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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maximum and minimum singular values
I was studying the Stephen Boyd's textbook on convex optimization and have a question. The book says the following:
"The singular values of A, $\sigma_1 \geq \sigma_2 \geq ... \geq \sigma_n$ are the ...
3
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4answers
80 views
Is second derivative of a convex function convex?
If $f$ is twice differentiable and convex, is it true that $f''$ is a convex function ?
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2answers
156 views
+100
A robust convex optimization problem
Consider a function $ f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R} $ such that $\forall x \in \mathbb{R}^n$ the map $f(x,\cdot)$ is convex, and $\forall y\in \mathbb{R}^m$ the map ...
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2answers
44 views
Finding convex conjugate of a bounded function
The convex conjugate of a function $f:\mathcal{X}\mapsto \mathbb{R}$ is formally defined as $$f^\star\left(y\right)=\sup_{x\in\mathcal{X}}\ \left\langle x,y\right\rangle-f\left(x\right).$$
In cases ...
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0answers
29 views
Indirect Concavity
Let $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $g(x_1,x_2) = e^{\min{(x_1,x_2)}}$ at each $ x_1, x_2 \in \mathbb{R}^2$. Find whether or not $g$ is indirect concave on $\mathbb{R}$.
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Convex functions on real vector spaces
So I'm trying to solve the following problem,
Suppose that $f$ is a non-zero convex real-valued function on a real vector space $V$ with $f(0) = 0$
Show that there is a linear functional $g$ on $V$ ...
2
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2answers
35 views
indirectly convex
Let g:$\mathbb{R^2}$$\rightarrow$$\mathbb{R}$ be defined by g(x)=Max{$x_1$,$x_2$} at each x=($x_1$,$x_2$)$\in$ $\mathbb{R}$. Determine whether or not g is indirectly convex on $\mathbb{R^2}$.
...
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1answer
17 views
Example on Correspondences
Giva an example of correspondence $F : \mathbb{R} \rightarrow \mathbb{R}$ such that the closure of $F$ is $ \overline{F}: \mathbb{R} \rightarrow \mathbb{R}$, upper semi continuous on $\mathbb{R}$, ...
2
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0answers
40 views
Linear Difference Equations
Let $g_1, g_2 \in \mathbb{R}^{[0,5]}$ be defined by $g_1(x) = x$ and $g_2(x) = 1-x$ for each $x \in [0,5]$.
Find a second-order homogeneous linear difference equation on $[0,3]$ such that {$g_1, g_2 ...
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2answers
41 views
How can i show this inequality?
Let $n>1$ and $a_1,...,a_n \in \mathbb{R}^+$ be such that $\sum a_i=1$. For evey $i$, define $b_i=\sum_{j=1,j\neq i}a_j$. Show that
$\sum_{k=1}^n \dfrac{a_k}{1+b_k}\ge \dfrac{n}{2n-1}$
Thanks a ...
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1answer
265 views
Going in the direction of the gradient
First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$.
Now my ...
2
votes
2answers
55 views
convexity of norm
I want to show that $f(v)=\|v\|^p$ for $1\leq p<\infty$ is strictly convex.
In the simplest case when $p=2$, we have:
...
3
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0answers
38 views
Does convexity of a function guarantee tractability of finding its minimum?
Formulating a problem as a convex optimization problem usually implicitly considered to imply being able to find global minimizer of the objective. My question is that if it is true or not.
...
3
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1answer
50 views
Does convex and radially open imply open?
I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace.
Here the 'openness' we are talking about is from any normed space.
...
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2answers
184 views
Proof of Clarkson's Inequality
Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
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0answers
11 views
secant method twice on convex decreasing function
I have a continuous, decreasing and convex function $f$. Given an interval $[a, b]$ such that $f (a)>0 $and $f (b)<0$, if I apply the secant method twice, where the outcome point will be ...
2
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1answer
38 views
Maximum of quasi-convex functions
A function $f$ is quasiconvex if all its sub-level sets are convex (i.e., $\{ x: f(x) \le \alpha\}$ is convex for all $\alpha$.)
For a convex function $f$, it is true that $f$ acheives its maximum ...
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0answers
54 views
Convexity of polylogarithms
I want to prove the following proposition:
The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$.
And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
1
vote
1answer
206 views
Convex hull of sets defined by (in)equalities
If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to ...
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67 views
On a version of gradient descent
I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has coordinate-wise Lipschitz constant $M$ (meaning that for all ...
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0answers
33 views
Suggestions for a reference-level text on optimization theory?
I'd be interested in knowing if anybody has suggestions on an advanced but still self-contained reference on optimization theory, centered around linear and convex problems. The key feature of my ...
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1answer
27 views
How to prove the property of convex function in higher dimension
Suppose $f:\mathbb{R}^n\mapsto \mathbb{R}$ is convex, could anyone tell me how to prove the following fact?
(1) If $f\in C^1$, then for any $u,v$
$$
f(v)\geqslant \langle\nabla f(u),v-u\rangle
$$
...
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1answer
49 views
Convex analysis: relative interior in finite and infinite dimension
Let $X$ be a normed space. Given a nonempty convex set $C \subset X$, the relative interior of $C$, denoted by $\text{ri} C$, is the set of the interior points of $C$, considered as a subset of ...
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25 views
KKT conditions of this convex optimization problem
Consider a continuous function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that $\forall y \in \mathbb{R}^m$ $f(\cdot,y)$ is convex, and $\forall x \in \mathbb{R}^n$ $f(x,\cdot)$ ...
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2answers
32 views
computational strategy for solving convex-concave minmax problem
Assume f(x,y) is convex in $x$ and concave in $y$.
Then \begin{equation}\min_x \max_y f(x,y)\end{equation} is globally solvable, because f is convex in x (max of convex is convex.)
