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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continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
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what can be said about the solution of the following differential equation

$y(x)=f(x)+\log(1+y'(x))$ with $\lim_{x \to 0}f(x)=\infty,f(\infty)=0$ and $f(x)$ is contiuoues and decreasing. In particular, can we prove that the solution $y(x)$ is convex in general? from ...
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Largest and smallest shape enclosed within circles

There is a convex shape $C$. It is known that the largest disc contained in $C$ has radius $r$ and the smallest disc containing $C$ has radius $R$. What are the smallest-area and largest-area shapes ...
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Find those values 'a' which belongs to the Convex Hull

Find those values of 'a' for which (1,a,1) belongs to the convex hull of $$\{(0,0,0), (1,1,2),(2,4,-6), (1,3,8)\}$$ Give me hints as much as you can, I would like to understand the mindset rather ...
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33 views

Convex hull of a set of points

Let $a_1,a_2...a_r \in R^n$ be points in $R^n$. Prove:$$CH(\{a_1,...,a_r\})=\left\{\sum_{i=1}^r\alpha_ia_i|\sum_{i=1}^r\alpha_i=1,\alpha_i\ge0\right\}=:K$$i.e. the convex hull of the $a_i$ is the set ...
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Conditions for unique solution of a maximization problem?

Let $S\subseteq \mathbb{R}^2$, $d:=(d_1,d_2) \in S$, and $s:=(s_1,s_2)$ a generic point of $S$. Assume that there exists $s \in S$ such that $s_1>d_1$ and $s_2 >d_2$. Consider the following ...
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Subdifferential of a continuous function is non-empty

Prove that if $X$ $-$ normed vector space, $x_0 \in \text{int }A$ and convex function $f$ is continuous in $x_0$ then $\partial f(x_0) \neq \emptyset$. $\partial f(x_0) = \{x^*\in X^*:\forall x \in ...
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Sum of k-largest eigenvalues of a symmetric matrix as an SDP

I found the following statement from a google search. If $S_k(\mathbf{X})$ is the sum of the $k$ largest eigenvalues of a symmetric $m\times m$ matrix $\mathbf{X}$, then,$$S_k(\mathbf{X}) \leq t$$ is ...
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in normed space hyperplane is closed iff functional associated with it is continuous

E is a normed linear space . i have two questions Q1 why the complement of H is nonempty Q2 How then the functional is continuous?? Thanks
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Is there Lipschitz property for subdifferential?

I'm trying to bound the quantity $\langle \nabla \Psi(x),\bar{x}-x \rangle$ above, with the bound depending on $\|x-\bar{x}\|$ and perhaps also of $\|x-y\|$ for fixed (but not varying) points $y$. ...
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When does a continuous function defined on a closed and bounded convex set has a fixed point?

For a function $f$ defined from a domain $K$ to itself, we have a point $x$ in $K$ is said to be a fixed point of $f$ if $f$ maps $x$ to itself. When the domain K is a compact convex set with some ...
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Approximating a set of convex quadratic inequalities by a convex polytope

I have a convex set of the form $$Z = \{x|x^TQ_ix+b_i^Tx+c_i\le0,i=1,\ldots,m\}$$ where each $Q_i\succeq0$, that I wish to approximate by a set of the form $$\hat Z = \{x|Ax\le b\}$$ We can further ...
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the hyperplane is closed iff functional is continuous

i have two questions Q1 what is the importance of symmetric neighbourhood here in the proof? Q2 how if l(N) is unbounded then l(N) is R ? Thanks
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Is it true that $A \geq B$ implies $\|A\|_2 \geq \|B\|_2$ for $A,B \geq 0$?

All matrices are real and not necessarily symmetric. Denote by $A \geq B$ the condition that $(A-B)$ has eigenvalues with non-negative real parts. Denote by $\| \cdot \|_2$ the $L_2$ matrix norm. Is ...
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Show that scalar-valued function of a matrix is convex

Consider the mapping $$f(X) = g\left(\frac{b}{a^TXa}\right),$$ where $g$ is a convex function, $b$ is a strictly positive scalar, $a$ is a real vector, and $X$ is restricted to be symmetric and ...
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What is the definition of “convex” and “relaxation” concepts in clustering?

I have following text from a paper i am trying to understand: I don't understand what does below sentence refers to as being convex/non-convex The problem is that even though the objectives ...
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38 views

Prove $\Xi (I - P)$ has eigenvalues in the non-negative real half-plane.

