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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Intuitive explanation of why a chord of a convex function has to be a straight line

I was trying to understand the definition of convexity better. A simple definition of convexity is: $$f(tx_1 + (1 -t)x_2) \leq tf(x_1) + (1-t)f(x_2)$$ $\forall x_1,x_2 \in Domain(f)$ Intuitively, ...
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68 views

Strong convexity on sets?

Consider the definition of convex functions: $$ f(tx+(1-t)y) \le t f(x)+(1-t)f(y) $$ It is easy to show the definition of the convexity on sets with respect to the above definition (Specifically for ...
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258 views

Strictly convex if and only if derivative strictly increasing?

Suppose $f$ is a real-valued function that is differentiable on an open interval $I$. It is well-known that $f^{\prime}$ is increasing on $I$ if and only if $f$ is convex on $I$. Is the following ...
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489 views

Affine functions as equality constraints in convex optimization problems

I am studying on an introduction to convex optimization problems. When defining a convex optimization problem, we have a convex object function, $f(x)$, a set of convex functions $g_i(x)$ where the ...
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Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
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212 views

Convexity and concavity of discontinuous functions

QUESTION F(x) =-x for x>=0 and F(x)=x for x<=0 Is the function convex/(strictly), concave/(strictly) I have attempted the answer but got strictly concave but isnt a discontinuous function meant ...
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65 views

Interior of sum of sets equals sum of interior of summands

I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold: ...
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336 views

Existence of a unique maximizer of a strict quasi-concave function defined over a convex set

Set $S \subset \mathbb R^2$ is compact and convex. A typical element of $S$ is $s=(s_1,s_2) \in S$. Also, $d \in \mathbb R^2$ is a fixed element such that there exists $s \in S$ such that $s \gt d$. ...
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109 views

Elementary proof of an inequality with $e^x$ when $|x|<1$.

Assume $|x| <1 $ and we already know $0 \leq e^x - 1 - x$. Note that last inequality was god given, we know it is true, but we do not know how it was proved. Can we deduce from here that $e^x - 1 - ...
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19 views

If $s_1 +s_2 \gt 1$ and $(t_1,t_2)$ be a convex combination of this with $(0.5,0.5)$ then show that $t_1t_2 \gt 0.25$

Let $(s_1,s_2)$ be such that $s_1 + s_2 \gt 1$. Let $(t_1,t_2)=((1-\epsilon )(0.5) + \epsilon s_1 , (1- \epsilon)(0.5) +\epsilon s_2)$, where $0< \epsilon \lt1$. I need to show that for $\epsilon ...
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158 views

Epigraph of closed convex hull of a function

$\newcommand{\co}{\overline{\operatorname{co}}}\newcommand{\epi}{\operatorname{epi}}$ Let $X$ be an n.v.s and $f: X \to \mathbb{R} \cup \{+\infty\}$ and define $$\co f(x) \doteq \sup_{\substack {x^* ...
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36 views

Showing convexity, having trouble showing positive definiteness

I am interested in showing the convexity of $$-\log(-f(\pmb{x}))$$ for $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{-}$ and $f$ convex. If we let $\nabla f$ denote the column vector where the $i$th ...
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138 views

Axiomatic Bargaining: Nash's Solution

The following text is from the book: Bargaining and Markets by Osborne and Rubinstein, Academic Press Inc. Page 17 under the chapter The Axiomatic Approach: Nash's Solutions:. Two individuals can ...
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41 views

Confusion about Concavity

Simple question. The function g of a single variable is defined by g(x) = f(ax + b), where f is a concave function of a single variable that is not necessarily differentiable, and a and b are ...
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For what $X$ do we have $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
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46 views

Prove concavity without testing the second derivative..

