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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Boundary points of probability simplex

I have a very simple question for which I know the answer but I can not prove it! What are the boundary points of a probability simplex? I know every probability vector with one zero component lies ...
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Proof that the image of an Itō integral is convex if the driving Wiener process is in a metric ball

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $A := \int_0^1 f(t)\,d W_t$ be the Itō integral of an $L_2([0,1])$ deterministic function $f$ with respect to the Wiener process $W$. ...
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Prove that $x \rightarrow \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt$ is convex

To put it bluntly I'm stuck proving proving the subsequent inequality $$ \forall x>0, \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt \int_0^\infty \frac{t^2 e^{-tx}}{1+e^{-t}}dt \geq {\left ( ...
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Finding the dual cone

Looking at the example here, I'm trying to understand how the author finds the dual cone $K^*$. The question asks to find the dual cone of $\{Ax | x \succeq 0\}$ where $A \in \mathbb R^{m\ ...
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Does local convexity imply global convexity?

Question: Under what circumstances does local convexity imply global convexity? Motivation: Classically, a twice differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ is convex if and only ...
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Litterature: Convex Geometry using differential forms

I've been looking for papers or books that discuss/use differential forms in convex geometry. I have been very unsuccessful in my search and was wondering if anybody here had come across such ...
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Stronger than strict convexity, bounded hessian?

I've encountered a condition similar, but slightly stronger, to that of a function being strictly convex. The condition is $\phi(\lambda x+(1-\lambda)y)\leq \lambda \phi(x)+(1-\lambda)\phi(y) - ...
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Proof that the set $\{ x \in \mathbb{R}^n: \max\{w_1^Tx, \ldots, w_N^Tx\} \geq \gamma\}$ is not convex in general

Let $w_1, \ldots, w_m$ and $x$ be vectors in $\mathbb{R}^n$, and $\gamma$ be some constant in $\mathbb{R}$. How can I prove that the set $\{ x \in \mathbb{R}^n: \max\{w_1^Tx, \ldots, w_N^Tx\} \geq ...
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Proving that a Hessian Matrix is positive definite

I'm currently stuck on a problem for my Artificial Intelligence class. The assignment is provided at the following link: http://courses.engr.illinois.edu/cs440/HW1.pdf The problems that I'm stuck on ...
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Legendre transform for HJB PDE

I am trying to understand section 2.3 of the following article: http://www.princeton.edu/~sircar/Public/ARTICLES/montreal.pdf . I think I understand that $H_v(t,g(t,z)) =z$, because by definition of ...
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Intersection of half planes vs union?

Can someone explain to be why we are taking intersection instead of union? Because taking the union means we are also taking the union of ALL the $y$s in $S$ no?
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Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
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Conjugate of a function

Yes this is homework. Given $f(x) = 1^{T}(x)_+$ where $(x)_+ = \max\{0,x\}$, what is $f^*$? I know that the conjugate of a function $f$ is $f^*(y) = \sup (y^Tx - f(x))$ but I do not know how to show ...
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convexity of matrix “soft-max” (log trace of matrix exponential)

In convex optimization it is often convenient to use the following smooth approximation to $\max\{x_1, \ldots, x_n\}$: $$ f_\lambda(x_1, \ldots, x_n) = \frac{1}{\lambda}\log \sum_{i = 1}^n{e^{\lambda ...
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1answer
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Determining if a function is convex

Yes this is homework. For which values of $a$ is the function $f(x)=e^{-a \sqrt x}$ with $\mathbf dom f = \mathbf R_+$ convex? The possible answers are: $a \le 0$ $a \ge 0$ $-1 \le a \le 1 $ ...
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Convexity of product of two given functions

Let $f(x)$ be a convex function of $x$ in a given positive interval. Also assume $f(x)\geq 0$ everywhere in that interval. Is the function $g(x)=xf(x)$ convex in that interval?. Is it possible that ...
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Composition of convex function and affine function

Let $g: E^{m} \rightarrow E^{1}$ be a convex function, and let $h: E^{n} \rightarrow E^{m} $ be an affine function of the form $h(x)=Ax+b$, where $A$ is an $m \times n$ matrix and $b$ is an $m \times ...
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Why is the feasible set of utility values (in bargaining problem) convex?

