Questions tagged [convex-analysis]

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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18 views

I’m studying engineering but I’d like to get more involved in pure mathematics.

I now realized I don’t like engineering that much and I love maths and I’m good at I. I want some book recommendations to get involved more in mathematics (especially convexity and fuzzy logic)
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22 views

Supremum of a strictly convex function is always infinity?

Suppose there is a strictly convex continuous function $f$: $R^n$ $\rightarrow$ $R$. Is the supremum of $f$ always infinity? Does there exist any bounded strictly convex function?
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11 views

Quadratic form can be represented as a convex combination of $\frac{n(n+1)}{2}+1$ ones

Question : In $\mathbb{R}^n$, consider an inner product $(\ ,\ )$. Here any linear map $L$ has the form $L(x)=(V,\ x)$ for some $V$ When $\|\ \|,\ \|\ \|^\ast$ are norms in dual relation, then we have ...
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2answers
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How to prove that $\frac{\cosh (2 \pi x)}{x^2}$ is convex?

I want to prove that this function $$f(x)=\frac{\cosh (2 \pi x)}{x^2}$$ is convex $(0<x<10)$, and find its minimum. The first and second derivatives are a follows $$f'(x)=\frac{2 (\pi x \sinh (...
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2answers
28 views

Theorem for the Optimization of Linear Function over a Bounded Polyhedron

In optimization theory, I often see people say that the minimum a linear function over a compact convex set is attainable at some extreme point of the feasible set. I have no problem with its proof, ...
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14 views

Smoothness (i.e. Lipschitz continuous gradient) of supremum

Define $$ \mathcal{P} = \{p \in \mathbb{R}^n \ | \ \sum_{i=1}^n p_{(i)}=1, \ p_{(i)} \geq 0, \ \sum_{i=1}^n \phi(p_{(i)}) \leq \rho \}, $$ where $p_{(i)}$ is the $i$'th element of the vector $p$, $\...
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25 views

Fenchel conjugate of convex combination of two norms

Let $\tau \in [0,1]$. Let us define the norm $\Omega(x)=\tau\|x\|_1+(1-\tau)\|x\|_{1,2}$; where $$ \|x\|_{1,2}=\sqrt{\sum_{g \in \mathcal{G}} \left(\sum_{i \in g} |x_i|\right)^2} $$ is the exclusive ...
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41 views

Proof verification: $f$ is convex iff $f'$ is monotonically increasing

This is (the first half of) exercise 14 in Baby Rudin Let $f:(a, b) \to \mathbb{R}^1$ be differentiable. Prove that $f$ is convex iff $f'$ is monotonically increasing. ($\Rightarrow$) Assume $f$ is ...
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1answer
24 views

Finding a Lower Envelope of a Compact Convex Set in $R^2$

Consider the compact convex set $S$ in $\mathbb{R}^2$ with three extreme points $(0, 0)$, $(1, 1)$, and $(0, 1)$, i.e., $S$ is a triangle. To find the lower convex envelope $f(t)$ of $S$, where $t\in[...
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1answer
32 views

Derivation of the Convexity Relation for Multivariable Function

It is easy to derive the convexity relation: if $f(x)$ is convex, $f(t x_{1} + (1 - t) x_{2}) \le t f(x_{1}) + (1 - t) f(x_{2})$ has to be satisfied where $f(x): M \subseteq \mathbb{R} \rightarrow \...
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1answer
29 views

Proving a transformation preserves a property

Let $f(x_1,x_2,\ldots,x_n)=(x_1/(\sum_{i=1}^n x_i),x_2/(\sum_{i=1}^n x_i),\ldots,x_n/(\sum_{i=1}^n x_i))$ be in $\mathbb{R}^n_{++}$, and $X$ be a set of points in $\mathbb{R}^n_{++}$. I was trying to ...
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1answer
23 views

Weakly convex and continuous implies convex [duplicate]

We have that a function $f$ is called weakly convex on $I$ if for all $x_1, x_2\in $ it holds that $$f\left (\frac{x_1+x_2}{2}\right )\leq \frac{1}{2}\left (f(x_1)+f(x_2)\right )$$ If a function $f$ ...
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1answer
47 views

Is the $\arg\min$ of a strictly convex function continuous?

