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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Convexity in each argument and directional derivative

Let $f(x,y)$ be a continuous function, convex in each argument separately. Does this imply the existence of one-sided directional derivatives in any direction? For example, does there exist (finite or ...
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Separating convex sets in a tvs $X$.

I got doubt with the proof of this theorem. Let $X$ be a tvs, $A,B \subset X$ with $A$ an open convex set and $B$ convex such that $A \cap B = \emptyset$. Then there exists $f \in X^*$ (where ...
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Union of all sets of optimal solutions to a perturbed linear programming problem

Please let me know if you have some ideas on how to approach this proof? I got stuck part-way through. The following linear program is a function of $\theta$, $ \begin{array}{ll} \min & c^\top x ...
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Proving function defined by algorithm is convex

I'm working on my bachelor thesis and I'm trying to prove a conjecture, but I seem to miss the hint that helps me. I have an algorithm that defines a function $f:\mathbb{R}_{\geq ...
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62 views

Prove that function is convex

How can it be proved that the function $f(x) = \ln \bigl(\sum\limits_{i=1}^{n} e^{x_{i}}\bigr) $ is convex?
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To prove the existing and uniqueness of a solution

Let function $f$ be differentiable and convex in $R^{n}$. How can it be proved that $\forall \lambda > 0$ solution of system equations $f'(x) = -\lambda x$ exists exclusively ($\exists \hspace{3mm} ...
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+50

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ...
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1answer
25 views

Concavity of a function

While I am reading a book I couldn't follow the following step. " By concavity of the function $x \sqrt{\log\frac{1}{x}}$ for $x \in (0,1)$ we have that " $O(x \sqrt{\log\frac{k}{x}})$ = ...
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37 views

Sufficient conditions for convexity using the right derivative

We have a function $f:[0,1] \rightarrow \mathbb{R}$ that is continuous on $[0,1]$ with a non-decreasing right derivative everywhere in $(0,1)$. Is this definitely sufficient to show that $f$ is convex ...
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1answer
83 views

A a.e. strongly convex function

Suppose that $f=f(x)$ is strongly convex a.e. for $x\in\mathbb{R}$, i.e. there exists $\epsilon>0$ such that $f''(x)\geq\epsilon>0$ a.e. for $x\in\mathbb{R}$. Then there exists ...
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1answer
95 views

Proof that a set is convex

$$k\in \mathbb{R}_{+}$$ $$M=\left \{ (x_1,x_2)\in \mathbb{R}_{++}^2 \mid x_1 x_2\geq k\right \}$$ Prove that the set $M$ is convex. A hint is given (quoted from the text): We could choose to ...
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Begin study of convex algebraic sets in complex projective space

Where should I begin the study of convexity of (semi-)algebraic sets? In other words, projective varieties defined by polynomials of complex variables. The long-term goal is to study optimization in ...
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24 views

Dual convex pairs

I am currently trying to understand a certain proof. The author uses the term dual convex pair for a pair $(\phi,\psi)$ of convex functions defined on subsets $X,Y$ of $\mathbb R^n$ satisfying: $$ ...
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1answer
46 views

A question about norm for bounded linear transformations

Let $H$, $K$ be Banach spaces, and let $A: H \rightarrow K$ be a bounded linear transformation. Its norm is defined by: \begin{equation} \|A\| = sup\{\|Ah\|_K: \|h\|_H \le 1\} \end{equation} How to ...
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Convex subsets, Normed spaces, Separating hyperplane

I'm trying to understand the proof of a theorem taken from a textbook. Theorem: If $S_1,S_2$ are disjoint non-empty convex subsets of a real vector space $X$ (which may be infinite dimensional) then ...
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1answer
24 views

Deriving projection operator for an affine set

Given an affine set $Ax=b$, the Projection operator to this set is $$P(z) = z - A^{T}(AA^{T})^{-1}(Az-b)$$ which is also affine. How is this derived?
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Robust LP question using box uncertainty model

I am trying to solve this robust LP problem by writing it as a QP $$\min_x x^TSx : \mu \leq r^T x , Ax \leq b$$ Under Box uncertainty model: $$R = \{r : \| r - \hat{r}\|_\infty \leq \rho\}$$ Here ...
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1answer
38 views

What is the shape of all convex combinations of $\geq$ five vectors in $\mathbb{R}^3$?

