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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If f(x, y) is convex, is g(x)=f(x, c) convex, for any constant c?

If $f(x, y)$ is convex (concave) defined on $\mathbb{R}^2$ and $g(x)=f(x, c)$, $c\in \mathbb{R}$, then is $g(x)$ necessarily convex (concave)?
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Are these equivalent definitions of faces of convex sets?

I consulted several books and found that the definitions of faces and exposed faces of convex sets are a bit messy. Many books just treat them as the same stuff. In book "foundations of ...
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Are these inconsistent definitions of extreme subset of convex set?

I need some elementary knowledge of polyhedral and I am consulting several books. However, I found the definitions may not be consistent. I am not sure if I understand them right. The question is ...
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Does every LCS has a convex balanced local base?

Does every LCS--locally convex (topological vector) space has a convex balanced local base? Then it implies every LCS is topologized with a countable number of separating seminorms. So there seems ...
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Proof of Jensen's inequality - why is this progression valid

I'm reading a proof of Jensen's inequality in the following version: Let $f: I\rightarrow\Bbb{R}$ be a convex function defined on an interval $I$. Then: $$\forall{x_1,...,x_n\in ...
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Convex Optimization: Advantages of Symmetric Primal-Dual Algorithms?

This is a follow-up to an answer on a previous question on PD algorithms: http://math.stackexchange.com/a/1193928/36257 I have done some research learning the mechanics of how Infeasible PD ...
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“No cone of this fan contains a nonzero linear space”

On page 393 of this paper, Speyer and Sturmfels produce a quotient of a particular geometric object (which happens to be a polyhedral fan), and claim that, as in the question title, "no cone in this ...
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Geometric description of composition of two convex polygons? [duplicate]

The composition was defined as follow: $(a,b)$ $\in$ $(R;S)$ $\Leftrightarrow$ $\exists c | (a,c) \in R \\ $&$ (c,b) \in S$ . Also the composition was defined as the binary product of two ...
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Is $\mathbf{y}^*$ a local minimizer of $f(\mathbf{h}(\mathbf{y}))$?

Let $f(\mathbf{x})$ be a twice differentiable function, where $\mathbf{x} \in \mathbb{R}^n$. Let $\mathbf{x}^*$ be a local minimizer of $f(\mathbf{x})$. Consider a differentiable and invertible ...
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How can I solve $y \in (N_X + \nabla f)(x)$ via projection?

I a aware that if I'm trying to solve for $x$ the problem $y \in [\lambda I + N_X](x)$ where $y$ is a known vector, and $N_X$ is the normal cone given by $N_X(x) = \{u : \langle u, x - y\rangle ...
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How to characterize the boundary of a convex set?

I am working on a part of a paper related to topological properties of boundary points. It is important to me to realize the topological and algebraic behavior the boundary points of convex sets. I ...
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Covering of convex sets

This is basic geometry question: Let $a,b,c,d,e$ be five distinct points in $\mathbf{R}^3$ and denote the convex hull of $X$ with $\mathrm{conv}(X)$. How can I prove formally that ...
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Is this sequence of concave functions unbounded?

Let $h_1, h_2,$ etc. be a sequence of positive real numbers such that $$\sum_nh_n = \infty.$$ Let $x_1, x_2,$ etc. be a sequence of real numbers in $(0, 1)$. Let $f_0, f_1,$ etc. be of sequence of ...
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Additional assumption to have a convex image

Let $f\colon \mathbf{R}^2 \to \mathbf{R}^2$ a continuous injective function. In general, it is not true that the image $f[X]$ is convex whenever $X \subseteq \mathbf{R}^2$ is convex. Is there some ...
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Sign of difference of two convex functions

Suppose you have two continuous, convex functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$. Suppose that both $f$ and $g$ are minimised at $x=0$ with $f(0)<g(0)$. ...
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Maximization of probability that all packets are successful simultaneously

I have packet streams $1...k$ and, streams with Prob(err) $p1...pk$. The $p$'s are consts $>0$. I'd like to maximize the probability all make it simultaneously while I'm allowing at most $N$ ...
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Non-empty interior of the convex hull of $f(a),f(b),f(c)$

Let $a,b,c$ be three points in the plane which are not collinear. Let $f\colon \mathbf{R}^2 \to \mathbf{R}^2$ be a continuous injective function. Show that the interior of $$ ...
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Why is this an nuclear norm?

