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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Proof that Convex Function with alternate variable is convex

Given that say, $f(x)$ is convex for $x>0$. Would the proof that $f(a-x)$ where $a>x$ is convex proceed as shown below. Please verify the correctness and/or alternative approaches that can ...
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24 views

Convex Hull = Boundary+Segments

If $A\subseteq\mathbb{R}^n$ is an non empty set and $H$ is the convex hull of $A$, how can I prove that the boundary of $H$ consists only of points that lie in the boundary of $A$ and segments that ...
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1answer
26 views

Is a twice differentiable function whose only extrema is a minimum automatically convex?

I have a twice differentiable function $H(x)$ for which I have already proven that: $\exists !x^*:H'(x^*)=0$ and furthermore that $H''(x^*)>0$ (the only extremal is a minima). $H''(x)$ is ...
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Prove a convex function

I have to prove that if $f:A \to \mathbb{R}$ is convex and $c \ge 0$ then $c \cdot f:A \to \mathbb{R}$ is convex. I know that function $f:A \to \mathbb{R}$ is convex if for $\forall x,y \in A$ and ...
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2answers
121 views

Convexity of the ratio of the standard normal PDF by its CDF

Is there some way to show that the following function $\psi$ is concave or convex? Here, $\phi$ and $\mathbf{\Phi}$ are the standard normal PDF and CDF, respectively. ...
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1answer
28 views

A convex hull of a union of convex sets [closed]

Let ${A_1},{A_2},....{A_n}$ be convex sets in a vector space and suppose $x \in \operatorname{co}({A_1} \cup {A_2} \cup \dotsb \cup {A_n})$. Is it true that $x = {t_1}{a_1} + \dotsb + {t_n}{a_n}$ such ...
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45 views

Proving a convex function [closed]

I'm given a function $f:A \to \mathbb{R}$ which is twice continuously differentiable on $A \subseteq \mathbb{R}^n$. $A$ is a convex set. Show that $f$ is convex. Any ideas on how to prove this?
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What are the extreme points of the closed unit ball of $C$? .

What are the extreme points of the closed unit ball of C(the space of all continuous functions on the unit interval), with the supremum norm?
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25 views

Rectificable curve as a boundary of a convex set

Let $K\subseteq\mathbb{R}^2$ be a convex compact set. Is it true that $\partial K$ (the boundary of $K$) is a rectificable curve (i.e. it has length)?
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43 views

how to prove convexity of the function below?

For a graph $G$ consider the following function, $$f=\sum_{(i,j) \in G ,(i,k) \notin G } \max(0,c+ \left\|e_i-e_j\right\|_2^2-\left\|e_i-e_k\right\|_2^2)$$ where $e_i \in\mathbb R^n$ ($n$ dimension ...
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Why there is nonconstant linear funectional $\Gamma $ on $X$ such that $\Gamma (A) \cap \Gamma (B)$ contains at most one point?

Let $X$ is a real vector space(without topology). call a point $x \in A \subset X$ an internal point of $A$ if $A-x$ is an absorbing set.Suppose $A$ and $B$ are disjoint convex set in $X$ and $A$ has ...
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1answer
21 views

Is the following function concave? (or log of it)

I have a function $f(x_1, x_2, ..., x_M) = \displaystyle \prod_{i = 1}^N \frac{(\sum_{j = 1}^{M} a_jx_jI_{ij})^2}{\sum_{j = 1}^{M} a_jx_j}$ in domain $\{{\bf x} \in {\bf R}^m \setminus {\bf 0} \ ...
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Sequence of convex non increasing sets convergence

I have a question for you. I was wondering whether a non increasing sequence of convex set converges to a convex set. Here my question made more precise: Let $\{S_k\}_{k=1}^\infty$ be a sequence of ...
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1answer
42 views

KKT conditions for a convex optimization problem with a L1-penalty and box constraints

I am having some trouble deriving / understanding optimality conditions for a convex optimization problem of the form: $$\begin{align} \min_{x\in\mathbb{R}^d}~ & f(x) + C.\|x\|_1 &\\ &x_i ...
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How to show the optimal condition of $f(\alpha) = \frac{R^2+G^2\sum_{i=1}^k \alpha_i^2}{2\sum_{i=1}^k \alpha_i}$

Consider the following function: ($\alpha>0$) $$f(\alpha) = \frac{R^2+G^2\sum_{i=1}^k \alpha_i^2}{2\sum_{i=1}^k \alpha_i}$$ It is a quadratic (in $\alpha$) over linear (in $\alpha$); therefore, ...
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21 views

Compact convex subset and hyperplanes

Suppose $K$ is a compact and convex subset and $x^*$ a point in $\mathbb{R}^n$. Suppose there exists $y\in \mathbb{R}^n$ such that $$\langle x^*, y\rangle> \sup_{x\in K} \langle x, y\rangle$$ ...
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321 views

Integral of an increasing function is convex?