Let $S := \{x \in \Bbb{R}^n \mid x \ge 0, \sum_{i=1}^n x_i = 1\}$ be the set of mixed strategies. For a bimatrix game with pay-off matrices $A$, $B$ we denote $C := \{ (u, v) \mid \exists (x,y)\in ...
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Convexity of $\frac{1}{f}$ over the set where the concave function $f$ is positive

$S \subset R^n,~~f : S \rightarrow R $ is a concave function. $S^{'}= \{ x \in S: f(x)>0 \}. $ Prove that $\frac{1}{f}$ is a convex function on $S^{'}.$
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1answer
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How to compute the image of a polyhedron under a linear transformation

Suppose $P$ is a polyhedron whose representation as a system of linear inequalities is given to us: $$ P = \{ x ~|~ Ax \leq b\}$$ Define $P'$ be the image of $P$ under the linear transformation which ...
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3answers
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Convex functions - two questions

I have two questions regarding convex functions: First question: Let f be convex function on closed interval [a,b]. Prove that f has maximum in x=a or x=b. I understand that $\forall ...
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Prove or disprove the concavity of the function [closed]

Prove or disprove the concavity of $f$ over the following two domains. $$f(x_1,x_2)=10-2(x_2-x^{2}_{1})^{2}$$ defined either over $$S_1=\{(x_1,x_2) : -1\leq x_1 \leq 1, -1 \leq x_2 \leq ...
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Finding the determinant of a $ k \times k$ matrix (Hessian matrix)

Given $H(x_{1}, x_{2}, x_{3}) = \begin{bmatrix} -2 & \frac{1}{2} & 0 \newline \frac{1}{2} & -2 & 0 \newline 0 & 0 & -4 \end{bmatrix}$, I want to find (I think) the leading ...
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Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
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Relation between convex functions

I formed the following conjecture and, since I can't find counterexamples, am trying to prove it. Let $f, g :[0,x_{max}]\rightarrow {\mathbb R}^{+}$ such that $f',g'>0$ $f'',g''>0$ ...
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Stochastic gradient descent for convex optimization

What happens if a convex objective is optimized by stochastic gradient descent? Is a global solution achieved?
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28 views

Calculate Dq(x)

Let A be a symmetric $m \times m$ matrix, and $q(x)=x\cdot Ax$ a quadratic form on $\mathbb{R}^m$. Question: Calculate $Dq(x)$; write your answer in vector notation. Does anyone knows the answer on ...
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convexity of a Hessian matrix.

Suppose I have $f(x_{1},x_{2}) = x_{1}^2 + x_{2}^2, S = \mathbb{R}^2$. How do I determine whether the function is concave or convex based off of the Hessian of what is above? I know the Hessian is ...
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On the convexity of a particular discontinuous function.

Let $f:D\to\mathbb{R}$ be defined as follows: $$ f(\mathbf{x})=\frac{a-(\mathbf{x}_N\cdot\mathbf{x}_0+x_{n+1})}{\sqrt{\mathbf{x}_N^T\mathbf{A}\mathbf{x}_N}}, $$ where ...
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Uniqueness of the solution to a quadratic opt problem

Consider a positive definite matrix $\boldsymbol H$, the known vectors ${\boldsymbol b}$ and ${\boldsymbol a}_i$. Now the minimization problem is casted with respect to the vector ${\boldsymbol x} $ ...
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How to find a curve inside a non-convex

I want to connect two points in a space within the space. If the space is convex, I can simple draw a line between them. But how about a non-convex space. How can I find a curve connecting these two ...
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Intersection of affine subspace of $\mathbb{R}^n$ with $[0, 1]^n$

Suppose I have an affine subspace $V \subseteq \mathbb{R}^n$, say given by a rank-$r$ system of $m$ equations in $n$ variables. I'm interested in two questions: Is there a straightforward way to ...
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Show that the set of points that are nearer $a$ than $b$ with respect to $\lVert \cdot \rVert_2$ is convex

I am trying to show the above statement: Show that the set of points that are nearer $a$ than $b$ in the sense of Euclidean $\lVert\cdot\rVert_2$ are convex. My attempt This follows from the ...
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Convex Hull of cyclic Permutations

It is known that the convex hull of permutation matrices yields exactly the stochastic matrices. I am interested in the convex hull of cyclic permutation matrices. Trivially this is a subset of the ...
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On Convexity of product of a convex and a bounded function.

