# Tagged Questions

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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### Optimization with box constraints - via nonlinear function

I have the following convex optimization problem over $\mathbb{R}^n$ with box constraints: \begin{align}\text{minimize }&\;f(x)\\ \text{subject to }&\;x \in [-1,1]^n\end{align} I can see ...
1answer
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### Is there a way to measure how (non)convex a function is, maybe analogous to condition number?

Consider the functions $f(x) = \sin x$ and $g(x) = (x+1)^2 (x-1)^2$. We know that $f$ has an infinite number of local minimizers and is nonconvex on a non-compact subset of $R$. We know that $g$ has ...
5answers
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### How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?

Let $A$ be a real $m \times n$ matrix. How do you prove that $\{ Ax \mid x \geq 0, x \in \mathbb R^n \}$ is closed (as in, contains all its limit points)? The inequality $x \geq 0$ is ...
1answer
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### Dual polyhedron & dual cone

From Wiki: Def. of dual of polyhedral (polytope): polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. EX: 2. Def. of dual ...
4answers
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0answers
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### Is sum (convex combination) of quadratic function/aggregator quadratic?

We know convex combination of concave/convex functions are concave/convex. While convex combination of two quasi-convex/quasi-concave functions necessarily quasi-convex/quasi-concave. Common ...
1answer
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### How Do I Check Convexity Using The Actual Definition?

Suppose $f:\mathbb{R}^{+}\rightarrow\mathbb{R}$ is defined as follows: \begin{eqnarray} f(u)=\text{sgn}(\rho)\left(u^{\rho+1}-1\right),~u\geq 0, \end{eqnarray} where $\rho\in (-1,\infty)$, $\rho\neq 0$...
2answers
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### Poof- A function is convex iff it is convex when restricted to a any line that intersects its domain.

When I read the Convex Optimization, Boyd I noticed a statement about determining a function to be convex or not. It is: "A function is convex iff it is convex when restricted to a any line that ...
1answer
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### Probability that the convex hull of random points is a triangle

Question: Consider a fixed number $k > 3$ of random points in the plane, each independently distributed according to a 2D standard normal distribution. What is the probability that the convex hull ...
7answers
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### Is every monotone map the gradient of a convex function?

Recently in a seminar someone mentioned that monotone maps are equivalent to gradients of scalar convex functions, but it's not clear to me why this is true. One direction of the equivalence is ...
0answers
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1answer
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### 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 ...
1answer
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### Convex functions, prove another definition.

I have the next problem: Suppose $f$ is continuos then $f:\mathbb{R}^n\to R$ is convex iff $\forall x,y \in \mathbb{R}^n \left( \int_0^1 f(x+\theta (y-x))d\theta \leq \dfrac{f(x)+f(y)}{2}\right)$. The ...
1answer
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### Find minimal point of convex set

I'm having some convex set $P \subset \mathbb{R}^n_+$ and a linear-time indicator procedure $I_P(x)$ that allows for each given point $x \in \mathbb{R}^n_+$ to say whether it lies inside $P$ or not. ...
1answer
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### Convex compact set must have extreme points

I am reading a paper and there is such description as title. Why? I have an example: $(0,1)$. This is a convex set but not closed, so I cannot find an extreme point. However if convex and compact,...
1answer
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### How many times can strictly convex functions intersect?

Some time ago, I saw a post related to the number of times that two convex (and continuous) functions' graphs can meet. In general, infinitely many times: one can think, for instance, of $g(x):=x^{2}$ ...
1answer
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### Is the optimizer of a strongly-convex cost function bounded?

Let f(x) be strongly-convex. Can its minimizer be unbounded? I suspect not. Can we obtain a bound on it in relation to the strong-convexity constant? I believe an equivalent formulation of this ...
1answer
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### how do i show that a function $f(x)$ is convex given that the inequality holds

So i have to prove that the inequality below is true if and only if f is convex and Lipschitz continuous. i have the first part down which is to assume f is convex and show the inequality. But i cant ...
1answer
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0answers
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### A set is affine if and only if its intersection with any line is affine. [duplicate]

How can we prove that the a set is affine if and only if its intersection with any line is affine? In fact, I want to know if there is such theorem that the intersection of two affine set is affine?
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### How to give a formula of the perimeter of a $r$-neighborhood of a smooth set in $2D$?

Let $A$ be a simply connected open set in $\mathbb{R}^2$ with smooth boundary. Define $$A^r := \{x \in \mathbb{R}^2: d(x, A) \le r \},$$ where $d$ is the distance function. Let $P$ denote the ...
1answer
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