Let $X\subset \mathbb{R}^n$ and $Y\subset \mathbb{R}^m$ be compact and convex sets, and let $f:X\times Y\rightarrow \mathbb{R}$ be a continuous function. Suppose that for each $y$, $f(x,y)$ is ...
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1answer
26 views

Projection onto a closed convex set in a general Hilbert space

Let $E$ denote a real Hilbert space and suppose $G \subset E$ is a nonempty closed convex set. I know that in this case, there is a unique nearest point in $G$ to each $x \in E$. Call this point $P_G(...
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59 views

Prove that function is convex, first order smoothness,second order smoothness

I have the following function: $$\mathbb{R}^n \ni (x_1, \cdots , x_n) \mapsto \ln( \sum_{k=1}^{n}{exp(x_k)})$$ I know how to prove if a function is convex, but I have trouble with this specific one. ...
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1answer
15 views

When is the sum of a quasi-convex function and a convex function quasi-convex?

I know that the sum of a convex function and a quasi-convex function is not necessarily convex, it is easy to construct counter-examples like $f(x)=-x$ and $g(x)=x-1/2|x|$. However, I have a setting (...
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2answers
72 views

Does convexity at a single point imply convexity w.r.t finite convex combinations?

Let $\phi:\mathbb (0,\infty) \to [0,\infty)$ be a continuous function, and let $c \in (0,\infty)$ be fixed. Suppose that "$\phi$ is convex at $c$". i.e. for any $x_1,x_2>0, \alpha \in [0,...
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2answers
74 views

Does convexity at a point imply existence of one-sided derivatives?

Let $\phi:\mathbb (0,\infty) \to [0,\infty)$ be a continuous function, and let $c \in (0,\infty)$ be fixed. Suppose that "$\phi$ is convex at $c$". i.e. for any $x_1,x_2>0, \alpha \in [0,...
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20 views

first order condition for quasiconvex functions

I need to prove the following statement. Let $ f:\mathbb{R}^{n}\to \mathbb{R}$ be a differentiable function. If $\forall x,y\in$ dom$(f)$, $f(y)\le f(x)\Rightarrow \nabla f(x)^{T}(y-x)\le 0$, then $f$...
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quasiconvex functions and their minimum [closed]

Let $f$ be a differentiable quasiconvex function. Show that the condition $\nabla f(x)=0$ implies that $x$ is a local minimum of $f$.
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72 views

Is the following function about nuclear norm strictly convex?

The nuclear norm (trace norm) of a matrix $X \in \Bbb R^{m \times n}$ is defined as $$\|X\|_* := \sum_{i=1}^{\min(m,n)} \sigma_i(X)$$ where $\sigma_i(X)$ are the singular values of $X$. Now there is a ...
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1answer
47 views

Equality for Convex Functions

A function $f$ is convex on $\mathbb{R}$ if for all $x\in \mathbb{R}$ $\lambda \in [0,1]$ $$ f(\lambda x+(1-\lambda) y)\leq \lambda f(x)+(1-\lambda)f(y), $$ Similarly we can define a convex ...
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1answer
17 views

Uniqueness of fitted values from the lasso

For some λ ≥ 0, suppose that we have two lasso solutions β hat, γ hat with common optimal value c*. I need to show that it must be the case that Xβ = Xγ meaning that the two solutions must yield the ...
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1answer
37 views

Is $\frac{f'}{f}$ bounded for $f$ convex, $f>c$?

Let $c>0$, $f\colon \mathbb{R} \to [c,\infty)$ be differentiable and convex. Do we have $$ \left\|\frac{f'}{f}\right\|_{\infty} < \infty ?$$ This seems to be true in simple examples, but I am ...
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7 views

Cones of continuous convex functions in the set of continuous functions on a compact set

I'm trying to understand the properties of the cone of continuous real-valued convex functions (called it $K_1$) and the cone of continuous convex piecewise linear functions (called this $K_2$) in the ...
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1answer
17 views

$\nabla f(x)^T(y-x) \geq 0$ if $x$ is optimal for a convex $f(x)$.