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a real-valued, $n$-dimensional function defined as follows: $$ f(\mathbf{x}) = g(\mathbf{x})h(\mathbf{x}), $$ where $g:\mathbb{R}^n\to\mathbb{R}$ is convex, and ...
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Voronoi-Diagram - split concave polygon

I have a concave polygon c and a set of points p[]. What I need is to have a set of polygons splitted by the voronoi-diagram. My problem is the following: I already found a library to calculate the ...
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convex/concave problem.

I want to show that if $y = f(x) > 0$ is a concave function on $\mathbb{R}$, then $z = \frac{1}{f(x)}$ is a convex function. Since $f(x) > 0$ then if we applied the second derivative test, ...
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There is a closed hyperplane.

$\textbf{Question: }$ If $M$ is an open convex set in normed linear space $R$ and $x_{0}\not\in M$, then there exists a closed hyperplane which passes through the point $x_{0}$ and does not intersect ...
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Is this set convex ?2

Is this set convex for every arbitrary $\alpha\in \mathbb R$? $$\Big\{(x_1,x_2)\in \mathbb R^2_{++} \,\Big|\, x_1x_2\geq \alpha\Big\}$$ Where $\mathbb R^2_{++}=[0,+\infty)\times [0,+\infty)$.
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The convex conjugate of a quadratic form with positive semi definite matrix

I want to find the convex conjugate (Legendre transform) of a quadratic for $1/2x^{t}Qx$ when $Q$ is positive semi-definite. If Q is non-singular, the solution is easy - the gradient is $y-Qx=0$ so ...
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Dual of a rational convex polyhedral cone

A rational convex polyehedral cone $\sigma\subseteq\mathbb R^n$ is a set of the form $$ \sigma=\operatorname{Cone}(u_1,\dots,u_k) := \left\{ \sum_{i=1}^k r_i u_i \,\Bigg|\, r_i\ge ...
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showing $y\to |y|^{p}$ is convex $p\geq 1$

$y\to |y|^{p}$ is convex only for $p\geq 1$ and $y\in \mathbb{R}$. This function is nondifferentiable but we can see that the second derivative is nonnegative in each interval $(-\infty,0)$ and ...
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Convexity and Affineness

In reading about convex optimization, the author states that all convex sets are affine. Are affinity and convexity equivalent? If I understand, both definitions incorporate the notion that a set is ...
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Space where the separation theorem doesn't hold

I have read the proof of the next separation theorem: Let X b a normed space in R and A a convex and open set that contains 0. Let b be a point wich is not in A then there exists f in X* such that ...
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strictly convex function when plotted but second derivative not unambiguously positve

I have a function $$ z(x) = (Kx)^{x/(1-x)}, x \in (0,1)\text{ and }K>1 $$ When I plot the function it has a U-shape. However when I take the second derivative wrt $x$ I have the following ...
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Convex Function to Given (Three) Data Points

Assume that a function $h(x)$ is decreasing and convex given interval $[l,u]$. I'd like to get a function which connects three points, say $(a,h(a)), (b,h(b)), (c,h(c))$, where $l\leq a<b<c\leq ...
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Find the Polar of a set.

Let $A \subset \mathbb{R}^{n}$ be non-empty. The set $$ A^{\circ} = \lbrace c \in \mathbb{R}^{n} |\; \langle c,x \rangle \leq 1 \quad \forall x \in A \rbrace $$ is called the polar of $A$. I'm ...
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Linear combination of convex set is convex

A set $U$ is a convex set if whenever $\mathbf{x},\mathbf{y}$ are points in $U$, the line segment joining $\mathbf{x}$ to $\mathbf{y}$,$$\mathcal l(\mathbf{x},\mathbf{y})=\left\{t\mathbf ...
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infimum of a convex function over an open domain

Let $f: \cal D_0 \to [0, \infty]$ be a convex function on a compact set $\cal{D}_0$ and let $\cal D \subseteq \cal D_0$. I think the following holds: $$ \inf_{x \in \cal{D}}\ f(x) = \inf_{x \in ...
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Dual of a convex function $f:\mathbb{R} \to \mathbb{R}$: existence, solution to ODE

Let $f(x)$ be a smooth strictly convex (i.e. $f''(x)>0 \,\,\,\,\,\forall x)$ funtion of $x\in \mathbb{R}$. Define the dual function $F(p)$ as $$F(p)=\max_x [px-f(x)].$$ Make a sketch ...