# 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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### Proving the concavity of a function

I want to prove that the function $x \mapsto \Phi(\Phi^{-1}(x) + \lambda)$ defined for $x \in [0,1]$ is concave for any $\lambda \geq 0$. $\Phi$ is the cumulative distribution function of a standard ...
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### Inequality involving the Hessian matrix of a convex function

Let $f \in C^2(\mathbb{R}^d)$ be a convex function with Hessian $H$. Is it true that $$(x^T H(x) - y^T H(y)) (x-y) \ge 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$$ for all $x,y \in \mathbb{R}^d$? ...
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### Medial axis of non-convex polygon

I used CGAL 4.8 - 2D Straight Skeleton and Polygon Offsetting, which covers the convex case. Now the polygon can be non-convex and simple. How to compute the medial axis in this case? This ...
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### Can Extragradient method be expressed with proximal steps?

As we know, for solving saddle point problems, forward-backward algorithm is generally not guaranteed to converge. But extragradient method converge Structured Prediction via the Extragradient Method ...
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### Proof of Convexity Using Quasiconvexity

Suppose $f_{1}, \ldots, f_{n}$ are continuously differentiable functions defined on $\mathbb{R}$ such that for each $i\in \{1,\ldots,n\}$, we have: (i) $f_{i}(v_{i}^{0})=0$ and ...
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### 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., ...
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### Dual of a maximization problem

We have a positive, smooth, increasing concave function $f:\mathbf{R}^n\to \mathbf{R}^+$ and $k$ smooth, increasing constraint functions $f_i:\mathbf{R}^n\to\mathbf{R}$. I've recently encountered two ...
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### When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
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### Closure of intersection of convex sets

Let $C_i$ be a convex set in $R^n$ for $i\in I$, suppose sets $ri \, C_i$ have at least one point in common, then how to prove this: $cl\bigcap\{C_i\mid i\in I\} = \bigcap\{cl \, C_i \mid i\in I\}$
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### Monotony and convexity of $U(t) = w(t) - t. w'(t)$

Let $\mathcal{D} = (\mathbb{R}^{+*})^2$. $c \in ]0,1[$. Moreover we have $\theta < 1$ and $\theta \ne 0$. We consider the following function (which is called CES or constant elasticity of ...
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### Existence of direction for polyhedral set

I refer to Lemma 4.42 of this lecture notes on linear programming, about the relationship between the boundedness of a polyhedral set and its directions. Let $P$ be a non-empty polyhedral set. ...
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### Representation of a point in terms of extreme points and extreme directions in a LP feasible region

We know that every point in a feasible region $(X)$ of a Linear Program can be represented as a convex combination of the extreme points of $X$ plus a non-negative combination of the extreme ...
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### Concave up theorem for $f:A \to\mathbb R, A \subseteq \mathbb R$ - True or false?

I am a 1st year-2nd semester student of the department of Applied Mathematics @ NTUA. Some days ago, I had a really interesting conversation with one of my old school mathematics teacher about a ...
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### Is function $f(x) = \left( ( {\bf w} -x {\bf v})^T ( {\bf w} -x {\bf v}) \right)^k$ convex for $k \ge 1/2$

Let ${\bf v}$ and ${\bf w}$ be column vector of dimension $n$. Is function $f(x) = \left( ( {\bf w} -x {\bf v})^T ( {\bf w} -x {\bf v}) \right)^k$ convex for $k \ge 1/2$ ? I want to show this via ...
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### Is $h(x,y) = g(x)f(y)$ convex, if $g(x)$ and $f(y)$ are convex in $x$ and $y$, respectively?

I want to check if $h(x,y) = g(x).f(y)$ is convex, given that g(x) is convex and decreasing in $x$ on its and $f(y)$ is increasing and convex in $y$. Is there any way to check/prove the convexity of ...
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### Good text book recomendation

I would like to do some reading about a technique called sequential convex programming. There is a lot of material about sequential quadratic programming out there, including books (Nocedal & ...
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### If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?