The convex combinations of two linearly independent vectors in $\mathbb{R}^3$ span a line. The convex combinations of three linearly independent vectors in $\mathbb{R}^3$ span a solid triangle. The ...
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1answer
34 views

Proving upperbound using convexity

The original question is to prove $$\frac{1}{n}\sum_{i=1}^n x_i \leq \log{(\frac{1}{n}\sum_{i=1}^n e^{x_i})} \leq \max_{1 \leq i \leq} x_i$$ I show that $$x_{max} = \max_{1 \leq i \leq} x_i$$ ...
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62 views

Is it always possible to partition a fat shape to fat shapes?

The slimness factor of a geometric 2-dimensional shape is defined (for this question) as the ratio of the side-length of its smallest enclosing square to the side-length of its largest enclosed ...
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26 views

Prove that a given function is convex

Consider the following convex set: $$S = \{m \in \mathbb{R}^N : m_i \geq 0 \text{ }\forall i=1, \ldots, N \wedge \sum_{i=1}^Nm_i = 1\}$$ and following function $f : S \rightarrow \mathbb{R}$: ...
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Gaussian measure of a convex hull with respect to scaling.

consider the following problem: Let $m\geq 1$ and consider $a_1,\ldots,a_{m+1}\in \mathbb{R}^m$. Let $A=(a_1,\ldots,a_{m+1})$ be the matrix having the $a_i$ as columns. Let ...
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property of Convex body on the plane

Let $K$ be a convex body on the plane with smooth curve. Observe the triangle $\triangle ABC$ that contains $K$ with minimal perimiter and let $X,Y,Z$ the points on $BC,AC,AB$ that belong to $K$. I ...
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1answer
23 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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33 views

Simplex - Help with a proof

I am trying to prove the following (basic) claim about simplex. If you check my proof and help me with the part where I stuck, I would appreciate it very much. Let $X$ be a finite set and define ...
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1answer
46 views

Difference of concave functions

Suppose that there are two concave functions $f_1(x)$ and $f_2(x)$ defined on $x\geq0$. In addition, the functions are positive, smooth, bounded ($|f_2|\leq b_2,|f_1|\leq b_1$ such that $b_2 = ...
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86 views

What functions are support functions of convex sets

Given a closed, convex, non-empty set $K\subseteq\mathbb{R}^n$ the support function $h_K:\mathbb{R}^n\to (-\infty,\infty]$ is defined as $$h_K(y) := \sup_{x\in K} \langle x,y\rangle$$ It is easy to ...
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Is the volume of a convex polytope efficiently computable from the vertices

Is there an efficient method to compute the exact volume of a bounded full-dimensional convex polytope, given the coordinates of its vertices (V-representation)?
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1answer
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Convexity and Jensen's Inequality for simple functions

Suppose $\varphi$ is convex on $(a,b)$. I want to show that for any $n$ points $x_1,\dots,x_n \in (a,b)$ and nonnegative numbers $\theta_1,\dots,\theta_n$ such that $\sum_{k=1}^n \theta_k = 1$ we are ...
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Composition of non-monotonic convex function

Given the following composition of functions: $h:\Bbb R^k\rightarrow\Bbb R$ $g:\Bbb R^n\rightarrow\Bbb R$ $f(x)=h(g_1(x),g_2(x),...,g_k(x))$ There are known rules which guarantee ...
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255 views

How to prove that the following function is convex?

I want to prove convexity of the following function: $$f(x) = log_x \left(1 + \frac{(x^a-1)(x^b - 1)}{x-1}\right)$$ for any fixed $a, b \in (0, 1)$ and: $x\in(0,1)$ $x\in(1, \infty)$ I'm trying ...
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1answer
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Decreasing Function Projected onto Simplex

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, defined as $f(x) := a x + b$, where $a<0$ and $b \in \mathbb{R}_{\leq 0}^{n}$. Note that $f$ is decreasing: $$ x \geq y \Longrightarrow f(x) ...
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Minimization of log-sum-exponential function subject to constraints.

