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)

1
vote
2answers
31 views

Is a convex cone a convex polyhedron?

Say that I have a convex cone $C=\{t|Ax = t, x\geq 0\}$. where $x\in R^n$, and $t\in R^m$, $A\in R^m\times R^n$. Can I say that this is a convex polyhedron? and why? EDIT: Just in order to avoid ...
1
vote
1answer
43 views

Lagrange's theorem and convex functions

Let f:U⊂ $\mathbb{R}^n$--->$\mathbb{R}$ a $C^1$ function with U being convex and an open set. Let g: U ⊂ $\mathbb{R}^n$ ---> $\mathbb{R}^m$ (with m smaller than n) an affine application. Let M={x $ ...
0
votes
1answer
20 views

Locally convex spaces - is any space that contains a locally convex space as a subspace, also locally convex?

Given $E$, a locally convex space (l.c.s.) and $E\subseteq F$ where $F$ is another subspace of a larger vector space. The inclusion is strict since I know there exists a $y\in F\backslash E$. I have ...
0
votes
0answers
13 views

Characterization of projective convexity

Let $K$ be a closed set in projective space $\mathbb P^n$. Is it true that $K$ is "projectively convex", i.e., its intersection with every line is connected, if and only if it is the projectivization ...
2
votes
1answer
109 views

Solution for $\min_{x^Tx=1} x^TAx-c^Tx$

$\min_{x^Tx=1} x^TAx-c^Tx$ looks like a simple QPQC problem. If $A$ is positive semi-definite, can I get the solution by first getting $x=A^{-1}c$ and then projecting $x:=\dfrac{x}{||x||}$ to make ...
0
votes
1answer
22 views

Proof of property of unique nearest point equal to convex.

In the picture below, I really don't know how to use the compactness to get there will be a ball $C$ with maximal radius. I report this book for my classmate and teacher. I tell they the $r$ has a ...
5
votes
1answer
142 views

Can someone confirm this sketch of a proof for the existence of smooth partitions of unity on an unbounded convex open euclidean set?

EDIT!! The problem originally described (see below) has been reduced to the correctness of a simple extension of an argument from Rudin's PMA. Feel free to skip to the proposed solution, below. As ...
5
votes
2answers
6k views

How to determine whether a function of many variables is convex or non-convex?

A function $f$ is convex if $$f(\theta x + (1 − \theta)y) \leq \theta f(x) + (1 − \theta)f(y)$$ for all $x, y \in \mathcal{D}(f)$, the domain of $f$, and $\theta \in [0, 1]$. How do I determine ...
0
votes
0answers
20 views

If $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are positive, non-increasing and convex functions, then $F(x,y) = f(x)g(y)$ is quasiconvex.

Hypothesis: $\forall x_{1},x_{2}\in \mathbb{R}, \forall \lambda \in [0,1], f(\lambda x_{1} + (1- \lambda) x_{2}) \leq \lambda f(x_{1}) + (1- \lambda) f(x_{2})$ $\forall x_{1},x_{2}\in \mathbb{R}, \...
0
votes
1answer
26 views

Extreme points of unit ball of $l_1(\mathbb{N})$

Let $K$ be the closed unit ball of $l_1(\mathbb{N})$ over real numbers. Show that $$ Ext(K)= \{\pm e_n: e_n=(0,\ldots,1,0,\ldots)\}. $$ My attempt: I could prove that $\{\pm e_n: e_n=(0,\ldots,1,0,\...
0
votes
0answers
14 views

Solve convex optimization problem with objective function having power \alpha >1

I am not able to figure out how to get the explicit solution of the following minimization problem: $$\min_{\mathbf{w}\in \mathbb{R}^n} = \sum_{i = 1}^{k}b_i\left(\mathbf{w}^T\mathbf{Z_i}\right)^{\...
1
vote
0answers
37 views

Need to prove that convex property is the intersection of an increasing and decreasing property for graph

I need to prove that any convex property for graphs can always be expressed as the intersection of an increasing property and decreasing property for graph, specifically that: $\forall A\subset B\...
0
votes
0answers
20 views

general framework for proof of convexity in least square problem

Let $f(x_i,\theta) : X \times \Theta \rightarrow Z$ be a parametric function with parameter $\theta$ that we wish to fit to set of samples $S =\{(x_i,z_i)\}_{i=1}^N$, where $(x_i, z_i) \in X \times Z$....
3
votes
1answer
49 views

Convex set: extreme points and distance to the origin

I'm fairly sure the following is true, although I wouldn't mind being proven wrong. If true, I would like to see an elegant proof, as my attempts are kind of messy. Let $K\subset\mathbb R^2$ be a ...
0
votes
1answer
68 views

Convex set without zero

Let $\emptyset \neq A \subset \mathbb{R}^n$ be a convex set with $0 \notin A$. Then there exist a $v \in \mathbb{R}^n$ such that $v \cdot a \geq 0$ for all $a \in A$ and there exists $a_0 \in A$ with $...
0
votes
0answers
27 views

convexity of composite function

There is a composite function F(x)=f(g(x)): R -> R where f(v): R^2 -> R and g(x): R -> R^2. It is given that f(v) is convex on v and v = g(x) gives a map R -> R^2. If x is in a convex set, is F(x) ...
1
vote
0answers
117 views

