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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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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54 views

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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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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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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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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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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177 views

DE solution's uniqueness and convexity

I am lost and don't know how to prove the following: If $M$ is a positive definite symmetric square matrix and if $\overrightarrow {v}(t)$ is a solution of: $$\overrightarrow {v'}(t) = ...
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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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1answer
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Characterize polytopes resulting from cutting a convex polytope by a hyperplane

We have a convex polytope $P$ for which we know its set of vertices. Using this set we characterize the H-representation of $P$ as: $\mathbf{A}\mathbf{x} < \mathbf{b}$. If a hyperplane defined by ...
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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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23 views

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 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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1answer
34 views

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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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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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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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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solving the primal problem via dual

On pp. 248 of Boyd and Vandenberghe: suppose 1) strong duality holds, 2) the dual optimal is attained at $(\lambda^*, \nu^*)$, 3) the dual function $L(x, \lambda^*, \nu^*)$ has the unique minimizer ...
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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} ...
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When is the convex hull of two space curves the union of lines?

I am interested in the convex hull of two space curves. Let $A,B\subset \mathbb R^3$ be two spaces curves. I am interested in when $\mathrm {con} (A \cup B)$ equals to $$\bigcup _{a\in A,b\in B} ...
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Minimizing a Concave Function over a convex set

Here is the optimization problem that I am trying to solve. Thanks in advance for all help/insight provided. Let $T:[-2,2]^N\to\mathbb{R}_{-}$ be a concave function of its arguments. Given ...
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Prove boundedness of 2nd derivatives

Let $f \, \colon \mathbb{R}^d \rightarrow \mathbb{R}^d$ be a smooth and convex function. Assume $f$ behaves asymptotically as a cone at infinity, i.e., $ \lim_{R \rightarrow \infty} \frac{f(R x)}{R} ...
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24 views

Convexity and monotonicity

Let $f(n)$ be non-negative real valued function defined for each natural number $n$. If $f$ is convex and $lim_{n\to\infty}f(n)$ exists as a finite number, then can we conclude that $f$ is ...
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42 views

How to prove a point in a set is an extreme point of the set ?

Def: an extreme point of a set $K$ is the point that cannot be expresssed as a convex combination of other points in $K$. Apart from the definition, what else arguments can we use to prove that a ...
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45 views

Prove that $\int_{0 \le u \le 1,\Omega}g^2(x)udx$ in term of $u$ is convex

I am having a cost function and I want to know whether convex or not. Could you explain help me my problem? My problem is that given a cost function such as $$F(u)=\int_{0 \le u(x) \le ...
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Convexity of the complex ellipsoid

Let $p_1,\dotsc,p_n$ be positive integers. Define the complex ellipsoid $$\Omega(p_1,\dotsc,p_n)=\left\{(z_1,\dotsc,z_n)\in\Bbb C^n:\sum\limits_{i=1}^n{\left|z_i\right|^{2p_i}}<1\right\}.$$ I ...
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Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
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Open embedding of affine toric varieties implies Cone is face of the other

let $\tau, \sigma \subseteq N_{\mathbb{R}}$ be two rational, strongly convex polyhedral cones with $\tau \subseteq \sigma$. Now we get an inclusion $S_{\sigma} \to S_{\tau}$ inducing an inclusion ...
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Proving that $\text{ri rge}\,A=\text{ri conv rge}\,A$

"If $A:\mathbb R^n\rightrightarrows\mathbb R^n$ is maximal monotone,then $\text{ri rge}\,A$ is convex". This is a proposition in auslender's book about the asymptotic cones. We can prove that ...
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How to check whether the following function is concave or convex or neither.?