In the Convex Optimization book by Boyd and Vandenberghe, it states that if $x \in X$ and $X$ denotes the feasible set, and $f(x)$ is a convex objective function, then $x$ is optimal IFF $$\nabla f(x)^...
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1answer
26 views

Minimizing a non-convex function through its component-wise convex functions

Let $f(x,y)$ be a continuously differentiable function from $\mathbb{R}^2$ to $\mathbb{R}$. I do not know whether $f$ is convex. But I do know that for any fixed $x$, $g_x(y)=f(x,y)$ is strictly ...
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1answer
18 views

Higher order Jensen-like expansion upper bound

If $Z$ is a random variable with fine moment generating function, what is a good way to upper bound $$|\log \mathbb{E}e^Z- \mathbb{E}Z- \frac{1}{2}\mathbb{E}Z^2|$$ This looks like a third offer Taylor ...
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1answer
24 views

How to prove the inverse image under an affine function is convex, if the image is convex?

Theorem in section 2.3.2 of Boyd & Vandenberghe's Convex Optimization: If $f:R^k \to R^n$ is an affine function and an set $ S \subseteq R^n$ is convex, the inverse image of $S$ under $f$ defined ...
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16 views

Duality in deterministic stochastic control and convex conjugate

I am currently reading the book "Stochastic Multi-stage Optimization" and trying to solve the Stochastic Optimal Control problem given in this book in the framework of duality. The problem ...
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15 views

component Lipschitz constant

We had the following definition in class: Definition Suppose $f: \mathbb{R}^{n} \to \mathbb{R}$ continuously differentiable and $\nabla f(x)$ Lipschitz-continuously with constant $L > 0$, i.e. \...
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1answer
41 views

Looking for a function that preserves concavity

Consider the function $$ f(x,y) = g(x) y $$ where $g$ is some other function. We can restrict ourselves to $y\geq 0$ and $0\leq x\leq 1$. I would like to find a function $g$ with the following ...
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0answers
29 views

Prove convexity (product of two functions)

Is there a way to prove convexity of $$ f\left(\frac{x^{\mathsf T}\sigma}{\sqrt{x^{\mathsf T}\Sigma x }}\right)\sqrt{x^{\mathsf T}\Sigma x}$$ where $\sigma$ is a vector, $\Sigma$ is positive definite, ...
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1answer
18 views

Finding whether the given minimization problem is a convex problem

I have the following problem which attempts to minimize the temperature deviation from a given set point in a home. Minimize, $ \sum_{t=1}^{N}{(T_{c}-T_{t})^{2}}$ where, $ T_{0} =T_{s}$ $ T_{t} = T_{t-...
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16 views

Guarantees for one concave function greater than the other concave function

Suppose we have two functions $f(x) = a \sqrt{x}+b\cdot x$ and $g(x)$ which can be any concave function, and $x\geq 0$ for both functions $f(x)$ and $g(x)$. Is it possible for us to compute a solution ...
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24 views

Determining convex function from all its expectations

Suppose I have a finite alphabet $\mathcal X$ and a convex function $h:\mathcal S(\mathcal X)\to \mathbb R$ from the simplex of probability distributions on $\mathcal X$ to the reals. Let $p_X$ be a ...
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2answers
75 views

Does midpoint-convexity at a point imply midpoint-convexity at other points?

This question is a follow-up of this one. Let $f:\mathbb R \to \mathbb [0,\infty)$ be a $C^{\infty}$ function satisfying $f(0)=0$. Suppose that $f$ is strictly decreasing on $(-\infty,0]$ and strictly ...
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1answer
30 views

Does midpoint-convexity at a point imply midpoint-convexity at a larger point?