Taylor's series question on bounds

$f(x) = f(y) + f'(y)(x-y) + \frac{f"(y) (x-y)^2}{2} + ....$ is the taylor's series. I am aware of bounds which specify the error in approximating $f$ with a polynomial. I want sufficient conditions ...
0
votes
0answers
18 views

An inequality of a multivariate function: $f(x_1,…,x_n) \geq \frac{1}{n}\sum_i^n f(x_i,x_i,…,x_i) $

Let us assume we have a non linear function $f : \Bbb R^{n+} \to \Bbb R ^+$, and let $x = \{x_1, x_2 , ..., x_n\}$, $x_i \in \Bbb{R}^+$, further define $\bar{x} = \frac{1}{n}\sum_i^n x_i$. My ...
0
votes
0answers
27 views

A necessary and sufficient condition for $f(x,y) = \phi(x²+y²)$ be a convex function

Let $f(x,y) = \phi(x²+y²) , \phi \in C^2$ and $\phi$ non-decreasing. Proof that $f$ is convex in the disk $x²+y² \leq a² \iff 2u \phi''(u) + \phi'(u) \geq 0 $ $\forall u \in [0,a]$ Here is my ...
0
votes
1answer
36 views

Show non-convexity of a function with vector input

How does one go about proving non-convexity of the function d? $$ d(v) = 1/2*||F(v)- p||^2 $$ $$ F(v)=\sum_{i=1}^n l_i*\begin{pmatrix} cos(\sum_{j=1}^i v_j) \\ sin(\sum_{j=1}^i v_j \end{pmatrix} $$ ...
0
votes
0answers
24 views

Subdifferential optimality conditions

I need help with subdifferential optimality. Let $f(x_1, x_2)=x_1^2 + x_2^2 + |x_1 -x_2 - y|$. Find: \begin{align} \min_{x_1, x_2} f(x_1, x_2) \end{align} This is convex, so must have unique ...
0
votes
0answers
24 views

minimal representation of convex hull

Here is a question about the convex hull. Let $S$ be a set and $\bar{S}$ be the closed convex hull of $S$, i.e., $\bar{S}$ is the smallest convex set that contains $S$. Then is the following claim ...
2
votes
0answers
73 views

LASSO constrained and penalised forms: first - order conditions?

The LASSO problem can be formulated in two ways: 1) Constrained formulation: $$ \|Xw-y\|^2\to\min_{w}\\ \text{s.t. } \sum_{i}|w_i|\leq{t}. $$ 2)Penalised formulation: $$ \|Xw-y\|^2+\lambda\left(\...
0
votes
0answers
35 views

rconvex image under nonlinear function

Let $X\subset R^3$ be a compact and convex set, and let $f: X\rightarrow R^3$ be a nonlinear function, with $f\in C^k$. What are the tools to investigate if the image $K=f(X)$ is also convex, in the ...
1
vote
1answer
64 views

Required conditions for product of two submodular functions to be submodular?

3.32 of Boyd's Convex Optimization book says: Products and ratios of convex functions. In general the product or ratio of two convex functions is not convex. However, there are some results that ...
1
vote
0answers
30 views

Is this function really not concave or convex in any range?

Consider the function $f(x,y)=\frac{y}{1+e^x}$ where $0<y<1$ and $x \in \mathbb{R}$. If you plot this function, it looks like this: Also note that for a given value of $y$ the function $f$ ...
0
votes
0answers
23 views

Sum of convex and concave functions when one is greater than the other

Given two $C^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ such that $f(x)>g(x)\text{ }\forall x\in\mathbb{R}$. Moreover, we know that $f(x)$ is convex while $g(x)$ is concave. Now, let's define $...
0
votes
0answers
17 views

Comparison between secant and derivative in a convex function

Imagine that we have a function $f:\mathbb{R}\to\mathbb{R}$ which is convex, that is $f''>0$. We also know that $f'''<0$, that is its first derivative function is concave. Now, we can define its ...
1
vote
1answer
35 views

Properties of convex function with Lipshitz continuous gradient (Prof. Nesterov's textbook)

I am reading the Prof. Nesterov's textbook: Introductory lectures on convex optimization - a basic course p.57 I have problem in the following: My ...
0
votes
0answers
26 views

If $f$ is log-convex then $f$ is convex

Here's my attempt: $f$ is log-convex. Then: $\log f(\lambda x + (1-\lambda)y )\leq \lambda \log f(x) + (1-\lambda) \log f(y)$ As $e^x$ is increasing, we can apply it to the inequation without ...
0
votes
1answer
14 views

Quasiconvexity (in the sense of Morrey) implies Rank-One convexity

I am trying to understand why Quasiconvexity implies Rank-One convexity. In a standard proof of this fact a sequence of functions is constructed, which converges weakly to zero in $W^{1,p}.$ in ...
0
votes
0answers
25 views

A function is convex if and only if its gradient is monotonous.