Let $\pi$ be a vector such that all its elements sum to 1. i.e, $\sum_1^n \pi(i) = 1$ where $\pi(i)$ denotes the i$^{th}$ component and $n$ is the length of the vector. Let $D$ be a diagonal matrix ...
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N-Functions (Nice Young functions)

A mapping $\Phi:[0,\infty)\to[0,\infty)$ is termed an N-function (nice Young function) if (i) $\Phi$ is continuous on $[0,\infty)$; (ii) $\Phi$ is convex on $[0,\infty)$; (iii) $\lim\limits_{t ...
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Second-order quadratic model with bias term

I have 3 points in the 3-d space and I would like to estimate the parameters of a second-order quadratic model with a bias term $z=f(x,y)=ax^2+bxy+cy^2+dx+ey=\theta^TQ\theta+\eta^T\theta$ where the ...
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does constant convexity assures global minimum

I have the following question: Consider a function $f:R^n \longrightarrow R$, s.t.: there is a point $x_0 \in R^n$ s.t. $\frac{\partial f}{\partial x^k} =0$ $\forall k$. the hessian matrix ...
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Why is the constraint $\|w\| = 1$ non-convex?

Related: Why is this function, related to SVM derivation, non-convex? I am studying notes which cover the derivation of SVM. The intuition is the geometric margin should be maximized in order to ...
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Clustering of vectors via inner product relationship

This might be an odd question, but suppose I have a lot of vectors $a_i\in\mathbb{R}^{3}$ (not necessarily unit) and for some unit vector $u\in\mathbb{R}^{3}$ I find $$ \sum_{i=1}^m ...
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Orthogonal Projection of an arbitrary point on the rectangle and the Lorentz cone

I want to know how to find the orthogonal projection of an arbitrary point on the rectangle and the Lorentz cone. I need to know the derivation or the concept not just the formulae which are available ...
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How $x^4$ is strictly convex function?

I hear that $f(x)=x^4$ is a strictly convex function $\forall x \in \Re$. However, strict convexity condition is that the second derivative should be positive $\forall x \in \Re$. For the mentioned ...
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how to show that$ n<{2n \choose n}$ in sets

what can be some methods to prove and explain $$n<{2n \choose n}$$ is true , Iam having diffuculty is proving and explain it though it seems easy . please can anyone help me with my small problem ...
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Estimation of Quadratic form parameters and convexity/concavity surface

I have 3 points in the 3-d space with their coordinates $(x~y~z)^T$. I would like to find the expression of the $\textbf{concave}$ quadratic surface that form those 3 points, i.e., $z=f(x,y)=ax^2 + ...
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Why is this function, related to SVM derivation, non-convex?

I'm working through a support vector machines tutorial. In eventually deriving the solvable objective function, the following objective function (to be maximized) was proposed, but dismissed as ...
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Deriving Dual Averaging from (Sub)gradient Descent

Here the presenter tries to derive a simple Dual Averaging from (sub)gradient descent. I have a little problems understanding the steps. (Sub)gradient descent: Loop through: $$ x_{k+1} = x_k - t_k ...
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Can a compact set of $\mathbb{R}$ have some properties and not being convex

The question is related to this one On a condition when bounded sets in R n is convex ?. Suppose that $n > 1 $ and that $C \subset \mathbb{R}^n$ is a compact (closed and bounded) set having a ...
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351 views

Convex function and its epigraph PROOF

Can someone help prove this statement. Consider a function $f:R^{n} \to R$ and epi $f$ = {$(x,t) \in R^{n+1}$: $x \in R^{n}$, $t \geq f(x)$} A function is convex if and only if its epigraph is a ...
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On a condition when bounded sets in $\mathbb R^n$ is convex ?

Is it true that a bounded set in $\mathbb R^n$ , $n>1$ , is convex iff every straight line through an arbitrary interior point of the set intersects the boundary of the set in exactly two points ? ...
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If the boundary of a convex set in $\mathbb R^n$ ($n>1$) is connected , is it necessarily also path-connected ?

If the boundary of a convex set in $\mathbb R^n$ ( where $n>1$) is connected , is it necessarily also path-connected ?
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Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
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Half space representation of a convex polytope

We know that the half space representation of a polytope is given by: $Ax<b$. Consider a convex polytope in $\mathbb{R}^3_+$ with vertices given by the following set of points: ...
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Dividing given rectangle into regions with certain finite points being given

The problem I am facing is as follows. Suppose we have a unit square and we have been given certain no. of finite points, n (For eg n=5) which are randomly spread in the square. Now, we want to ...
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1answer
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Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ?

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ? if yes , then is it also path connected ?