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.

learn more… | top users | synonyms (3)

0
votes
0answers
21 views

Formal definition of a face of a polyhedron

Given an $n$-dimensional convex polyhedron, an $(n-1)$-dimensional face of it can be defined as an intersection of the polyhedron with a supporting hyperplane. What is the formal definition in the ...
0
votes
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 ...
1
vote
1answer
12 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., ...
0
votes
0answers
18 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 ...
0
votes
2answers
17 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 ...
0
votes
1answer
33 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 ...
1
vote
2answers
471 views

when is the epigraph a convex cone?

The problem is from Stephen Boyd's textbook, which I couldn't solve. The question is "when is the epigraph of a function a convex cone?" The solution says that it is when the function is convex and ...
0
votes
0answers
37 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 ...
1
vote
0answers
16 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.
1
vote
0answers
29 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?
0
votes
1answer
22 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}_{> ...
1
vote
0answers
10 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 ...
0
votes
1answer
29 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 ...
2
votes
0answers
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 ...
1
vote
1answer
66 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 ...
1
vote
2answers
35 views

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 ...
0
votes
1answer
21 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.
0
votes
0answers
27 views

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 = ...
0
votes
0answers
38 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 ...
2
votes
1answer
51 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 ...
1
vote
2answers
58 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( ...
10
votes
3answers
15k views

How to check convexity?

How can I know the function $$f(x,y)=\frac{y^2}{xy+1}$$ with $x>0$,$y>0$ is convex or not?
3
votes
1answer
45 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 ...
0
votes
0answers
52 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 $f$ ...
3
votes
1answer
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 ...
0
votes
0answers
32 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 ...
1
vote
0answers
19 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 ...
9
votes
1answer
693 views

Is every convex-linear map an affine map?

Let's say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$. ...
8
votes
1answer
1k views

Farkas Lemma proof

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
2
votes
1answer
74 views

Property of log-concave function

In S.Boyd's lecture: And in his vedio, he said: You are allowed one positive eigenvalue in the Hessian of log-concave function. http://web.stanford.edu/class/ee364a/videos/video04.html (at ...
1
vote
0answers
35 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 ...
1
vote
0answers
58 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 ...
1
vote
1answer
31 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 ...
8
votes
2answers
85 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 ...
3
votes
0answers
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 ...
1
vote
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?
0
votes
0answers
30 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 ...
8
votes
2answers
120 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 ...
2
votes
0answers
20 views

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 ...
0
votes
0answers
34 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 ...
0
votes
0answers
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) = ...
1
vote
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: $sup_x [ f(x)+sup_x g(x)] = sup_x[f(x)] + sup_x[g(x)] = sup[f(x)+g(x)]$.
0
votes
1answer
17 views

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 ...
1
vote
0answers
47 views

why dual of $l_1$ norm is $l_\infty$ and vice versa

This might be a very dumb question but I am having a hard time to understand why dual of $l_1$ norm is $l_\infty$ and vice versa. The dual of a norm is denoted $\lVert\cdot\rVert_*$, defined as $$ ...
0
votes
0answers
17 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 ...
1
vote
2answers
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 ...
2
votes
2answers
29 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 ...
0
votes
1answer
35 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?
0
votes
0answers
19 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 ...
0
votes
0answers
19 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 ...