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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Proving that $0 \in A \implies h_A = j_{A^\circ}$

Where $h_A$ is the support function of $A$ and $j_{A^\circ}$ the Minkowski functional of the polar set of $A$ There is a "proof" in my course which I don't understand: " Let $x \in A$ and $t>0$ ...
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Limite of the distance of the iteration to each set

Let $C_1$, $C_2$ convex and closed sets such that the intersection is noempty. I want to show that the iteration $x^{k+1}=f(x^k)$ generated by the function $f: \mathbb{R^n} \to \mathbb{R^n}$ defined ...
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Existence of global minimum

Could someone help me with this problem? Let $C$, $D$ convex and closed sets such that the intersection is empty. I want to show that the function $f: \mathbb{R^n} \to \mathbb{R}$ defined by $f(x) = ...
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Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
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Projection onto convex set defined by $\|\mathbf{t} -\mathbf{W}^T\mathbf{y}\|^2 \leq k$

I want to use the method of Projections Onto Convex Sets, and for the problem at hand I need to find a closed form solution for $\mathbf{P}_C$, the projection onto set $C$, defined as: $$C = \{ ...
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A convex function has a lower bound?

Suppose that $f=f(x)$ is strictly convex for $x\in\mathbb{R}$, i.e. there exists $\epsilon>0$ such that $f''(x)\geq\epsilon>0$ for $x\in\mathbb{R}$. Does there exist $\delta>0$ such that ...
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105 views

Does every convex-linear map have an affine extension?

There is one step in a proof which I don't manage to show, although it seems to be very easy. Let $A, B$ be real vector spaces, let $S \subset A$ be a convex set and let $\text{aff}(S)$ be its affine ...
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+50

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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Isoperimetric inequality, isodiametric inequality, hyperplane conjecture… what are the inequalities of this kind known or conjectured?

Question: Which inequalities similar to the famous isoperimetric inequality is known? conjectured? I recently learned about some inequalities which are all similar to the famous isoperimetric ...
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convex hull function in matlab

Is there anyway to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
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Is the set of convex bodies include in a closed ball compact?

I consider the set $\mathcal{K}_B$ of convex bodies (convex and compact) which are include inside the unit closed ball of $\mathbb{R}^d$. I endow this set with the Hausdorff distance. Is it compact?
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+100

Subdifferential boundary conditions

Let $\Omega \subset \mathbb R^d$ be a domain and $\Gamma$ be its boundary. Assume also that we have a convex (proper, lower semicontinuous) function $\phi \colon \mathbb R^d \to \mathbb R \cup ...
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Mid-point convexity does not imply convexity

A function $f: X \rightarrow \mathbb{R}$ is said to be mid-point convex if for all $x, y \in X$, we have $f(\frac{x + y}{2}) \leq \frac{f(x) + f(y)}{2}$. Can you please give an example of a function ...
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How prove this inequality generalized from 1969 IMO problem 6

Let $x_{1},x_{2},y_{1},y_{2},z_{1},z_{2},w_{1},w_{2} $ are all positive numbers, and such $$x_{1}y_{1}z_{1}-w^3_{1}>0,\; \text{ and }\;x_{2}y_{2}z_{2}-w^3_{2}>0.$$ show that ...
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Proving an inequality for convex and increasing functions [closed]

Can anyone show or disprove the following? Let $f:R^+ \rightarrow [1,+\infty]$ be a strictly increasing, convex nonasymptotic and differentiable function and $f'(0)>1$, then for $\forall \ ...
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1answer
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Is a constrained optimization problem equalivant to its Lagrangian form?

For the following problem: $\text{min:}\ f(x)\\ s.t. \ g(x)\leq t$ Is the above problem equalivant to the following problem? $\text{min:}\ f(x) + \lambda g(x) \\ s.t. \ \lambda\geq0$ where $t$ and ...
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1answer
42 views

What's the solution for $\max_{x\in(0,1]}: \{-1-x\}$

What's the solution for the following optimization problem? Is the constraint set convex? $$\max_{x\in(0,1]}:\{-1-x\}$$
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1answer
19 views

Confusions of Convex Set

I am interested in the properties of convex set in $\mathbb R^n$ and want to clarify the three statements below $A$ is an open convex set. Can we get the conclusion that $\bar{A}$ is convex? ...
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Sum of quasi-concave function and linear function

Let $F(k)$ have the following properties: $$ F(k) \in C^1 \\ F'(k) > 0 \\ F(0)=0 \\ \lim_{k \to \infty} F'(k) = 0, \lim_{k \to 0} F'(k) = \infty \\ F ~\text{strictly quasi-concave}. $$ Then, let ...
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1answer
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Analog of Birkhoff's theorem for doubly stochastic matrices

Birkhoff's theorem states that extreme point of the set of doubly stochastic matrices are permutation matrices. An $n \times n$ matrix $A$ is doubly stochastic if each row and column sums to 1. What ...
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2answers
30 views

Critical points and Convexity?

Function $f(x)$ has no critical points in $M$, can we say $f(x)$ is either convex or concave over $M$?
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No critical points means convex or conave? [closed]

If we don't know whether $f(x)=0$ is convex or concave or not, but we know under certain constraint sets there is no critical points of $f(x)$ inside meaning the solution of $df(x)=0$ is outside the ...
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1answer
29 views

Compact and convex discrete set

I am working with discrete sets but I have a doubt: is the set $\{ 0,1\}$ compact and convex? And the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$?
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Formal definition of convexity for multivariate function?