Let $P$ be a stochastic matrix (square, non-negative,rows sum to one). Let $\Xi$ be a diagonal matrix with any left principal eigenvector of $P$ on the diagonal and zeros elsewhere (stationary ...
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Convergence rates of the radius of largest hypercube within a random convex hull

Let $X_1,\cdots,X_n$ be i.i.d drawn from uniform distribution on $[0,1]^d$, and let $$\eta_n:=\inf\{\eta\in (0,1):[\eta,1-\eta]^d \subset \mathrm{conv}\{X_1,\cdots,X_n\}\}.$$ Then it is easy to show ...
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Uniform lower bound on convex functions bounded in $L^2$ norm

Consider a class of (proper closed) convex function on $[0,1]^d$, which we shall denote $\mathcal{F}$. If every element of $\mathcal{F}$ is bounded in $L_2$, say $$\int_{[0,1]^d} |f(x)|^2\ dx\leq 1,$$ ...
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49 views

Properties of a differentiable and strictly convex $f:(a,b) \to \mathbb{R}$

Let $f:(a,b) \to \mathbb{R}$ be a differentiable and strictly convex function I tried to explore some of the properties of such a function. For all $x,y \in (a,b)$ with $x \neq y$ I could apply ...
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convexity of function built from piecewise linear convex function?

Let $B(x):[0,1] \rightarrow [0,1]$ be piecewise linear increasing convex function with $B(0)=0$ and $B(1)=1$. (Think of the power of the neyman-persron test). Let $E(x)=-log(B(1-e^{-x}))$ a logaritmic ...
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Quasiconvexity of the composition of two functions

Consider $A: \mathbb{R}_{\geq 0}^n \rightarrow \mathbb{R}_{\geq 0}^{n \times n} $ and $B \in \mathbb{R}_{\geq 0}^{n \times m}$, and $c \in \mathbb{R}_{\geq 0}^n$. Assume that, for all $y \in ...
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Convexity of set $\{x\mid\operatorname{dist}(x,S) \leq \operatorname{dist}(x,T)\}$

Is it possible to prove analytically the convexity of the set: $\{x\mid {\rm dist}(x,S) \leq {\rm dist}(x,T)\}$ where $S,T \subseteq \Bbb R^n$, and ${\rm dist}(x,S) = \inf\{\|x-z\|_2 \mid z \in S\}$? ...
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Seperation of convex compact sets with affine halfspaces

Let $C_1,C_2,...,C_m$ compact convex sets s.t. $\bigcap C_i = \emptyset$. I want to show that in that case there exsist affine halfspaces $H_i$, such that for every $i=1,2...,m$, $C_i \subset H_i$ ...
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45 views

Type of convex function?

I want a convex function $f:\mathbb{R} \to \mathbb{R}$ with the following property: given points $x,d \in \mathbb{R}$, and $\alpha \in (0,1)$, we have $$f(x + \alpha d) \geq \alpha f(x + d).$$ Is ...
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Solution set of a quadratic inequality

Let C $\subseteq$ $\Re^n$ be the solution set of a quadrtatic inequality, C = $\{x \in \Re^n | x^TAx +b^Tx + c \leq 0\}$. $A \in \Re$, b $\in \Re^n$ and c $\in \Re$. We want to show: That C is ...
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Uniquness and Exisstence of One Theorem

I need a short and nice Proof for Uniqueness and Existence of the following theorem: Suppose (H, <0,0> ) is a Hilbert space, and M is a closed convex set and $x \in H$, then there is a unique ...
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Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
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Looking for numerical methods for finding roots of convex vector function ${\bf f}({\bf x})={\bf 0}$

Consider the function ${\bf f}:\mathbb{R}^n\to\mathbb{R}^m$ defined as ${\bf f} = (f_1,f_2,\ldots,f_m)$ where each $f_i:\mathbb{R}^n\to\mathbb{R}$ is twice-continuously differentiable convex in ${\bf ...
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Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
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Convexity of the quadratic form of a matrix

I am going through a research paper where they have made a statement without proof. They have mentioned that ...
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Prove a certain matrix is positive semidefinte.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi$ to be a diagonal matrix with a principal left eigenvector of $P$ on the main diagonal and zeros ...
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Convex conjugate of $\ell_1$ and $\ell_2$ norm

