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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Complex analysis question regarding Cauchy's integral formula and holomorphic functions

Let $U \subseteq \Bbb{C} $ be an open convex subset of $\Bbb{C}$ We also assume that $\partial U$ is smooth. We fix a point $z_0 \in U.$ Using Cauchy's integral formula, show that $$\lvert ...
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Convexity of Log Determinant of Function

Given a function $g(x): \mathbb{R} \to \mathbb{R}^{N x N}$, under what circumstances is $f(x) = - \log \det g(x)$ a convex function? Assume that each of the $N^2$ entries in $g(x)$ is convex over ...
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Area of the lattice generated from $(n, n\sqrt{2} \mod 1)$

I plotted $\Big\{ (n, n \sqrt{2} \, \mathrm{mod} \,1) \;\Big| -50 \leq n \leq 50 \Big\}$ and even though the $n \sqrt{2}$ is a line, the pattern that emerges is a lattice. What is the basis of this ...
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Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, ...
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Convex hull of $3$ dimensional set reduced to $2$ dimensional set

Let $S = \{(f_1(t), f_2(t), f_3(t)) : t \in \mathbb{R}\}$ and suppose $f_3(t) \geq 1$ for all $t \in \mathbb{R}$. Is finding the convex hull of $S$ in some way equivalent to find the convex hull of $T ...
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Prove that function is convex

Let $f\colon [a,b] \rightarrow \mathbb R$ be continuous and convex. Let $m \colon [a,b] \rightarrow \mathbb R$ and $m(x) = \max \left\{f(y): y \in [a,x] \right\}$. Prove that $m$ is convex I ...
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Prove that all terms of a sequence of functions are convex.

Let $\ f_{n}: [0,1] \rightarrow \mathbb R, \quad f_{n}(x) = \left(e^{x}\right)^{1/n}.$ Is there a natural $n$ such that $f_{n}$ is concave on $[0,1]$? So second derivative is ...
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Dual Cone Construction $\{z \; | \;z \perp v \text{ for some } v \in \Lambda \}$

In a linear algebra computation, in order to estimate the second eigenvalue we consider a collection of vectors. Let $\Lambda$ be a cone in $\mathbb{R}^d$ then $$ \Lambda' = \Big\{z \;\Big| \;z ...
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Describing set of points where a convex function is differentiable

I've been told that the set of points at which a convex function $f: \mathbb R^n\rightarrow \mathbb R$ is differentiable is an $F_{\sigma}$ set, and I was hoping someone could help me see this. ...
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Condition for product of increasing and decreasing functions to be quasiconcave?

Is there any condition for product of increasing and decreasing functions to be quasiconcave? More specifically, I am having in mind a condition for $F(x)\cdot(1-G(x))$ to be quasi concave where $F$ ...
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Line segment in the unit sphere

I want to prove the following statement Let $X$ be a normed linear space, with linearly independent vectors $x,y$, such that $\|x\|=1=\|y\|$, with $\|x\|+\|y\|=\|x+y\|$, then there is a line ...
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Application of convex functions in economy [closed]

I have read in some texts that convex functions has application in economy. I want to see some clear examples of such applications.
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Proof of Separation Thm, intermediate step

This is an intermediate step required to prove the Separation Thm (https://en.wikipedia.org/wiki/Hyperplane_separation_theorem) Let $S,T\subseteq \mathbb{R}^n$ be nonempty, convex. Say ...
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Is the following function strongly convex?

I've tried to check the Hessian and see whether the following function is strongly convex w.r.t the euclidian norm however this function is not diff. at the origin. $g(x)=||x||_2+||x||_2^2$ Thanks
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If every convex function is of bounded variation?

The properties of convex functions are of interest. I would like to know that if every convex function is of bounded variation?
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For polytopes, does union and linear transformation commute?