The following figure is from a paper: "The Convex Geometry of Linear Inverse Problems" Just on p.3. As in the figure, the red lines are $2 \times 2$ symmetric unit-Euclidean-norm rank-one ...
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Examples of uniformly convex function

$f$ is uniformly convex with modulus $\phi$ if for any $\alpha \in [0,1]$ and any $x,y$ in the domain, $$f(\alpha x + (1-\alpha)y) \le \alpha f(x) + (1-\alpha) f(y) - \alpha(1-\alpha)\phi(\|x-y\|)$$ ...
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Show that a specific cone in the plane has two generators

Let $u \in \mathbb{R}^2$ and $\{v_1, \dots, v_s\} \subseteq \mathbb{R}^2$ ($s \geq 2$) such that $\{v_i, v_j\}$ is linearly independent if $i \neq j$ and $(u, v_i) > 0$ for all $i \in \{1, 2, ...
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Blackwell's informativeness criterion

Let $a=(a_1,\dots,a_i,\dots,a_n)$ be a probability vector, i.e. $\forall i: a_i\ge0$ and $\sum_i a_i=1$. Suppose $b$ is another $n$-dimensional probability vector. Is it true that there always ...
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Convex Optimization: do Primal Dual methods need to start with strictly feasible point?

I'm learning about Primal-Dual interior point algorithms from Boyd & Vandenberghe Convex Optimization, ch.11.7. Now the text mentions: In a primal-dual interior-point method, the primal and ...
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Convexity of a subset of $R^3$ based on its bounding surface

Let D be a closed region in $R^3$ whose boundary S is a closed, smooth, orientable surface. The tangent plane at every point on S intersects it in a connected set $M$. Is D (the interior of S) a ...
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For convex, smooth function with $\phi''(1)>0$ we have $\phi(s)-\phi(1)\geq \phi'(1)(s-1)+c(s-1)^{2}$

For convex, smooth function $\phi:\mathbb{R}_{+}\to \mathbb{R}$ with $\phi''(1)>0$ we have $\phi(s)-\phi(1)\geq \phi'(1)(s-1)+c(s-1)^{2}$ for constant $c>0$. Attempt By Taylor's theorem ...
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51 views

Convex function on closed interval: boundary points?

$I=[a,b]$, let the function $f:I\rightarrow\mathbb{R}$ be convex. (1) Is it possible to prove the existence of the limits: $$\lim_{x\rightarrow a^+}f(x) \ \ \ \ \ \lim_{x\rightarrow b^-}f(x)$$ If ...
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Dense face in Schwartz space

I have a lema and an example for which i don't understand how are they not in contradiction. I am not sure is there something obvius that i am missing. Any help would be much appreciated. Here it is: ...
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29 views

Log-Determinant Concavity Proof

Can you please help me understand how he gets the equation marked by red from the above one ?
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Non-empty intersection with the interior of a convex set

Let $C$ be a convex set in the plane with non-empty interior, and let $x \in C$ be a point on its bourdary. Prove that, for each open neighborhood $N$ of $x$ there exists $y \in N$ which belongs to ...
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Fourier coefficient of convex function

On $I = [0, 2π]$ consider the function $f : I → \mathbb{R}$ to be convex. Define: $$a_k\pi := \int_0^{2\pi}f(x) \cos(kx)\,dx$$ Show that the convexity of $f$ implies that $a_k ≥ 0$ when $k ≥ 1$. ...
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Nearest point on Spherical Cap

Let $A \subset \mathbb{S}^n$ be a spherical cap. More specifically, there exists a point $v \in \mathbb{S}^n$ and $\epsilon > 0$ such that $A = \{u \in \mathbb{S}^{n}\mid v\cdot u \geq \epsilon\}$. ...
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Exercise 2 in Hatcher, section 1.2: the union of convex sets is simply connected

Here is my problem : I have convex subsets $X_1, \dots, X_n$ of $\mathbb R^m$ such that $X_i \cap X_j \cap X_k$ is never empty. I have to show that $\bigcup\limits_{i=1}^nX_i$ is simply ...
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weighted geometric mean

Ask for a hint to show following concave: $h(y) = y_1^{\theta1}...y_m^{\theta m}$ with $\theta_1+...+\theta_m=1$ and $\theta_i \geq 0$ If I do not want to use Hessians to show, any better way to ...
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About the strictly convexity of log-sum-exp function

The log-sum-exp function $f: \; \mathbb R^n \to \mathbb R$ is defined by $$f(x)=\ln \left (e^{x_1}+\cdots + e^{x_n} \right).$$ It is well-known that this function is convex, but I wonder that ...
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Convex Convolution [duplicate]

Let $f: R^n\to R\cup\{\infty\}$ to be a convex function. Let $f_{\epsilon}(x)=\frac{|x|^2}{2\epsilon}$. Show that: $$\lim_{\epsilon\to 0}\inf_{x=y+z}f_{\epsilon}(y)+f(z) =f(x)$$ The $\leq$ part is ...
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Basis of the vector space generated by a convex cone

Let $C$ be a pointed convex cone in a vector space, meaning that: $C + C \subset C$, $\mathbb{R}_+ \cdot C \subset C$, $C \cap (-C) = \{ 0 \}$. If $V$ denotes the vector space generated by ...
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Is exp(-x) convex?