Let $f$ be a real valued differentiable function defined for all $x \geq a$. Consider a function F defined by $F(x) = \int_a^x f(t) dt$. If f is increasing on any interval, then on that interval F is ...
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64 views

Proof that a log-of-sum-of-exponentials is a convex function

It's well known in statistical mechanics that the following is a convex function of the vector $\theta$: $$ A(\theta) = \log \left( \sum_{i=1}^\infty e^{\theta \cdot f(i)} \right) $$ where $f(i)$ is ...
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Good graphic tool for drawing the convex hull of two planes?

If I want to draw the convex hull of two 2D planes, what kind of tool box should I use ? The graph of functions in page 2,3 of the following file are quite nice, anyone can guess what kind of ...
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1answer
87 views

Intersection of a convex polygon and a moving circle

I have a straight line which intersects a convex polygon in 2D plane. There exists a circle with constant radius. The center of circle is moving on this line. So at first the polygon and circle don't ...
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202 views

Crossing of two convex functions with same asymptotic slopes

Suppose you have two continuous, positive convex functions $F(x)$ and $G(x)$, $x\in\mathbb{R}$ such that: $$\lim_{x\rightarrow\pm\infty}F'(x)=\lim_{x\rightarrow\pm\infty}G'(x)=\pm 1$$ and that ...
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37 views

Two duality theorems

Suppose $X$ is a Hilbert space with norm $||.||$ and $K$ is a weak compact and convex subset of $X$. The supporting functional: $$h(x^*)=\sup_{x\in K} \langle x^*, x \rangle$$ The indicator ...
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Intuition on primal convergence in dual subgradient method

It is well known that the subgradient method applied to the Lagrange dual of a convex problem may produce a sequence converging to the dual optimum, but the primal iterates produced by this sequence ...
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Centers of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
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Uniqueness of supporting hyperplane for a face of a cone

In William Fulton's 'Introduction to Toric varieties' he says - " When $\sigma$ spans $V$ and $\tau$ is a facet of $\sigma$ then there is a $u \in \sigma ^{\vee}$ unique upto multiplication by a ...
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1answer
55 views

Jensen inequality for convex functions - infinite countable number of weights

Does Jensen inequality for convex functions hold if there is countable infinite number of weights? For example weights are given by sequence $(\frac{1}{2^n})_{n\in\mathbb{N}_1}$ ? If not, where is ...
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If the hessian of a function is positive definite everywhere, is it convex everywhere? [duplicate]

G'day! If the hessian of some multivariable function is positive definite everywhere, does that necessarily imply that the function is convex everywhere?
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Increasing Function and Convex Set Question

Consider a function $0 \le f(x) \le 1$ which is increasing in $x \in [a,b]$, I was wondering can I say that $f(x) \le \epsilon$ for $0< \epsilon <1$ defines a convex set? I think the ...
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Directional derivative of difference of two convex functions

I would like to find the references and the proof for the following fact: Let $g,h:\mathbb{R}^n\rightarrow\mathbb{R}$ be two convex functions and $f=g-h$. Suppose that $\bar{x}\in\mathbb{R}^n$ such ...
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Convex Hull algorithm.

Working on making a Convex Hull algorithm. I need to figure out how to iterate the remaining points to find the shortest angle as marked below in the picture. I am ...
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Classify extreme points of multivariate implicit functions when cross derivative is not available

I have the following problem: Let $f(x,y)$ be a function defined on $[0,1]^2$ I want to prove that $f(x,y)$ has no local minimum for $x>y$. I have no idea about the sign of the cross derivatives ...
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Calculating the convex conjugate of the function $f(x)=\lim_{n\to \infty}\left(-\frac{1}{n}\log \sum_{k=1}^n e^{a_k\cdot x+b_k}\right)$.

The convex conjugate also known as Legendre–Fenchel transformation of a convex function $f:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}$ is function $f^\ast:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}$ definite ...
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1answer
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Extreme points of the set of positive regular borel measures on a compact Hausdorff space

I have some troubles with a specific proof of a (Bochner-type) theorem in Rudin's book "Functional Analysis". More specifically, let $X$ denote a compact Hausdorff-Space and let $M$ denote the set of ...
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Legendre–Fenchel transformation $ f^{\ast}(x^\ast)=\sup_{x\in\mathbb{R}^n}\{\langle x,x^\ast\rangle -f(x)\} $

The convex conjugate also known as Legendre–Fenchel transformation of a convex function $f:\mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$ is definite by $$ ...
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When is this set convex and compact?