Consider a function $F(L)=(L-L^2a)T^{L-1}$, where $0<L<\frac{1}{a}$. The constants $a$ and $T$ may take values over $]0,1[$ and $[0.01,0.1]$, respectively. The first derivative of $F$: ...
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58 views

About hyperplanes on the boundary (with no $C^1$ regularity ) of compact convex sets

I am reading a paper and the authors use the following property: "Let $K$ a compact and convex set in $R^n$ with nonempty interior. Let $x_0 \in \partial K$ and suppose that the boundary is not ...
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36 views

Confusion with a proof about the continuity of convex functions

I studying convex analysis and in my book I have the following statement and proof: Lets assume that $f:S\rightarrow \mathbb{R}, \;S\subset \mathbb{R}^n$ is a convex function. Then $f$ is ...
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About supporting hyperplanes of convex sets

Let $K \subset R^n$ a convex set, and $x \in \partial K$ such that that there exists a closed ball $B(x_0,R) \subset K$ of positive radius with $x \in B(x_0,R) $. My intuition tells me that there ...
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How to show that the vertices of a convex hull are given by these specific subsets…

We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$. Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ ...
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77 views

Strictly convex set

When a function is strictly convex it has many desirable properties, most notably that it admits a unique minimum. I was wondering if there is anything desirable about a strictly convex set (meaning ...
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186 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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641 views

Projected Area vs. Surface Area of a 3D Set

(In what follows, I'm making up the nomenclature as I go along, so please pardon anything nonstandard.) Suppose I have a set of points $A \in \mathcal{R}^3$ which is compact, convex, and simply ...
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83 views

Optimisation over matrix entries

I was looking to write the KKT conditions to solve this optimisation problem. $$\min_{\substack{\sum_j x_{ij}\le k_i \\ i=1,2,\ldots N}} a^\top (I-X)^{-1} b $$ Since there are $N^2$ decision ...
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52 views

When is the normal cone to a closed convex set in a locally convex set maximal monotone?

Let $X$ be a locally convex set with the following property: (P) $\forall C\subset X$ closed convex, the normal cone $N_C$ is maximal monotone as a multi-valued operator from $X$ to its topological ...
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A question about convex open set in a topological vector space.

Supose $E$ is a topological vector space(may not be Hausdorff). $U\subset E$ is an open set such that $U+U=2U$. How to show $U$ is convex? I can see if $E$ is $T_1$,then $E$ should be Hausdorff. ...
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135 views

Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
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If -log(f) is convex, is f automatically convex?

Say I want to know if $f(x)$ is convex. Can I apply any convex function, strictly increasing function to it and preserve convexity? Say $f(x),g(x)$ are convex and strictly positive and I want to know ...
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104 views

an interior point of a convex set

How can we prove a point is an interior point of a convex set, considering we don't have all of the extreme points of the given convex set ? or How can we find an interior point of a convex set, ...
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1answer
64 views

Hausdorff Distance between Subdifferential sets

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
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51 views

Quantitative aspect of Caratheodory theorem

Let A be a compact convex set in n-dimensional space. [ Of principal interest is n > 2 . ] A result of Caratheodory states that A is equal to the union of its simplices (i.e. simplices with all ...
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84 views

checking for convexity/concavity of a function

i'm having some problem in establishing the convexity/concavity of the following two functions. Check for the concavity/convexity of the following functions: (a) ...
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171 views

Legendre transform concave function

Let $f$ be a concave function and define $f^*(y) := \inf_{x}(yx-f(x))$. Is this in any sense related to the Legendre transformation? -If yes, is $f^*$ also concave? Is this transformation invertible ...
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Characterization of concave function on $\mathbb{R}^N$

I need help with the following problem, I have no idea how to proceed: Let $u \colon \Omega \subseteq \mathbb{R}^N \to \mathbb{R}$ a continuous function, where $\Omega$ is open, connected and ...
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An empty subdifferential

Can you give me an example of function $f$ defined on an Hilbert space, real valued (extended with $+ \infty$), lower semi continuous, convex and proper for which $\operatorname{dom}(\partial f)= ...
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70 views

Help with an inequality involving a convex function

Let $a< f(x) < b $, $x \in \Omega $, $\mu(\Omega )=1 $, and set $t=\int f d \mu $. Then $a < t < b $. Suppose $\phi $ is a convex function on $(a,b) $ then by definition of convexity ...
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tangent cone of a projection

