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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(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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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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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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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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1answer
87 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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1answer
142 views

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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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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33 views

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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1answer
37 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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152 views

Nonnegative solution of a linear system

Given three collections of parameters $\epsilon_1 > ... > \epsilon_N$, $(a_1,...,a_{N-1})$ and $(b_1,...,b_N)$ that satisfy the following conditions $\forall i, a_i \geq 0, ...
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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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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 ...
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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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1answer
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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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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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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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163 views

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 ...