I would like to minimize the following function: $f(x)=log(e^{-x_1}+..+e^{-x_n})$ Subject to: $\sum_{i=1}^{n}{x_i}=1$ $0 \leq x_i \leq 1$ So far I have discovered the following: If all the ...
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an elementary inequality about convex function

Given $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is convex, then we have $f(y)\geq f(z)+Df(z)\cdot(y-z)$ where we fix a point $z\in B(x,r/2)$ Integrate the above inequality directly with respect to $y$, ...
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Showing a function is concave

Given $F(\underline{x}) = Ax_1 + Bx_2 + \ln(a^2-(x_1^2+x^2_2))$ on $S=\{\underline{x}\in\mathbb{R}\mid x_1^2+x_2^2<a^2\}$ with $A,B,a\in\mathbb{R}$, show that $F$ is concave on $S$. Since we have ...
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1answer
31 views

How to prove that $f$ is convex function

Let $f:(a,b) \mapsto \mathbb{R}$ be continuous function such that $f(\frac{x+y}{2})\leq \frac{1}{2}f(x) + \frac{1}{2}f(y) \;\; \forall x,y \in (a,b)$ Show that $f$ is convex function. Please give ...
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Notion of a concave function and proving ln is concave

I've just checked that the definition is right, a function is convex if: $(1-t)f(x_1)+tf(x_2)\ge f((1-t)x_1+tx_2)$ which is odd because this is ... well I was taught (very young age) that concave ...
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Minimization of product function subject to constraints

I want to minimize the following function: $\prod_{i=1}^{n}{x_i}$ Subject to the following constraints: $\sum_{i=1}^{n}{x_i}=1.1+(n-1)(0.1)$ and $0.1 \leq x_i \leq 1.1$ How should I go about it? ...
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Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
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Convex set of polynomial coefficients

Assume we have an infinite order polynomial $f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...$. and we know all roots of this polynomial cite outside the unite circle. It is obvious that latter condition ...
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What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
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Extreme points by intersection of extreme rays and hyperplane

I just met one question and have no idea about the proof, hope someone can give me some ideas on how to attack this question. Given a graph $G=(V,E)$ with $|E|=n$. Define a set $S$: ...
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$\mathrm{Prox}_{f}(x)$ and $\mathrm{Prox}_{af}(x)$

Let $a\in \mathbb{R}$, and $f$ is a convex function $f: \mathbb{R}^n\rightarrow \mathbb{R}$. $\mathrm{Prox}_{f}(x)=y_1$ and $\mathrm{Prox}_{af}(x')=y_2$. Because I know $\mathrm{Prox}_{f}(x)$. And ...
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Effect of proximal projection using a divergence measure, on the maximizer of the function

Suppose we have a probability distribution $p(\mathbf{x})$ and we know : $$ \mathbf{x}^* = \arg\max_{\mathbf{x}} p(\mathbf{x}) $$ Suppose we do a projection of this distribution onto another family ...
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Dual of dual cone of nonconvex closed cone

let $K$ be a nonconvex closed cone, then $K^{**}=conv(K)$ should this hold? I am not quite sure about it. Thanks.
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convex closed and unclosed functions and (lower semi)continuity

I'm grappling a bit with lower semicontinuity and convex functions. Let me consider convex functions as functions to $\mathbb{R}$ and not to $\mathbb{R}\cup \{\pm\infty\}$. By Rockafellar's book ...
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1answer
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The conjugate relation between two functions.

Suppose $K\in \mathbb{R}^{m\times n}$, $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$. Let a function $F: \mathbb{R}^m \rightarrow \mathbb{R}$, $F(y)$. Let $y=Kx$, $G(x)=F(Kx)$. Suppose $G(x)=F(Kx)$, ...
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How to show this converges in probability

$A_n(s)$ is a sequence of convex random functions defined on an open set $S\in \mathbb{R}^p$ which converges in probability to some $A(s)$ for each $s$. I'm trying to show that $\sup_{s\in K} \big| ...
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1answer
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Equality constrained Quadratic Program

Consider the QP $$ x^* = \arg \min_{\displaystyle x \in \mathbb{R}^n{\geq 0}} \ \frac{1}{2} x^\top P x + q^\top x \ \text{ sub. to: } A x = b, $$ where $P \succ 0$. Without the non-negativity ...
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1answer
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Are there necessary and sufficient conditions for Krein-Milman type conclusions?

This, the third of three self-answered questions, contains a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The second question is here. ...