conv}(X)\cap\text{conv}(Y)\neq\varnothing$ I would guess that the intersection is a convex hull of some other ...
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1answer
23 views

Convexity of the Pareto front: formal definition

Does anyone have a reference to a formal definition of what convexity of a Pareto front in multiobjective optimisation means? All literature I've seen uses the term without defining it.
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27 views

A lower semi-continuous convex function being not continuous on its domain

Let $f : \mathbb{R}^N \longrightarrow \mathbb{R} \cup \{+\infty \}$ be a lower semi-continuous convex proper function. Let $dom f$ be the domain of $f$, i.e. $dom f:= \{ x \in \mathbb{R}^N \ | \ f(x) ...
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Get the global minimum with functions convex in a subset of the domain; numerical methods.

I have a $C^\infty$ function $f:\mathbb{R}^n\to \mathbb{R}$ that is positive and known to have a zero and a global minimum in an unknown point $x$. Furthermore, $f$ is convex in the set $$ S = \{x'...
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1answer
71 views

To show a closed convex set $S \subseteq R^n$ is bounded if and only if $S$ contains no rays.

I want to show that a closed convex set $S \subseteq R^n$ is bounded if and only if $S$ contains no rays. Where $r \in S$ is a ray of $S$ if $x \in S$ implies that $x+\mu r \in S$ for all $\mu \in R_{...
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1answer
41 views

Strong duality of SDPs

On pp. 654 of Boyd's book, it is claimed that strong duality holds between the SDPs B.2 and B.3 (at the bottom of this page). Does it require additional assumption that one of them is strictly ...
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39 views

A question about Hilbert Spaces and convex sets

I am struggling with this and could really do with some help: Let $H$ be a Hilbert space over $\mathbb{R}$, $\{v_n\}$ be a sequence of vectors in $H$, and $C$ be a convex subset of $H$ containing $\{...
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72 views

Jensen's inequality; what's the need for the probability measure?

Jensen's inequality states that if: $\mu$ is a probability measure on $\Omega$, $f$ is integrable function (on $\Omega$) and $\phi$ is convex on the range of f then: $\phi \left( \int_{\Omega}g(x)\,d\...
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1answer
47 views

Is $f(x)$ convex if $\log f(x)$ is convex?

One of the convex composition rules states that $h(g(x))$ is convex if $h(x)$ is convex and non-decreasing, and $g(x)$ is convex. Now I want to go the other way - I know that $\log(f(x))$ is convex ...
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90 views

Convex and continuous function on compact set implies Lipschitz

Let the function $f: C \rightarrow \mathbb{R}$ be convex and continuous, where $C \subset \mathbb{R}^n$ is a compact set. Prove or disprove that $f$ is Lipschitz continuous on $C$. Comments: If $f$ ...
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1answer
54 views

what is the closed form solution for $\min_x ||y-x||^2_2+\lambda ||x||_2$

$y$ and $x$ are vectors. $\|\cdot\|_2$ is Euclidean norm. In one paper I read, they say the closed form solution is $x=\max\{y-\lambda \frac{y}{\|y\|_2}, 0\}$. I don't know why $x$ need to be non-...
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55 views

$f'$ decreasing everywhere but not defined in one point. Is $f$ concave?

Small issue: Suppose that $f:[a,b] \rightarrow \mathbb{R}$ is a continuous function, differentiable except on a finite set of points, let say in one point $y$. For $x<y$ and $x>y$ we have $f''&...
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1answer
47 views

Is the set of probability density functions convex?

Given is the set of probability density functions defined as $P:=\left \{ p(x)\mid p(x)\, is\ a\ probability \ density \ function \right \}$ Is $P$ a convex set? I am not sure that here i have to ...
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1answer
52 views

What is the closed form for this norm $\|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv$?

I read a paper, which has the equation above $\|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv$. Is this a well known norm and what is the closed form for this norm. Also what does this norm describe? Thanks in ...
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26 views

Sufficient condition for self-concordance?

I've revised a previous question that was ill-formed. Consider the following two definitions. Def'n 1 (Lipschitz continuity of Hessian): A function $f:\mathbb{R}^n\to\mathbb{R}$ is said to have a ...
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0answers
46 views

Log convexity and the gamma function

I am writing an essay on the gamma function. I have learnt and understood convex theory and how the log-convex nature of the gamma function makes it a unique extension of the factorials (Bohr-Mollerup)...
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106 views

Can a “continuous” convex combination not be element of the convex hull?

Short version of question: can a "continuous" convex combination not be element of the convex hull? I am not a mathematician, so please excuse me if I am not precise. I consider first, e.g., 4 ...
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1answer
34 views

Boundedness of sublevel sets of an integral function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an increasing function, i.e., $x<y \Rightarrow f(x) < f(y)$ for all $x,y$. Assume that $\lim_{|x| \rightarrow \infty} |f(x)| = \infty$ Define $F(x)...
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53 views

convex polygon triangulation

Suppose we have given a convex polygon on $n$ vertices $P= \{ a_1, \cdots , a_n \}$ in the plane (arranged clockwise). How can we prove that there exist atleast two indices $i$ such that circle $a_{i-...
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1answer
52 views

Is this $\underset{x<1,y<1}{\sup}f(x,y) = \underset{x<1}{\sup} \underset{y<1}{\sup}\ f(x,y) $ correct?