Given a function $f$, we can define a function $f^{*}$, called the convex conjugate (also known as the Fenchel conjugate) of $f$ as follows: $$ f^*(\vec{z})= \sup_{\vec x \in \mathbb ...
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uniformly convex domain and uniformly convex function

I want to ask a question about uniformly convex domain: Suppose $\Omega$ is unifromly convex, i.e. for each point $x_0\in\partial\Omega$, with regualrity $C^2$ smooth. Uniform convexity of domain ...
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349 views

Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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convexity of piece wise function

I have a piece-wise function defined on the real line. The pieces are connected continuously, and the second derivative of each piece is strictly positive. Does this means that the function is convex? ...
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Minkowsky Theorem

Theorem: Let $L$ be an $n$-dimensional lattice in $\mathbb R^n$ with fundamental domain $T$, and let $X$ be a bounded symmetric convex subset of $\mathbb R^n$. If $Vol(X)>2^nVol(T)$ then $X$ ...
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Properties of functions having the form $g(x,t) = t f(\frac{x}{t})$

I have been frequently coming across the function $g(x,t) = t f(\frac{x}{t})$ in my course on convex optimization. A friend of mine mentioned that it is the perspective function, but the book on ...
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Does every strongly convex function has a stationary point?

Say does every differentiable $\mu$-strongly convex function $f:\mathbb{R}^n\mapsto\mathbb{R}$, with $\mu>0$ have a point where its gradient is $0$? If not so which is the minimum you can impose ...
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Property of Newton step for self-concordant convex functions

Suppose $f(x)$ is a convex and self-concordant function minimized at $x^*$. I have two starting points $\tilde{x}_0$ and $\hat{x}_0$ such that $|\hat{x}_0-x^*| \le |\tilde{x}_0 - x^*|$. We also know ...
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Segment ordered density conjecture.

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset ...
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Using Taylor's theorem and Lagrange form of the reminder to prove the second order condition for convexity

I try to prove the second order condition for convexity. So far' I've done the following: First, I prove second order => convexity: Let $f$ be a function with positive semi-definite Hessian. Using ...
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How to determine if a vector belongs to the conical hull of a set of vectors?

Let $\mathbf{p}_i$ be a finite set of finite-dimensional real vectors with non-negative components with the property that, for any $k$, $\mathbf{p}_k$ cannot be expressed as a linear combination with ...
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Proofs regarding about all Second Derivative Test cases (Inconclusive & Single Variable)

This is how I would prove f''(c) > 0 that f(c) has local min and I would easily flip the inequalities and state a conclusion for f''(c) < 0 that f(c) has local max. Quick Proof for f''(c) > 0 ...
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Is every convex function differentiable amost every where?

If a function $f: D \subset R^n \rightarrow R$ is convex, then for every $x,y \in D$ and $\alpha \in [0,1]$, $$ f(\alpha x + (1 - \alpha)y) \leq \alpha f(x) + (1 - \alpha)f(y). $$ I konw a convex ...
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Every concave function that is nonnegative on its domain is log-concave?

This is a statment from Wiki. I'm not sure why this is true: If: $f(\theta x+(1-\theta y) \geq \theta f( x) + (1-\theta)f(y)$ And $f(\cdot) \geq 0$ then: $$f(\theta x+(1-\theta y) \geq f( ...
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The polar of a set: Importance and Applications

Given a duality $(X,Y)$, for any subset $A\subset X$, we can define the polar set $A^{\circ}\subset Y$. In this sense, the polar relates sets and sets in the dual space. Since cones are sets, we can ...
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What's wrong with the following trace optimization?

I'm reading a paper that has used the augmented Lagrange function for optimization. I've tried to derive one subproblem but got a different answer from that in the paper. Could you help check it ...
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Minkowski sum of convex sets in the plane which are not polygons

Can the Minkowski sum of two convex sets in the plane which are not polygons be a polygon? Explicitly my convex set is of the form $ C= \{(x,y) \in \mathbb{R}^2 : x,y \geq 0 \text{ , } ...
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How to express a set as an intersection of halfspaces

I have a set S = {x $\epsilon$ $\mathbb R^n$| $x^Ty \le 1$, $\forall y \epsilon A$} Now, I want to prove that this set is closed and convex. I know that expressing this set as an intersection of ...