Let $f:(-\infty,0] \to \mathbb [0,\infty)$ be a $C^1$ strictly decreasing function satisfying $f(0)=0$. Given $c \in (-\infty,0]$, we say that $f$ is midpoint-convex at the point $c$ if $$ f((x+y)/2) \...
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1answer
91 views

Does this partial midpoint-convexity imply full convexity?

Let $f:(-\infty,0] \to \mathbb [0,\infty)$ be a $C^1$ strictly decreasing function. Definiton: Given $c \in (-\infty,0]$, we say that $f$ is midpoint-convex at the point $c$ if $$ f((x+y)/2) \le (f(x) ...
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15 views

Sub-differential of a convex function along a particular direction

Take a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. Choose an arbitrary direction $d \in \mathbb{R}^n$ and consider the restriction of $f$ to the line through $x \in \mathbb{R}^n$ in the ...
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1answer
18 views

Uniform lower bound of positive-definite Hessian on unit sphere

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be twice continuously differentiable with positive-definite Hessian (denoted by $\nabla^{2}f$), i.e., for all $x,y\in\mathbb{R}^{n}$, we have $y^{\top}\nabla^{2}f(x)...
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0answers
14 views

Is the function $A\phi(x)+\phi(x-c)$ log-concave on $[0,c]$ where $A\geq 1$?

Let $\phi$ be the Gaussian CDF, $\phi(t)=\int_{-\infty}^t\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \, dx$. We know that $\phi$ is log-concave. If we consider a combination of $\phi$ with a shift of $\...
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1answer
36 views

Does midpoint convexity at a single point imply full convexity at that point?

Let $f:[a,b] \to \mathbb [0,\infty)$ be a continuous function, and let $c \in (a,b)$ be a fixed point. Suppose that $f$ is midpoint-convex at the point $c$, i.e. $$ f((x+y)/2) \le (f(x) + f(y))/2, $$ ...
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1answer
22 views

The converse of some theorems about Minkowski functional

(From Lax's functional analysis) Let p denote a positive homogeneous subadditive function defined on a linear space $X$ over the reals. (1)The set of points $x$ satisfying $p(x) < 1$ is a convex ...
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3answers
46 views

Showing a function is convex/concave given another function is concave

Assuming $f(x)$ is not a linear function, given a concave decreasing function $f(x)$, I want to find whether $g(x)=f(x)\cdot x$ is convex or concave for strictly positive $x$. However, I'm having a ...
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1answer
67 views

Biggest convex set inside a concave unit ball

Denote the unit ball for the $p$-norm in $\mathbb{R}^N$ with $p \in (0,1]$, $$S_p^N = \Big \{ x \in \mathbb{R}^N,\ \Big(\sum \limits_{i=1}^N |x_i|^p\Big)^{1/p} \le 1 \Big\}$$ We want to find a convex ...
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1answer
25 views

Convex functions and minimizations [closed]

triyin to solve this exercise, any advices and hints on how to prove it are more than helpful. The problem Let $C \subset \mathbb{R}^n$ be a convex set, and $f: C \rightarrow \mathbb{R}$ is also ...
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2answers
22 views

Is the function convex?

Let's have the following function $f:\mathbb{R}^{2}\to\mathbb{R}$ defined by $|x+y|$, is it convex? We have $\lambda\in (0,1),x,y\in$ dom$(f)$, so $|\lambda x+(1-\lambda)y|\le \lambda |x| + (1-\lambda)...
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1answer
30 views

Proving a mathematical statement with projections

Let $a, x \in \mathbb{R}$, $S \subset \mathbb{R}^n$ a convex and closed set and $L = S + \{x\}$. Prove that $L$ is convex and closed and that $proj_L(a) = x + proj_S(a - x)$. Guys, im stuck with this ...
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2answers
25 views

proving an inequality involving projections

Let $S \subset \mathbb{R}^n$ be a convex and closed set. Show that, given $x, y \in \mathbb{R}^n$: $$\|\operatorname{proj}_S(x) - \operatorname{proj}_S(y)\| \leq \|x - y\|$$ This questions seems to be ...

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