I am stuck with what looks like a basic problem from convex analysis and I don't seem to have the answer. Any pointers or insights would be of great help. Suppose $K$ is a closed convex set in ...
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Prove that $\log I_{\nu}(x)$ is concave

As the title suggests I need to show that the log of the modified bessel function is concave. When I graph it, certainly seems to be the case. So far I have that: $$ y=\log I_{\nu}(x)\\ ...
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Describing the minimizers of this function

Consider the following continuous function over $x$, with $a,b>0$: $$f(x)=\begin{cases} ax-\sqrt{x} & \text{for }x\leq b^{2}\\ ab\sqrt{x}-b & \text{for }x>b^{2} \end{cases}$$Note that ...
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intuitive question about the boundary of a set

Consider $\Omega_1, \Omega_2$ two convex, bounded domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ suppose that $\operatorname{dist}(x, \partial \Omega_2) =\min_{y \in \partial \Omega_2} ...
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49 views

How to know if a function is not “too convex”?

In my math courses, I have never come across the idea of being "too convex", but this is from an economics course. Essentially, you have some function $P(Q)$, where $Q>0$. The model tells us to ...
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Why $\frac{\partial u((1-t)x+ty)}{\partial t} \mid_{t =0} \geq 0$, if $u(x) \geq u(y)$ and $u$ is quasiconcave and differentiable?

Let $u$ be quasiconcave and differentiable at $x$. If $u(x) \geq u(y)$, then how to show that $\frac{\partial u((1-t)x+ty)}{\partial t} \mid_{t =0} \geq 0$? $u$ is quasiconcave means that for all ...
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Conditional expectation of a random vector taking values in convex sets

on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ i have a random vector $X\in L^1_{\mathbb{P}}(\mathbb{R}^d)$ (integrable with values in $\mathbb{R}^n$), such that $\mathcal{P}-a.s.$ $$X\in ...
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Convexity problem

Let $S$ be a convex set. If $x\in$ int$S$ and $y\in$ cl$S$, show that relint[x,y] $\subset$ int$S$. I easily proved this for a case where y is in the interior of S, but am stuck if y is in the ...
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Is the function (sum-of-squares) / sum convex on nonnegative input?

Let $$f \colon \mathbb{R}_{> 0}^n \to \mathbb R$$ be defined by $$f(x_1,\dotsc,x_n) = \begin{cases} 0 &\text{if }x_1 = \dotsb = x_n = 0\text{,}\\ \frac{\sum_i x_i^2}{\sum_i x_i} ...
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196 views

An exercise on convex decreasing function properties

A function f$(x)$ defined for $x\geq0$. It is positive, decreasing, convex and log-convex: $\frac{d^2}{dx^2}\log[f(x)]>0$, $f(0)<1$. Can we prove that $f''(x)x+f'(x)>0$ for sufficiently ...
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$f:D\subset \Bbb R^2 \rightarrow \Bbb R$, where $D$ is a compact and convex set, reaches it maximum at $int(D)$

I'm trying to prove that if $D$ is a compact and convex (for every two elements of $D$, the line that connects them is contained in $D$) then: If $f:D\subset \Bbb R^2 \rightarrow \Bbb R$ and at ...
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Convexity of $S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ^{n}\right\}$

i've to prove that the following set is convex: Let $A \in \mathbb{C}^{n \times n}$ $$S=\left\{ \frac{\underline{x}^HA\underline{x}}{\underline{x}^H\underline{x}} , \underline{x} \in \mathbb C ...
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1answer
159 views

Minkowski sum of two polytopes via the halfspace representation

If i have two polytopes denoted by $P_1, P_2 \subset \mathbb{R}^d$, suppose their halfspace representations are respectively $H_1x \leq K_1$ and $H_2x \leq K_2$. Now, considering their Minkowski sum, ...
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68 views

How to find the convex hull of a given set?

$A=\{(0,0),(0,1),(1,0)\}$ $B=\mathbb{Q}^2$ $C=\{(x,\sqrt{x})\in \mathbb{R}^2:x\ge0\}$ I have to find Conv(A), Conv(B) and Conv(C). My attempt Conv(A) is the boundary (correction: obviously it ...