But can we find a ...
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1answer
123 views
What Stopping Criteria to Use in Projected Gradient Descent
Suppose we want to solve a convex constrained minimization problem. We want to use projected gradient descent. If there was no constraint the stopping condition for a gradient descent algorithm would ...
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0answers
52 views
Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as
$\Omega_f(x) \triangleq ...
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1answer
141 views
Global Min-Max Optimization
When is
\begin{equation}
\min_X \max_Y f(X,Y)
\end{equation}
globally solvable? (i.e. we can find global solution for the optimization problem?)
I am not looking for reformulations.
Is it only when ...
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1answer
22 views
Where the gradient of a convex function approaches zero
Does there exists a differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with a unique global minimum which we denote by $x^*$ such that there exists a sequence $x_k$, not ...
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1answer
48 views
Find function with given properties
Find a smooth function $g: \mathbb{R} \to \mathbb{R}$ that
domain $g$ is $\mathbb{R}$
range of $g$ is a subset of $\mathbb{R^+}$
$g$ is concave.
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1answer
210 views
Definition of a function being unimodal
For a function $f: \mathbb{R}^n \to \mathbb{R}$, I am looking for the definition of $f$ to be unimodal.
From Wikipedia
$f$ is unimodal if there is a one to one differentiable mapping $X = G(Z)$ ...
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1answer
209 views
What is the probability of having a pentagon in 6 points
If the probability that $5$ random points in the plane whose horizontal
coordinate and vertical coordinate are uniformly distributed on the
interval $\left(0,1\right)$ occur to be the vertices of a ...
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1answer
47 views
How to prove that this function is convex
My problem is that:
The domain is $\mathbb{R} ^n _{++}$ .
I need to prove that $f(x_1,...,x_n)=\sum_{i=1}^{n}x_i\cdot ln(x_i) -(\sum_{i=1}^{n}x_i)\cdot ln(\sum_{i=1}^{n}x_i) $
is convex.
I tried to ...
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1answer
60 views
Jensen's inequality and $L^p$ norms
Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
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1answer
144 views
Prove that the spaces have the same homotopy type
This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima.
"Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...
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1answer
35 views
Every exposed point is a extreme point
Let $C$ be a non-empty convex subset of $\mathbb{R}^n$.
We say that $x\in C$ is a extreme point of $C$ if for every $z,y\in C$ and $t\in [0,1]$ such that $x=ty+(1-t)z$ we have $x=z$ or $x=y$.
Or ...
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1answer
24 views
Relation about Gateaux differentiable and differentiable
Assume there is a multivariable function $f:\mathbb{R}^n\to\mathbb{R}$. I want to know the relation between Gateaux differentiable and differentiable. That is, if $f$ is differentiable, is it Gateaux ...
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1answer
30 views
Special type of convexity
Does anyone know if there is a name for a function $f:W\rightarrow\mathbb{R}$ ($W$ is a vector space) that satisfies
$$
f\left(\theta x + \left(1-\theta\right) y\right) \leq \theta^\alpha ...
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2answers
58 views
Lipschitz functions in $\mathbb{R}^n, \ \ \mathbb{R}^m$, extension
I've found the following lemma :
Let $\{x_1, . . . , x_k\}$ be a finite collection of points in $\mathbb{R}^n$
,
and let $\{y_1, . . . , y_k\}$ be a collection of points in $\mathbb{R}^m$, such that
...
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0answers
22 views
Extreme points and positive linear combinations
Let $C$ be a non-empty convex subset of $\mathbb{R}^n$. We say that $x\in C$ is a extreme point of $C$ if for every $z,y\in C$ and $t\in [0,1]$ such that $x=ty+(1-t)z$ we have $x=z$ or $x=y$.
Or ...
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1answer
19 views
Does the Border (Boundary) Points of a convex shape in the positive quadrant make a convex function?
Let $\mathbb{S}$ be a convex body in 2-D with some non-zero intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis such that $(c,y)\in ...
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0answers
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Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit
$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
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23 views
Convexifying Functions
I have the following question:
Imagine you have a function $F(x,y(x),y'(x)), F:\mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $ and this function is not convex.
Then you can ...
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2answers
83 views
Convexity of $\log \det(I+(P-\text{tr}(X))\cdot X)$
where $P-\text{tr}(X)>0$ and $X$ is a diagonal matrix with diagonal elements are $(x_1,x_2,\dots x_m)$
1
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1answer
48 views
KKT and Slater's condition
I was studying Stephen Boyd's text book and got confused in the KKT part. The book says the following:
"For any convex optimization problem with differentiable objective and constraint function, any ...
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0answers
104 views
strict convexity with a measure theoretic property
Suppose $(x_n)$ is a positive sequence of reals converging to $x$. Furthermore we have a measure space $(E,\mathcal{E},\mu)$ given, with finite measure $\mu$. There are a measurables nonnegative and ...
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0answers
18 views
Solution of a Quadratic Optimization Problem
Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem,
\begin{align}
...
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2answers
45 views
Given some points in the Euclidean space, find a plane satisfying some restrictions
In a 3-D Cartesian coordinates, suppose we are given $n$ points and their coordinate values in the form $(x,y,z)$. Obviously there are uncountable planes which divide those points into two groups, ...
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1answer
34 views
Robust feasibility with halfspace?
Consider a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, such that for all $x \in \mathbb{R}^n$ we have
$$ a_1^\top x + b_1 \leq f(x) \leq a_2^\top x + b_2 $$
for some given $a_1, a_2 ...
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1answer
40 views
About Balanced-Convex Hull of a Set
Definition. $\newcommand{\bco}{\operatorname{bco}}$Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, deonoted $\bco A$, is the intersection of all balanced-convex ...
