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.
2
votes
2answers
46 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
votes
0answers
36 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
votes
1answer
48 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.
...
1
vote
2answers
182 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 + ...
0
votes
0answers
10 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
votes
1answer
37 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 ...
2
votes
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
124 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 ...
-2
votes
0answers
28 views
Can anyone explain the proof of lemma 2.2 and proposition 2.1 by any means sketch, image or text! [closed]
http://beatrice-acciaio.net/down/trajectorial_doob_ABPST.pdf
Page: 4
Lemma 2.2
Proof of prop 2.1
1
vote
1answer
205 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 ...
4
votes
0answers
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 ...
1
vote
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 ...
0
votes
1answer
26 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
$$
...
1
vote
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 ...
0
votes
0answers
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)$ ...
1
vote
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 ...
0
votes
1answer
122 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 ...
3
votes
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 ...
0
votes
1answer
140 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 ...
0
votes
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 ...
0
votes
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.
1
vote
1answer
208 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)$ ...
9
votes
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
1answer
33 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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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
...
0
votes
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 ...
1
vote
1answer
18 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 ...
1
vote
0answers
32 views
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 ...
0
votes
0answers
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 ...
0
votes
2answers
81 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
vote
1answer
45 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 ...
4
votes
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 ...
0
votes
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}
...
1
vote
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, ...
1
vote
1answer
32 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 ...
1
vote
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 ...
1
vote
2answers
52 views
How is the concave closure operation defined?
I learned the in a vector space over an ordered field, the convex closure operation of a subset is defined as the smallest convex set that contains the subset. I was wondering how the concave closure ...
1
vote
1answer
25 views
Explain about convexity in geometry and in optimization.
My question is 'what is a difference between convexity in geometry and optimization?'
5
votes
1answer
198 views
Cones of positive semidefinite matrices generated by matrices of rank $1$
Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
10
votes
1answer
611 views
Minkowski Inequality for $p \le 1$
I've been trying to prove the concavity of a particular function which I reduced to proving the reverse Minkowski Inequality for $p \le 1$, $p \ne 0$ for arguments in $\mathbb{R}^{n}_{+}$. That is,
...
2
votes
1answer
46 views
How to prove that is a cone
I have to prove that the set $$K=\lbrace u\in C[0,1]\mid u(t)\geq a(t)||u|| \text{ on } [0,1]\rbrace,$$ where $a(t)=\displaystyle\frac{t(2p-t)}{p^2}$ with $\frac12<p<1$, and ...
2
votes
2answers
136 views
definition of strongly convex
There are several equivalent definitions for strongly convex.
For example, some literature said:
A function $f$ is strongly convex with modulus $c$ if either of the following holds
$$f(\alpha ...
4
votes
1answer
182 views
Can one define the derivative of a function using tangent cones? Does such a notion already exist?
I'm interested in finding an analogue of a derivative that applies to functions which are defined more general subsets of $\mathbb{R}^n$ than open subsets. In particularly, I'm looking at functions ...
0
votes
0answers
21 views
Distance between some set and its convex hull
What does "distance between some set and its convex hull" mean? Can anyone show, as an example, if I do Minkowski addition of sets, and find the Minkowski addition's convex hull, how does one find ...
0
votes
3answers
26 views
Show that the maximum of a set of convex functions is again convex
Let $f_1(x), f_2(x), \ldots, f_n(x)$ be a set of convex functions. We define $f(x)$ as
$$ f(x) = \underset{i}{\text{max}} \left\{ f_i(x) \right\}. $$
How do I show that $f(x)$ is also convex, and ...