Given two (convex) polytopes $P_1$ and $P_2$ and a linear transformation $T$, is it true that: $$T(P_1 \cup P_2) = TP_1 \cup TP_2$$ What if $P_1$ and $P_2$ are not convex?
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Convexity of Determinant of linear combination

Is it possible to show that the following is a convex function in $x$? $f(x)=\det(\sum_i x_i A_i)$ $A_i$ are real symmetric, positive definite matrices. Minkowski's inequality doesn't seem to do ...
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Upper bound using a convex function

Let $g, f: K\times S \to \Bbb R$ be convex and continuous functions on compact and convex sets $K,S \subset \Bbb R^n$. Does there exist a differentiable strongly convex function $h$ on $S$ (w.r.t. ...
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Is this function strongly convex?

Let A,B be two intervals in $R$ and let $f(x,y):A\times B\rightarrow R $ be a continues function. Assume that $f$ is convex in both $x$ and $y$ Is the following function $g:A \rightarrow R$ is ...
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Optimizing over intersection of polytopes inside polytope

I have the following situation, as shown in the following diagram: $W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) located within a regular simplex and having coordinates $\in ...
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If a function is strongly convex on a set $X$, then is $f + \delta_X$ strongly convex on the closure of $X$?

If $f:\Omega \mapsto \mathbb{R}$ is strongly convex on $X \subset \Omega$, then can one say that $f + \delta_\bar{X}$ is strongly convex on $\bar{X}$? (Here $\bar{X}$ is the closure of $X$ and ...
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Test for convexity

Consider the online learning setting where instantaneous loss is given by \begin{equation} \ell_t(f_t;(\mathbf{x}_t,y_t))=\max \left( {0,\left( \left( {\frac{N}{P}} \right){I_{(y_t = 1)}} + \left( ...
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A question related to convex and compact set

I encounter a problem related to convex and compact set, which is stated as follows. Whether or not the following claim is correct? Claim: Let $C$ be an arbitrary subset of $R^n$ such that $C$ is ...
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convex bounded, closed $\Longrightarrow$ compact?

I know that A subset S of $\mathbb{R}^n$ is compact if it is bounded and closed (Heire-Borel theorem), Howver, if S is convex, containing the origin, closed, is it compact? Thanks in advance!
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Notion of convexity on nonconvex domain?

Let $X\subset \mathbb{R}$ be potentially a nonconvex set, and let $f:X\rightarrow \mathbb{R}$ satisfy $\frac{f(x)-f(x')}{x-x'}$ increasing in $x$ and $x'$ for all $x,x'\in X$. Is there a name for this ...
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A metric space and polytopes

suppose we define a metric space $L_2(p)$ induced by a duality pair given by $\langle x,y\rangle =\sum_{j,k} p(j,k)[ x_1^j y_1^j+ x_2^k y_2^k]$ Where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ and $p$ is a ...
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Finding an unfrustrated set of local linear constraints with given minimal value

Let $ \{F_{i}\} $ be a finite set of linear functional on a convex closed subset $B\subset\mathbb{R}^{n}$ such that each $F_i$ is $k-local$ (acts on $k<<n$ variables only). Assume ...
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Prove that the set of extreme points of a compact convex set is not empty.

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...
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Where can I find the nowhere subdifferentiable example of rockafellar?

I'm told that Rockafellar gave an example of a real extended function defined on a locally convex space, whose subdifferential is empty at each point of its domain. The function is proper, lower ...
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Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Fix some positive integers $L$ and $k \leq L$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= ...
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Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
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One definition of strong convexity (from textbook of Prof. Bertsekas in 2015)

In strong convexity, there are a few definitions, one of them is: $f$ is strongly convex over $\mathcal{C}$ with coefficient $\sigma$ if $\forall x,y \in \mathcal{C}$ and all $\alpha \in [0,1]$, we ...
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Write a random variable as a convex combination of other 2

I'm trying to prove that if $f:[0,1]\to\mathbb{R} $ is continuous and convex, then the Bernstein polynomials are too. The hint that I've got is this: "Let $p_1 < p_2 < p_3<1$ and consider ...
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Is there a name for this property among variables?

I have a convex function of four variables, $f(w,x,y,z)$, which when solving for the symbolic arg min of one variable assuming the other three are known I got something similar to the following. ...
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Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball?

