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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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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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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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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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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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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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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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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Projection onto a matrix where the diagonals are identity matrix

I'm trying the understand intersection of convex sets given in "Convex Optimization - Boyd" which I'm also trying to code in cvx. The two convex sets I'm trying to find the intersection are given ...
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Efficient way to compute the strong convexity modulus of a function?

I have a strongly convex function $f:X\to\mathbb{R}$, where $X\subseteq \mathbb{R}^n$, with strong convexity parameter $\sigma>0$. By definition $f$ satisfies, for all $x,y\in X$ and $t\in[0,1]$, ...
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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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how to prove convex function for multy variables?pleese ansewr quickly [on hold]

If $f$, $f_y$ ,$f_z$ are continuous on $[a,b]\times R^2$, show $f(x*,y,z)$ is convex on $[a,b]\times R^2$ if and only if $$f(x, \theta y_1+(1- \theta) y_2,\theta z_1+(1- \theta )z_2) \le \theta ...
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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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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
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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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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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28 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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Relative Interiors of polyhedra

***Source article: Magnanti, T. L., & Wong, R. T. (1981). Accelerating Benders decomposition: Algorithmic enhancement and model selection criteria. Operations Research, 29(3), 464-484
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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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32 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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Convexity of a function on R^n

$f(x_1,x_2) = 2x_1^2 - 3x_1x_2 + 5x_2^2 - 2x_1 + 6x_2$ Is the function convex on $\mathbb{R}^n$? How to solve?
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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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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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relative interior and the affine map

In the convex analysis book by Hiriart-Urruty &Lemarechal, Proposition 2.1.12 states $ri [A(C)] = A(ri C)$. Where $ri$ is the relative interior and $A: \mathbb{R}^n \to \mathbb{R}^m $ is an ...
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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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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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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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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 ...
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A short question about the convexity of a function

Let $x$ and $y$ be two numbers; $0\leq x \leq 1$ and $0\leq y \leq 1$ satisfying $$\mathcal{X}\times \mathcal{Y}=\left\{(x,y):\sum^{\lfloor k\rfloor}_{i=0}\binom{n}{i}(1-y)^{i} y^{n-i} +\sum^{\lfloor ...
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is $R^N_{ ++}$ a convex set?

Is $R^N_{ ++}$ a convex set? I'm working on some optimization hw problems that have some functions of the type: $f:\mathbb{R}^2_{++} \rightarrow \mathbb{R}$ And it seems like in general whenever ...
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when does the Minkowski inequality for infinity norm become equality

I have two vectors $x, y \in \mathbb{R}^d$, it is well known as Minkowski inequality that: $|x+y|_\infty \leq |x|_\infty + |y|_\infty$, where $|x|_\infty= \underset{i=1..d}{\max} |x_i|$ with $x = ...
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When is the difference of two convex functions convex?

Assume that $X$ is a finite dimensional Banach space. I know that in general if two functions $f:X \mapsto \mathbb{R}$, $g:X \mapsto \mathbb{R}$ are convex then the function $(f-g):X \mapsto ...
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41 views

Prove that convex function on $[a,b]$ is absolutely continuous

In the book Roberts, Varberg, Convex functions, on pp.9-10 is proved that If $f:(a,b)\rightarrow \mathbb R$ is continuous and convex then $f$ is absolutely continuous on each $[c,d]\subset ...
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Why should the points that define a simplex be affinely independent?

I have a question regarding the definition of a simplex. I am quoting the definition of a simplex from Wikipedia, which is similar to the definition in my textbook too "Specifically, a $k$-simplex is ...
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The definition of convex body and the Hilbert cube

I currently have a question about the definition of convex body. The formal definition is: a convex body is a convex set which has non-empty interior. By non-empty interior, we meant for a set ...
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Algorithm - the longest chord whose supporting line contains a given point, in a convex polygon

"Let $P$ be a convex $n$-gon and $q$ a point in the plane. Find an algorithm to compute the longest chord whose supporting line contains q." When $q$ is external to $P$, I think I can prove the ...
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algorthm to find a farthest point in a convex polygon to an external point

Given a point $q$ external to a convex polygon $P$, propose an algorithm to compute a farthest point in $P$ to $q$. One can always have at least one vertex of $P$ in the set of farthest points of $P$ ...
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algorithm to find closest point in a convex polygon from an external point

Given a convex polygon $P$, and a point $q$ of the plane, external to $P$, what is the fastest algorithm to compute the closest point in $P$ to $q$. A linear algorithm of course works, computing the ...