Let a convex $ U \subset_{op} \mathbb{R^n} , n \geq 2$, with the usual inner product. A function $F: U \rightarrow \mathbb{R^n} $ is monotonous if $ \langle F(x) - F(y), x-y \rangle \geq 0, \forall x,...
20
votes
6answers
2k views

What are the main uses of Convex Functions?

Up till now I have just learned that the concept of convexity in functions of one variable is used to complete the graphs of functions, meaning to locate points of inflexion and see if the graph is ...
0
votes
0answers
23 views

Composition of convex and concave functions

I had a homework question: "Show that the function f(x,u,v) = -log(uv-xTx) is convex on domain {(x,u,v)| uv-xTx,u,v > 0}". EDIT: x,u,v are Real No.s One pdf I found online says: "We can express f as ...
0
votes
1answer
33 views

Minkowski sum and Polygons

The problem:.. Given two convex polygons $A$ and $B$, we can define Minkowski sum as A + B = {a + b: a $\in$ A, b $\in$ B}, where $a + b$ vector sum. Prove that: every vertex $p \in A + B$ is a ...
1
vote
1answer
27 views

Proximal-type support function properties - nonnegative & strongly convex (proof)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems The following part confuses me: $\\$ $\\$ ${\color{red}{E}}\...
0
votes
1answer
18 views

How to prove a two variable set is convex

$X=\{(x,y)\in R^2\ :\ 3\le 2x+3y\le 8\}$ i tried to solve it as: Let set $X$ is convex for $x_2,y_2\in X$ such that $\alpha x_1+(1-\alpha)x_2$,$\alpha y_1+(1-\alpha)y_2\in X$ Now, $3\le 2x+3y\le 8$ ...
1
vote
1answer
35 views

Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems I am confused about the green part of the following: $\\$ ...
1
vote
0answers
33 views

How to prove this function is quasi-concave? [closed]

Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?
3
votes
1answer
34 views

Generalization of Brouwer’s fixed-point theorem

Perhaps the most widely known version of Brouwer’s famous fixed-point theorem reads as follows: For any $n\in\mathbb N$, let $A\subseteq\mathbb R^n$ be a compact (with respect to the Euclidean ...
1
vote
1answer
42 views

Open ball around a point in the hyperplane in a locally convex space

Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$. It is my intuition that $V_z$ should ...
0
votes
0answers
17 views

Probability that a point lies in an uncertain convex hull

Given $n+1$ independent random vectors $X_i \sim N(\mu_i,\Sigma_i)$, where each $\mu_i \in \mathbb{R}^n,$ let $C$ denote the random region formed by taking the convex hull of a realization of the set ...
0
votes
0answers
29 views

Can we solve minimax in this way?

I am working to use proximal operators for solving a minimax optimization problem. It is known that if you use alternative optimization, the algorithm cycles, see an answer to this question convex-...
1
vote
1answer
57 views

Minkowski sum and vectors

Problem: Given two convex polygons A, B, we can define Minkowski sum, as A + B = {a + b: a $\in$ A, b $\in$ B}, where a + b vector sum. Prove that: for every external perpendicular u to an edge of A,...
0
votes
2answers
122 views

Which is bigger $e^{(a+b)}$ vs $e^a + e^b$?

I understand that exponential function is a convex function so for any convex function $\theta(a+b) > \theta(a) + \theta(b)$, but can someone provide a more formal proofs ?
1
vote
1answer
34 views

A convexity argument

Let $(\alpha_n)$ be a sequence of positive real numbers s.t. $\sum \alpha_n=1.$ Consider a sequence of complex numbers $(\beta_n)$ s.t $|\beta_n|=const$ for all $ n \in \mathbb{N}.$ Suppose that $\...
0
votes
0answers
53 views

convexity of a nonlinear function

I tried to search similar answers to my problem here, but unfortunately I'm a little bit lost on this subject, originally from a physical problem, but can be stated as: Let $A\subset R^3$ be a ...
2
votes
1answer
38 views

Functions mapping convex sets on convex sets

A function $f:\Bbb R^n\to\Bbb R^m$ is said that maps convex sets on convex sets if it satisfies: (1) the image $f(A)$ of any convex set $A$ is a convex set; (2) the preimage $f^{-1}(B)$ of any convex ...
0
votes
3answers
39 views

Convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant 1$

I would like to ask the convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant1$. We know that $y\geqslant\frac{1}{x}$ is convex for $x>0$. But if we transform the inequality into $$xy\geqslant1(x,...
0
votes
1answer
39 views

How to prove conv(M)+u=conv(M+u) and Aff(M)+u=Aff(M+u)

I don't know how to prove that if $ M \subseteq R^n, \forall u \in R^n $ then conv(M)+u=conv(M+u) and Aff(M)+u=Aff(M+u). conv is convex hull and Aff is affine hull. Yes it is a homework question, ...