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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Convex hull of set of sparse vectors?

I am trying to understand how one can define the convex hull of sparse vectors. I understand that for k sparse vectors can be described as a union of subspaces (such as in: ...
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regularity of a bivariate function

Consider a bivariate function $f(x,y)$ which is concave in $y$. Moreover, for any given $y$, let $x^*(y)$ be the solution to $f_x(x,y)=0$, and there is $f_x(x,y)>0$ for $x<x^*(y)$ and ...
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Proximal operator for the nuclear norm of Hankel (x)

I have a problem in hand for which I need to compute the proximal operator of the composite function $||Hankel(x)||_{nuc}$ where $x \in R^N$ and $||.||_{nuc}$ denotes the matrix nuclear norm. For a ...
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How to use convexity in this step?

I am trying to fill in the details of a proof about the following statement: If $f:\mathbb{R}^n\to \mathbb{R}$ be a convex function, if subdifferential of $f$ at $x$ is singleton, then $f$ is ...
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Proving that the second derivative of a convex function is nonnegative

My task is as follows: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of ...
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Continuity of convex function [duplicate]

Let $f$ be a proper convex lower semi-continuous function on $\mathbb R^n$, how can we prove $f$ is continuous in the interior of $Dom(f)=\{f<\infty\}$ ?
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If $f$ is midpoint convex, continuous, and two times differentiable, then for any $a, b \in \mathbb{R}$, there exists $c$ such that $f''(c) \geq 0$

One of my analysis texts states this as an exercise If $f$ is midpoint convex, continuous, and two times differentiable, then for any $a, b \in \mathbb{R}$, there exists $c \in [a, b]$ such that ...
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Non-trivial lower bound approximation of a convex function using the second derivative at the minimum

Say that I am given an infinitely differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$. I am wondering if I can construct a meaningful lower bound approximation of $f$ using it's ...
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49 views

Expectation of an increasing, bounded concave function of a non-negative random variable

Let $h:[0,\infty)\to [0,1)$ be a strictly increasing and strictly concave function. Let the argument of this function be a random variable $C$ with probability density function (pdf) $f_{C}(c)$ with ...
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Notion of positive third derivative for nondifferentiable functions?

Are there any notions that generalize the idea of a positive third derivative of a univariatve function to those for which the function is not necessarily differentiable? For example, a function ...
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Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
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Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
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How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
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51 views

finding discrète coordinate of Intersection of two convex polygon?

I seek for cartésien coordinate of vertex's of the intersection area between two polygons ? We have two convex polygon's P & Q such that : all vertex of P (resp. Q) are in 2D cartésien plane. I ...
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Improvement of an Inequality

It would nice if someone could help me with this problem. I am looking at an improvement to the classical Jensen's Inequality: $$\int_\limits{}^{} \phi(x) \mu \mathrm{d}x \geq ...
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Is the optimum of this problem unique?

I have a convex optimisation problem: $$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$ The KKT conditions are: $$\lambda=\begin{cases} ...
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How to calculate Jensen's Inequality

How would one show that $\mathbb{E}\Big[f(X)\Big] \geq f\Big[\mathbb{E}(X)\Big]$ for the convex function f(x)=x$^{2}$ ?
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(Updated) Geometric Illustration of Monotone and Maximal Monotone Maps

I am writing a note about the Monotone and Maximal Monotone maps from the following book http://link.springer.com/book/10.1007%2Fb97594 In this book we read a map ...
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1answer
26 views

Nonlinear discontinuous convex function

Let $X$ be a normed vector space. Construct nonlinear discontinuous convex function $f:X \rightarrow \mathbb{R}$. I tried something with $\frac{-1}{\|x\|}$ but had no success.
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25 views

Is this function jointly convex in its variables?

I have a function which I suspect is jointly convex, but have a difficult time proving it, especially since the Hessian is messy. The function is $f(y_i,i=1.2,\ldots,N)=\sum_i l_i w_i + y_i$, where ...
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Is there a unique saddle value for a convex/concave optimization?

Here is the question: Consider a function of two variables $f(x,y)$ on some compact domain. Let it be convex in $x$ and concave in $y$. Is it true that $f(x,y)$ has a uniqe saddle point ...
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Convexity definition when $\lambda \in \mathbb{R} \setminus (0,1)$

We are given the standard definition $$f(\lambda x + (1 - \lambda) y) \leq \lambda f(x) + (1-\lambda) f(y)$$ for $\lambda \in (0,1)$. I am trying to prove that the opposite inequality is true when ...
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Triangle containing most points from a set

Given a point set in $\mathbb{R}^2$, I need to find a triangle connecting three points of the set that contains the most points of the set. Points that lie on the connecting lines don't count. The ...
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Is the optimal solution of a strictly convex function over $\mathbb{Z}^d$ a rounded version of its optimal solution over $\mathbb{R}^d$

Consider a strictly convex function $f: \mathbb{R}^d \rightarrow \mathbb{R}$. Let $x^* = \min_{\mathbb{R}^d} f(x)$ denote the (unique) minimum of this function over $\mathbb{R}^d$. Similarly, let ...
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Show that $f(\mathbf x)=e^{Q(\mathbf x)}$ is a convex function.

Here's the problem: Let $A$ be a positive definite symmetric matrix and let $Q(\mathbf x)$ denote the associated quadratic form on $\mathbb R^n$. Show that $f(\mathbf x)=e^{Q(\mathbf x)}$ is a convex ...
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38 views

How do I integrate the inequality $ \frac{f(\frac{1}{2}+h)+f(\frac{1}{2}-h)}{2} \leqslant f(\frac{1}{2})$ over the range $h\in[0,1/2]$?