Let $M\in R^{M\times N}$, a function $f: M\rightarrow R$ is called convex on $M$ if $f\big((1-\lambda)X1+\lambda X2, (1-\lambda)Y1+\lambda Y2\big) \leq (1-\lambda)f(X1,Y1) + \lambda f(X2,Y2)$ For ...
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Convex cone question.

Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
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383 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
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1answer
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Certain property of convex functions…

I come to you with yet another qualifying problem we can't seem to solve... Let $f:$ $(0,\infty) \to \Bbb R$ be convex, and let $\lim_{x \to 0}f(x)=0$. Show that $g(x)$ = $f(x) \over x$ is increasing ...
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335 views

What Stopping Criteria to Use in Projected Gradient Descent

Suppose we want to solve a convex constrained minimization problem. We want to use projected gradient descent. If there was no constraint the stopping condition for a gradient descent algorithm would ...
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51 views

alternative definition of Affine map

Let $f:X\longrightarrow Y$ be a function on real vector spaces (note that $X,Y$ have arbitrary dimensions). If $T(x)=f(x)-f(0)$ is linear, $f$ is called an affine map. Prove that $f$ is affine if ...
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Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
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Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
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On the convexity of a particular form of integrals

EDIT: I made some critical corrections below. Let $\mathcal{H}\colon\mathbf{w}\cdot\mathbf{x}+c=0$ be a hypeplane in $\mathbb{R}^n$. Also, let $g\colon\mathbb{R}^n\to\mathbb{R}_+$, be a non-negative, ...
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How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
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Convex cones and positively homogenous and subadditive functional

Let $V$ be a linear topological space, $K \subsetneq V$ a convex, closed cone with $0 \in K$ and $k \in K \setminus (-K)$. Show that the functional $\varphi : V \to \mathbb R \cup \{ \pm \infty \}$ ...
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197 views

Checking convexity

I know that the function $(\mathbf{a}-\mathbf{b})'(\mathbf{a}-\mathbf{b})$ is convex in $\mathbf{a}$ ($\mathbf{a}$ and $\mathbf{b}$ are vectors, not scalars). Would ...
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Issues with quasi Newton method convergence

I have this issue with the convergence of the quasi newton method. I have a convex objective function which I need to minimize wrt some parameters. I generated some synthetic data using a defined ...
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1answer
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Motivating convex sets.

I am kind of TAing for a class of real analysis, and I would like to speak a little about convex sets tomorrow, and explain why they are important. What kind of examples could I give? I was thinking ...
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Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
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The Fundamental Theorem of Matrix Games, and the “indifference” method of solving games

In the following we will consider two-person zero-sum games. Let $A = (a_{ij})$ be the payoff-matrix of such a game. In this book the fundamental theorem of such games is states as: Theorem: Given ...
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Is convex hull of a finite set of points in $\mathbb R^2$ closed?

Is the convex hull of a finite set of points in $\mathbb R^2$ closed? Intuitively, yes. But not sure how to show that. Thanks!
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Min of concave symmetric function on a convex set

Consider the convex set $$C=\left\{ \mathbf{x}\in \mathbb{R}^N :0\le x_1\le x_2\le\dots\le x_i\le x_{i+1}\le \ldots\le x_N\le \frac 1{N-1}\text{ and } \sum_{k=1}^{N}x_k=1\right\}$$ I need to minimize ...
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Volume of Minkowski sum of a point and a hypercube

Let $A$ be a single point and $B$ a unit cube in $\mathbb{R}^n$, what is then the volume $\lambda \mapsto \mathrm{Vol}\big((1-\lambda)A + \lambda B\big)$? I am not exactly sure, what the set ...
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Concave function divided by a convex function. What is the result?

Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...
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Pseudoconcavity of a fractional function q(x)=f(x)/g(x) if f(x) is nonnegative concave and g(x) is positive convex?

I know from Chandra that $q(x)=\frac{f(x)}{g(x)}$ is strong pseudoconcave if $f(x)$ is nonnegative concave and $g(x)$ is strictly positive convex. Is there a theorem that states $q(x)$ is ...
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1answer
24 views

Graph of polytope and hyperplane

Suppose that $P$ is a compact and convex polytope in $R^d$ and let $G$ be the graph of $P$ ($V(G)$ are the vertices of $P$ and $E(G)$ are the $1$-dimensional faces - for example polyedral graphs are ...
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A question about pyramids (polytopes)

Let's use the following definition of a face: A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ ...
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52 views

Relation between mean width and diameter

Question: Let $A$ be a compact set in $\mathbb R^n$. Is it always true that $\text{mean-width}(A) \ge C \cdot \text{diam}(A)$ for some constant $C$ depending only on the dimension? If not, is it ...
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Quickly checking if an inequality holds on a convex region

Let $C$ be a given convex polygon in $\mathbb{R}^2$ containing the origin and let $a$, $\mathbf{b}$, and $Q\succeq0$ be a given scalar, vector, and matrix respectively. Is there a fast way to verify ...
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35 views

Why is pointwise maximum a convex function?

It seems like if you have a family of function $$g = \{a(x), \: b(x), \: c(x), \:d(x)\}$$ $$\text{given} \:\: f(x):= max(g),$$ $$\text{if} \: f(1) = a(1), \: f(2) = b(2), \: f(3) = c(3), \: f(4) = ...
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Prove: $f: (a,b) \rightarrow \mathbb{R}$ is convex iff for every triple $x_1,x,x_2 \in (a,b)$…

Assignment: Prove that a function $f : (a,b) \rightarrow \mathbb{R}$ is convex if and only if for every triple $x_1,x,x_2 \in (a,b)$ with $x_1 < x < x_2$ following inequality is satisfied: ...