$ S= \{(f_1,f_2)\in L^2(I)\times L^2(I)| f_1(x)+f_2(x)\leq 1, a.e.; 0\leq f_1(x)\leq a(x)\leq 1, a.e.; 0\leq f_2(x)\leq b(x)\leq 1, a.e. \}$ To make $S$ to be convex and compact, does $a,b$ need ...
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Relation between the number of halfspaces and the number of vertices of a convex polytope

Suppose we have an $n$-dimensional convex polytope $\mathbf{P}$ represented by an intersection of half-spaces as the following: \begin{equation} \mathbf{P} = \{ x \in \mathbb{R}^n \mid x \in ...
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$L_+^p(X,\mu)$ is a closed and convexe subset of $L^p(X,\mu)$.

I have a problem with an exercise: Let $(X,A,\mu)$ a measure place, $p\in[1,\infty)$ and $\mu(X)<\infty$.Prove that the set $$L_+^p(X,\mu):=\{f\in L^p(X,\mu):f(x)\geq 0\ \mu-\text{a.e.}\}$$ is a ...
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Convex quadratic problem solver gives different answers?!!

I'm not a mathematics girl but I'm pretty sure that the variance of a vector X should be a convex quadratic problem. my objective function is as follows: arg min var(sum(L) + X*L) x>0 vector X is ...
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Equivalent implications about convex functions

Consider $U\subset\mathbb{R}^n$ open and convex with $f\colon U\rightarrow\mathbb{R}$, $f\in C^1(U)$. Show that the following are equivalent: (i) $f$ is convex; (ii) For all $a, a+v \in U$, one has ...
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Solution of an equation and a system of inequalities

Consider an integer $n \geq 1$, a positive real number $A$ and a collection of nonnegative real numbers $\{a_{i,j}\}$ defined for $(i,j) \in \{1,\cdots,n\}^2$. I want to find necessary and sufficient ...
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Proof that f is convex

Consider $D\subset\mathbb{R}^n$ a convex set and $f_i:D\rightarrow\mathbb{R}$ convex functions in $D$, $i \in I$ is a any set of indexes. Suppose there is $\beta \in \mathbb{R}$ such that ...
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Understanding the subdifferential sum rule

A previous question asked: Given: $f$ and $g$ are lower-semicontinuous proper convex functions, $x \in \text{ri dom}(f) \cap \text{ri dom}(g)$, $h = f+g$, $p \in \partial h(x)$, Prove that there ...
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Is the ratio of a convex and linear function pseudoconvex?

Both functions are differentiable. I know from Chandra1 that the ratio of a nonnegative convex and a strictly positive concave function is pseudoconvex. Does this hold when the denominator is a ...
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Solution to a nonlinear problem at an extreme point

I have a convex optimization problem of the form: $$ \begin{aligned} \operatorname*{minimize}_{\mathrm{x} = (\mathrm{x}_1, \dots, \mathrm{x}_m) \in \mathbb{R}^{nm}} &\quad f(\mathrm{x}) = ...
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Convex Function

$f: U\subset\mathbb{R}^m \to \mathbb{R}$ is a convex function if $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$, for all $x,y \in U$ and all $t \in [0,1]$. If $f$ is convex and continuous function, and $f$ has ...
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Concavity of a positive homogeneous one function

Let $f:(0,\infty)\times(0,\infty)\rightarrow(0,\infty)$ be a twice continuous differentiable funcion such that (1) $f$ is homogeneous one, i.e. $f(tx,ty)=tf(x,y)$, for all $t,x,y>0$; (2) $\log f$ ...
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1answer
49 views

Convex function when λ∉[0,1].

f :R→R is convex, Prove, for every x,y∈R, and λ∉[0,1] f(λx+(1−λ)y)≥λf(x)+(1−λ)f(y). In definitoin of convex funcion λ belongs in [0,1], but here not.
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40 views

Logarithm of Gaussian function is whether convex or nonconvex?

I have a gaussian distribution such as $$P(x)=\frac {1}{\sqrt {2\pi}\sigma}e^{-\frac {(x-\mu)^2}{2\sigma^2}} $$ As my knowledge, $P(x)$ is non convex function interm of $x$. However, if I map it to ...
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95 views

Describing convex hulls in purely metrical terms

Let $X$ denote a Euclidean space; take $X = \mathbb{R}^n$ for concreteness. Now consider $x,y \in X$. Then the line segment joining $x$ and $y,$ hereafter denoted $[x,y]$, can be described in ...
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42 views

Convex functions-Comparison of derivative and second derivative

Let $\phi:(0,\infty) \to \mathbb{R}$ be a function with second derivative, strictly incresing and concave. Suppose that $f(t)=\phi(e^t)$ is convex. Then one can prove that $$ \lim_{x \to \infty} ...