Is the closure of every bounded convex set , with non-empty interior , in $\mathbb R^n (n>1)$ homeomorphic to a closed ball (by closed ball I mean $B[a,r]:=\{x \in \mathbb R^n : d(x,a)\le r\}$ , ...
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Convex function and second devirative

I would like to ask a question about the condition of a convex function. We know that a function $f(x)$ is convex if and only if $f''(x) \geq 0$. But what if a function has more than one variable? ...
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Convexity under diffeomorphisms

Let $K \subset \mathbb{R}^n$ be a compact convex subset with non-empty interior, and $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a diffeomorphism. Then is it true that $f[K]$ is convex?
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Is the function $f(x) = |x|$ convex?

I am asking because some Internet pages contradict. In Wikipedia, the definition I found of a convex function is: "Let $X$ be a convex set in a real vector space and let $f: X \to R$ be a function. ...
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Minimum of biconjugate of a nonconvex $f(x)$ is the minimum of $f(x)$ also?

In Fazel (2002) Matrix rank minimization with applications, Ch. 5.1.4-5.1.5, the author finds an analytic expression for the convex biconjugate of their nonconvex function; however, they state that ...
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The maximum of several affine functions is a polyhedral function

A function $f: \mathbb{R}^n \mapsto (-\infty,\infty]$ is polyhedral if its epigraph is a polyhedral, i.e. $$\text{epi}f=\{(x,t)\in \mathbb{R}^{n+1} | \ \ C\left( \begin{matrix} x\\ t \end{matrix} ...
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Relative Interior of a Convex Hull

Given pts $y_0,...,y_k \in \mathbb{R}^n$, their convex hull is Co($y_0,...,y_k$):={$\sum_{i=0}^k a_i y_i$ : each $a_i \geq 0$, $\sum_{i=0}^k a_i =1$}. Their affine hull is ...
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Are the stationary points of a strongly convex function unique in each dimension?

Consider a strongly convex function $~f: \mathbb{R}^n \rightarrow \mathbb{R^+}~$ with a unique minimum at the point $x^* \in \mathbb{R}^n$. I am wondering: if I have another point $y \in ...
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How to solve the convex combination problem of matrix?

Let $A \succeq B$ denote matrix $A-B$ is positive semidefinite, and here is the definition of redundant(all the matrix dimensions are $N\times N$ ): Given a set of matrix $\{B_i\}_{i=1}^{l}$, if ...
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Intersecting Simplices with Normballs

Let $e$ be the vector of all ones 's consider the standard simplex $$\Delta_m:=\{x\in\mathbb{R}^m_+: \langle x, e\rangle=1\}.$$ Then the truncated simplex $\Delta_m^d$ is given as ...
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Are convex functions enough to determine a measure?

Suppose we are talking about $\mathbb{R}^n$. We know that if $\mu$, $\nu$ are two finite Borel measures such that $$\int_{\mathbb{R}^n}f(x) \, d\mu(x)=\int_{\mathbb{R}^n}f(x) \, d\nu(x),$$ for all ...
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$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
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Is the projection function convex?

Define the following function $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ to be the projection function onto a convex and a closed set C $f(x)=\arg\min_{y\in C} ||x-y||_2^2 $ Denote $f_i(x)$ ...
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Is this function composition convex?

Say we have two functions $f:R^n\rightarrow R$ , $g:R^m\rightarrow R^n$. Given that $f$ is convex, under what conditions on $f$ and $g$ we will be able to say that the composition function ...
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Interior of polar cone and self-concordant function

Suppose $f$ is a self-concordant(see 9.6.2) barrier of a proper cone $K$ ( solid,convex, closed and pointed) in $\mathbb{R}^n$. It looks like the value of all $\nabla f$ is just the interior of the ...
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Convex Optimization Closed Form Solution

I'm currently studying for my exame in convex optimization. It covers problem formulation, first and second order conditions of optimality, unconstrained optimization, constrained optimization with ...