If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
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### Alaoglu & Krein-Milman to show bounded convex weak-$*$ closed subset has extreme point

Let $X$ be a normed linear space and $K$ a bounded convex weak-$*$ closed subset of $X^{*}$. I need to show that $K$ possesses an extreme point; however, I am not entirely sure how to do this. I ...
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### Are odd functions that are concave and increasing everywhere necessarily linear?

The title gives away pretty much the entire question except for the fact that the class of functions I am interested in is a subset of twice continuously differentiable functions. I think that if an ...
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### What value of 'a' will be the function is convex, concave or not either?

$$f(x,y) = -6x^2 + (2a+4)xy - y^2 + 4ay$$ The solution has to be : $$-2-\text{gyök}(6) \leq a \leq -2 + \text{gyök}(6)$$ I tried to define the derivation of the function accordance with $x$ , and ...
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### Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$\operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}.$$ (it is a cone in ...
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### Find a positive convex function $f$ defined on $[a,b]$, s.t. $f(a)\times f(b)=1$ and $\int_a^b{f'^2dt}=12$

Find a function $f:[a,b]\to \mathbb{R}$ which is convex on $[a,b]$ such that $\int_a^b{f(t)dt}=0$, $\int_a^b{f'^2(t)dt}=\frac{12}{b-a}$, and $f(a)f(b)=1$? Another similar question which states: Find ...
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### Caratheodory's theorem for a point in boundary

I am wondering whether the following holds: if $x$ in $\mathbb{R}^d$ lies in the boundary of the convex hull of a set $P$, then $x$ can be expressed as a convex combination of $d$ points in $P$. We ...
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### Convex hull of bounded set is bounded

I want to show that in a locally convex topological vector space $X$, the convex hull of a bounded set is bounded. Apparantly this does not hold if $X$ is not locally convex. So the fact that that ...
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### How can show the following function is log-concave?

Suppose that $g(x)$ is an increasing function and $0\leq g(x)\leq1$. I was working on a problem and it reduced to show that if $1-g(x)$ is log-concave then $$f(x)=(1-g^a(x))^b, a\geq 1, b,x>0$$ is ...
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### How to prove this inequality about the arc-lenght of convex functions?

Let be $f,F:[0,1] \to \mathbb{R}$ with $f,F \in C^2([0,1])$ two convex functions such that $f \le F$ in all points and $f(0)=F(0)$, $f(1)=F(1)$. Considering the plane-curves $\gamma(t)=(t,f(t))$ and ...
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### maximum of a concave function in a minkowski sum

Let: $f(x,y)$ - a strictly-concave function, monotonically increasing in $x$ and $y$; $A,B$ - two compact and convex sets in the positive quadrant; $C$ - their Minkowski sum, $A+B$; $(x_A,y_A)$ - ...
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### convex function on open interval

I have a quick question. If a continuous function $f$ is convex on $(a,b)$, then the following is true? Could you explain why or why not it is true? $for \,\,x\in(a,b)$, $t=\frac{x}{b}<1$ Thus, ...
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### How to get inequality representation of cone give by generators

To be as specific as possible, I have a subspace $I \subset R^{14}$ of dimension $3$ and an $11\times14$ matrix $A$ with linearly independent rows spanning the orthogonal complement of $I$. I have an ...
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### If $1^TR K Y diag(f) l=m$, how to upper bound $f^T l$

In the following $K$ is a kernel matrix, ( and so is positive semidefinite), $Y$,$R$ are diagonal matrices with elements $R_{ii}=+1$ or $R_{ii}=-1$, $Y_{ii}=+1$ or $Y_{ii}=-1$. $f$ is a diagonal ...
Let $\Lambda\subseteq R^n$ contains $m$ elements, where $\lambda_i$ is the $ith$ element, and $co(\Lambda)$ is the smallest convex cone contains $\Lambda$. Also, consider any point $u\in R^n$. Now, I ...