$\underset{x<1,y<1}{\sup}f(x,y) = \underset{x<1}{\sup} \underset{y<1}{\sup}\ f(x,y).$ Intuitively, I think the above equation holds for all $f(x,y)$. Am I right?
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31 views

A supremum problem

Let $a=\underset{\|u\|_c\leq 1, \|v\|_r\leq 1}{\sup}v^TY^Tu$. If $\lambda<a$, $\underset{u_m, v_m}{\sup}v_m^TY^Tu_m-\lambda\|u_m\|_c\|v_m\|_r=+\infty$. While if $\lambda > a$, then $\...
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59 views

A power inequality with convexity / majorization flavor

Let $a,b,c,d\in\mathbb R_+$ be nonnegative reals. Define the function $f:\mathbb N_+\to\mathbb R$ as $$ f(k):=\left(\frac{(a+b+c+d)^k+(a-b+c-d)^k-(a+b-c-d)^k-(a-b-c+d)^k}{4}\right)^{1/k} $$ Is it ...
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Duality between $Ax<b$ and another system, but Gordan's just misses

I am trying to show that $$Ax < b$$ Is feasible iff $$ A^T y =0 , b^Ty + s = 0, (y,s) \ge 0, (y,s) \ne 0$$ Is infeasible. Work So Far Now when I try to hit this with Gordan's Lemma I seem ...
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38 views

How can the ADMM algorithm be distributed?

I am reading Boyd's tutorial paper on ADMM. One question I had after I reached the formulation of the algorithm on pp.14 is: how can the alternating or sequential fashion of x-minimization step and z-...
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3answers
39 views

$sup_x [ f(x)+sup_x g(x)]= sup_x[f(x)+g(x)]$?

$sup_x [ f(x)+sup_x g(x)]= sup_x[f(x)+g(x)]$ is the statement correct? Can I prove like this: $sup_x [ f(x)+sup_x g(x)] = sup_x[f(x)] + sup_x[g(x)] = sup[f(x)+g(x)]$.
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1answer
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Showing that $ye^{x/y} = \max_{\alpha>0} \left[ \alpha(x+y) - y\alpha \log(\alpha) \right]$

In my optimization textbook, the author states without proof that $$ ye^{x/y} = \max_{\alpha>0} \left[ \alpha(x+y) - y\alpha \log(\alpha) \right]. $$ To be honest, this does not seem very obvious ...
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21 views

convex envelop on a unit ball

What does that exactly mean by "the function $g$ (continuous) is the convex envelop of the function $f$ (discrete) on a unit $\ell_\infty$ ball". I understand the part of "a function being the convex ...
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128 views

Opening and closing convex sets

It seems true that, given $K \subseteq \mathbb{R}^n$ a convex set with $K^\circ \neq \emptyset$, then $\overline{K^{\circ}} = \overline{K}$ and $\left ( \overline{K} \right )^\circ = K^\circ$. I am ...
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31 views

Tangents of a Strictly Convex Fuction

Let $f:\mathbb R \rightarrow \mathbb R$ be differentiable and strictly convex. Is it true that $x,y \in \mathbb R$ and $x \neq y$ imply \begin{equation} f'(x)(y-x) <f(y) - f(x) \end{equation} If ...
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1answer
36 views

Smallest convex body inscribed in $n$-cube with all its symmetries? [closed]

Consider the cube $[-1,1]^n$ and convex bodies inscribed in it, such that all these bodies have the symmetries of the cube. Is there a lower bound on the volume? Which shapes attain it?
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22 views

Smallest volume of a centrally symmetric convex body inscribed in $n$-cube

We consider several centrally symmetric convex bodies inscribed (intersecting all its facets) in an $n$-cube , $[-1,1]^n$, with volume $2^n$. For instance, such a crosspolytope (its polar dual) has ...
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2answers
35 views

Strictly Convex Implies Invertible Gradient?

If $f:\mathbb R^n \rightarrow \mathbb R$ is strictly convex and continuously differentiable, does this imply that $\nabla f$ is a one-to-one mapping? To be precise, can we say that $x, y \in \mathbb ...
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20 views

Applying homogeneity for a norm to prove triangle inequality

I am hoping to have someone help explain to me the steps of this proof of the triangle inequality for $L^p$ spaces, $p\ge 1$. Specifically the first two applications of the homogeneity property don't ...
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50 views

Computing Direction of Descent for General Fermat Problem

Suppose that $k \geq 3$, and let $X_1, \dots , X_k$ be $k$ points in $\mathbb{R}^2$, and let $w_1, \dots , w_k$ be $k$ positive real-valued constant weights. Given this, consider the function $g(p) = \...