I would like to know the formal steps and theory. I was told that, by integrating this inequality, I can achieve one of the definitions of a concave function in the interval [0,1]. Thanks for your ...
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Mean value for a concave function over $[0,1]$ VS $f(1/2)$

I am looking for a concave function $f(x)$ for which the integral over $[0,1]$ is bigger than $f(1/2)$. That is, a function which mean value between 0 and 1 is bigger than the middle value of the ...
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56 views

How do I prove function convexity? [duplicate]

I have the following task: Prove that if $ f : I \rightarrow \mathbb{R} $ is continuous ($ I $ is a range) and $$ \forall {x,y \in I} \qquad f\left(\frac{x+y}{2}\right) \leq \frac{f(x) + f(y)}{2} $$ ...
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Determine if$ f(x) = -|x + 2| \,\,\,\forall x ∈ [-2, 0]$ is convex

Having trouble with this homework question, Determine if $f(x) = -|x + 2| \forall x ∈ [-2, 0]$ is convex using the below definition of convexity. A function $f: X -\to\mathbb{R}^n$ is convex for ...
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Existence of Hessian of convex conjugate

Define convex conjugate of $f, f^*(x):=\sup_{y\in\mathbb{R}^n}\langle x,y\rangle-f(y)$. Then I want to prove this statement: Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Assume $f$ is ...
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Question about Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies - Kannan Lovasz Simonovits 97

I've been trying to understand this paper: "Random Walks and An $O^*(n^5)$ Volume Algorithm for Convex Bodies", Ravi Kannan, Laszlo Lovasz, Miklos Simonovits. Motivation: The paper is about ...
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Convex analysis of $h(x) = \log (f(x)) \ $ for $f \in C^2(\mathbb{R, \mathbb{R}_+})$

Problem: Let $f: \mathbb{R} \to \mathbb{R}_+$ be of Class $C^2$ such that $$g(x)= f(x)e^{cx} \text{ is convex } \forall c \in \mathbb{R} $$ Verify that $h(x)= \log(f(x))$ is convex My approach: ...
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Quasiconvex functions

Let $\lambda$ be any real number and $f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ given by $$ f(x,y) = \begin{cases} \lambda &\mbox{if } \quad0<x<1, y=1, \\ 1+\lambda y & \mbox{if } ...
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An inequality regarding convex functions

For a convex function $f(\cdot)$ for $x_4 \ge x_3 \ge x_2 \ge x_1 \ge 0$ and $t,t^{'}\ge 0$ we are given that \begin{equation} \frac{f(x_4)-f(x_3)}{f(x_2)-f(x_1)} \ge ...
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Proof regarding convex sets

A set of points is said to be convex provided that every pair of points in the set can be joined by a line segment that lies entirely within the set. Show that, if $ | ∇f(x)| ≤ M \space \space ...
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Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
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Two fundamental questions about convexity of a function (number2) [duplicate]

The second question is as follows (the first one is here): Consider a function of two variables $f(x,y)$ on some compact domain. Let it be convex in $x$ and concave in $y$. Is it true that ...
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57 views

Two fundamental questions about convexity of a function (number1)

The first question is as follows (see the second one): If $f$ is a convex function it is know that there is at most a single minimum. However the argument of the minimum is not guaranteed to be ...
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Convex downward function and its inverse function

How to prove that if function $f$ is convex downward and invertible then $f^{-1}$ is convex downward or convex upward? When is it downward and when upward?
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Polynomial and convex functions

Consider polynomials $\mathbb{R} \rightarrow \mathbb{R}$. I have to Give an example of polynomial that isn't convex downward nor convex upward. Give an example of polynomial that is convex downward ...
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Lipschitz constant of the convex function $f(x) - \frac{a}{2} |x|^2$

I was going through this blog post https://blogs.princeton.edu/imabandit/2013/04/04/orf523-strong-convexity/ It has been mentioned without proof that for a function ...
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Function going from $0$ to $1$ with minimal concavity

How small can I take $C>0$ such that there exists an $f\in C^2(\mathbb R;\mathbb R)$ satisfying the following properties: $f(x)=0$ for $x\leq 0$ and $f(x)=1$ for $x\geq 1$ $f'(x)\geq 0$ for ...
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Does every boundary segment of a convex polyhedron lies on one of its faces?

When I was reading this note, I found Theorem 3.1.5 said: Let $P\in\mathbb{R}^n$ be a polytope whose affine dimension is $d$. Then, every point on the boundary of $P$ lies in a facet of $P$. I have ...
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29 views

condition for uniform input to minimize convex function

Let $f:\mathbb{P}(X) \rightarrow \mathbb{R}$ be a convex function where $\mathbb{P}(X)$ is the set of probability distribution on (finite) set $X$. I am looking for condition on $f$ so I can said ...
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39 views

Convex hull of convex set boundary

A is a closed convex set with non-empty interior. Does A must equal to the convex hull of its boundary? I know this is false when A is half space. But what about other sets?
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convergence of infimal convolution

I am trying to show the following statement: Let $f:\mathbb{R}^n\to \mathbb{R}\cup\{\infty\}$ be a convex function. Let $f_\epsilon(x)=\frac{|x|^2}{2\epsilon}$. Show that $\lim_{\epsilon\to ...
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How can I solve $\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$ in a closed form depending on projection?

I'm trying to solve the minimization problem $$\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$$ where $b \in \mathbb{R}^n$, $X \subset \mathbb{R}^n$ is a convex set. And $A$ is a symmetric ...
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Can I assume $g$ is finite for proof involving infimal convolution

I am trying to show the following statement: Let $f,g:\mathbb{R}^n\to \mathbb{R}\cup \{\infty\}$ be two convex functions. Assume that there are constants $C_1,C_2>0,\alpha>1$ such that ...
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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 ...