Is $f(x)=e^{-x}$ a convex function? I know that $e^x$ is convex. If I take the second order derivative of $f(x)$: $$f''(x)=e^{-x}$$ Then we can see for all the $x$, $f''(x)>0$. I'm not sure ...
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(Convex) Reformulation of a program

Given $\{(x_i,y_i)\in \mathbb{R}^d\times \mathbb{R}\}_{i=1}^n$, consider the the following program: \begin{eqnarray*} \mathrm{min}_{\{\hat{y}_i \in \mathbb{R}\}_{i=1}^n,\{g_k \in ...
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Containment of zero matrix in a convex subset of ${\mathbb R}^{m\times n}$

Does anyone know the answer (and proof) to this question: Suppose $S\subset {\mathbb R}^{m\times n}$ is a convex and compact subset of $m\times n$ real matrices with respect to the Frobenius norm, and ...
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Equality form of second order Taylor series

I am reading a book on optimization wherein a statement using Taylor's expansion is made without proof. \begin{equation} f(\mathbf{y}) = f(\mathbf{x}) + (\mathbf{y} - \mathbf{x})^T\nabla ...
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How might one show that the Proximal mapping of the indicator function is the projection operator?

Let ${C \subset \mathbb{R}^n}$ be closed, convex, and nonempty. How might one show that the proximal mapping of the indicator function of $C$ is in fact the projection operator on to $C$?
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Is face closed?

A face of convex set $C$ is a convex subset $F$ of $C$ such that for $x,y\in C$ and some $\lambda\in \langle 0,1\rangle, \lambda x+(1-\lambda)y\in F$ implies $x,y\in F$. I was wondering if every ...
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Comparison of two convex functions at the points of equal derivative

Suppose that I have 2 increasing, convex functions $f_1$ and $f_2$ such that: $f_k:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ for $k = 1, 2$, $f_k(0) = 0$ for $k = 1,2$ and $f'_1(x) < f'_2(x)$ for ...
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Is exponential function strongly convex?

Assume $x \in \mathbb{R}$. In the wiki page, one property of strongly convex functions $f(x)$ is that it satisfies: $f''(x)\geq m > 0~\forall x$ with with parameter $m > 0$. Given $f(x) ...
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Derivative of a multivariable function

I have a fairly basic question that has perplexed me for a few hours now. I am trying to evaluate the derivative of a function $g(t):\mathbb{R}\rightarrow\mathbb{R}$ defined as \begin{equation} g(t) ...
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Eigenvectors of a quadratic form and iterative descent

I am interesting in using eigenvectors of a quadratic form to perform iterative steps to get the function value to a certain point. While other methods may be more common, my quadratic form is not ...
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what is the convex hull of the rank k psd matrix

Given the set $\{X|0\preceq X , rank(X)=k\}$. What is the convex hull (convex envelope) of this nonconvex set? If we further require $X=VV^T$, where $V^TV=I$, $V$ has the size $n\times k$. Then the ...
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Convex function with non-symmetric Hessian

Let $U$ be an open convex subset of $\mathbb R^n$ and $f:U\to\mathbb R$ a convex function on it. It is a well-known fact that if the second partial derivatives exist everywhere on $U$ and are all ...
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Gradient and Hessian of a function with Matrix Variables

What is the Gradient and Hessian of this function? $$ f(X)=\langle X,D\rangle-c \cdot \sqrt{\langle X,E\rangle}$$ where $X,D,E$ are all semi-definite matrices. Where Gradient becomes zero?
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Is this function of Semi-definite matrices, convex?

Is the following function is convex. Consider the following function $$ f(X)=\langle X,D\rangle-c \cdot \sqrt{\langle X,E\rangle}$$ where $X,D,E$ are all semi-definite matrices. Is this function ...
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If $\Omega$ is convex, then $K_{\Omega}$ is convex?

Let $\Omega\subset\mathbb{R}^n$ and $$K_{\Omega}=\{\lambda x|\lambda\ge0,x\in\Omega\}$$ Is it true that if $\Omega$ is convex, then $K_{\Omega}$ is also convex. Let $\gamma\in(0,1)$ and $z,t\in ...