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Questions tagged [convex-analysis]

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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Conditions of strong convexity for non-smooth functions

I was reading this paper for some results on the strong convexity for non-smooth functions but I'm not getting this proposition at all: Lema II (i) $f$ is strongly convex with parameter $\mu$. (ii) ...
Nicolas brj's user avatar
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Number of iterations needed for the method of steepest descent

The function $f(x,y) = 4x^2 + 2y^2 + 2xy -4x + 6y$ has a unique global minimizer at $(x,y) = (1, -2)$ Starting at $(5,2)$ how many iterations of the steepest descent method would it take, at least, to ...
WannaBeRealAnalysist's user avatar
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Hessian matrix negative definite except in a finite number of points

I'm dealing with optimization problems and I have found myself wondering on this statement: If $f$ is twice differenciable on $\mathcal{D}$ and its hessian matrix $\mathbf{H}_f$ is negative definite ...
Goug's user avatar
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3 votes
1 answer
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Modulus of convex functional is convex functional

I'm having a hard time trying to prove the following: show that if $p$ is a convex functional then $|p|$ is also a convex functional. Here a "convex functional" is a function $p:X\to \...
Masacroso's user avatar
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Does a continuous piecewise-affine function preserve the convexity of a set?

Consider a convex set $C\subset \mathbb{R}^{n}$ in the $n$-dimensional real vector space, and a continuous piecewise-affine function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$. Is the set $f(C) = \...
linfengleeeeeee's user avatar
-3 votes
0 answers
19 views

Sum of two-variable and single-variable convex functions to be convex [closed]

It holds that if $f(x)$ and $g(x)$ are convex functions, then $h(x)=f(x)+g(x)$ is also convex function. While I meet a problem in my research. I have a three functions $f_1(x)$, $f_2(x,y)$, $f_3(y,z)$,...
ln7's user avatar
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-2 votes
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Convexity of zero of partial derivative of a convex fonction. [closed]

Is there an integer d, an integer $1 \le i \le d$ and a convex function $f : \mathbb{R}^d \rightarrow \mathbb{R}$. Such that the set $\{x \in \mathbb{R}^d / \frac{\partial f}{\partial x_i}(x) =0 \}$ ...
MikeProgrammeur's user avatar
2 votes
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79 views

Number of Tverberg Partitions [closed]

Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect. For example, $d=2$, $r=3$, 7 points: Let $p_1, p_2,...
D. S.'s user avatar
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1 answer
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Helly's theorem for $n\geq d+3$

Helly's theorem : Let $C_1,\ldots,C_n$, $n\geq d+1$, be convex sets in $\Bbb R^d$. Suppose every $d+1$ have a common intersection. Then they all have a common intersection. Proof: We're given that ...
D. S.'s user avatar
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When does $A-B= B-A$ hold for bounded, closed, convex sets $A,B$?

When does $A-B=B-A$ hold for bounded, closed, convex sets $A,B\subset \mathbb{R}^n$? In 1D it is easy to see that this holds true if and only if the two intervals have the same center points. Is ...
Thorsten Hohage's user avatar
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Is a quadratic program convex with respect to its parameters? [closed]

Consider a quadratic program $$ \min_z \quad \frac{1}{2} z^T A z + b^T z $$ $$ s.t. \quad Cz \leq d $$ with $A$ positive definite. For any choice of $A$ and $b$ we get a solution $z^*$. Is the ...
FlavioRed's user avatar
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1 answer
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+100

What is the number of facets of a $d$-dimensional cyclic polytope?

A face of a convex polytope $P$ is defined as $P$ itself, or a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
A. H.'s user avatar
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0 answers
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Proof Clarification: $f$ is convex iff it's convex when restricted to every line intersecting its domain

About this answer: Proof: A function is convex iff it is convex when restricted to any line .. I need clarification about the proof the user @gerw gave. I managed to understand and perform the proof ...
J.N.'s user avatar
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1 answer
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Doubts on the procedure: convexity of a function

I'm writing here to ask for clarifications about my professor's solution of this exercise: state if $f(x, y) = \ln(x^a - y)$ is convex on $\mathbb{R}^2$, $a \in (0, 1)$. Now he says that for $f$ to be ...
J.N.'s user avatar
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1 vote
1 answer
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Fréchet subdifferential for weakly convex functions on Hilbert spaces

Let $\mathcal{H}$ be a real Hilbert space. The Fréchet subdifferential of a function $f\colon \mathcal{H} \to \mathbb{R} \cup \{\pm\infty\}$ is the set-valued operator $\partial f:\mathcal{H} \...
BasicUser's user avatar
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1 answer
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Unit ball with dual set [closed]

Let $X\subseteq \mathbb{R}^d,$ we define the set dual to $X$, denoted by $X^*$, as follows: $$X^*=\{y\in \mathbb{R}^d \mid \langle y,x\rangle\leq 1, \forall x\in X\}.$$ Geometrically, $X^*$ is the ...
D. S.'s user avatar
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-1 votes
1 answer
20 views

Do three points of inequality between convex functions imply inequality over an interval? [closed]

Say I have 2 convex functions, $f:\mathbb{R}\rightarrow\mathbb{R}$, and $g:\mathbb{R}\rightarrow\mathbb{R}$. I want to prove that $f(x) > g(x), \forall x \in [l, u]$. I know that $f(l) > g(l)$ ...
Brian's user avatar
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1 vote
2 answers
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Prove $\frac{d}{dx} \frac{f(x)-f(a)}{x-a} \geq 0$ if $f(x)$ is convex without twice differentiability.

I've recently been trying to understand some proofs about convex functions. The definition of convex I'm using is: Let $f(x)$ be a once differentiable function defined on $[a, b]$. $f$ is convex iff ...
Alp's user avatar
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-2 votes
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Does the set where a convex function fails to be second derivative have some geometric properties? [closed]

I know it is of Hausdorff measure 0 by Alexsandrov's theorem. And the second derivative of a convex function can be viewed as a Radon measure. Does it have some other geometic properties?
ccs's user avatar
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0 answers
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Set dual with half-spaces

Let $X\subseteq \mathbb{R}^d,$ we define the set dual to $X$, denoted by $X^*$, as follows: $$X^*=\{y\in \mathbb{R}^d \mid \langle y,x\rangle\leq 1, \forall x\in X\}.$$ Geometrically, $X^*$ is the ...
D. S.'s user avatar
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0 votes
0 answers
39 views

Are the Products of Positive Semi-Definite Matrices Always Semi-Definite? [closed]

Is $AT'KTATK^2T + AT'K^2TATKT $ semi-definite when $K$ and $K^2$ are positive? All the entries in A,T,K are real and positive, I know the the product of positive semi-definite matrices are not ...
bei xi's user avatar
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-1 votes
0 answers
50 views

Number of cells in hyperplanes [closed]

I follow up from this question1, question2.. An arrangements of hyperplanes in $\mathbb{R}^d$ is simple if the hyperplanes are in general position (for every $1\leq k\leq d+1,$ the intersection of k ...
D. S.'s user avatar
  • 282
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0 answers
38 views

Degrees of freedom in mirror descent

The mirror descent algorithm makes certain choices which appear somewhat arbitrary, and so for each of them I am hoping for an understanding (maybe a proof) of why they are necessary, or at least some ...
oat's user avatar
  • 1
0 votes
1 answer
50 views

Legendre transformation is a continuous map

I was reading an article about the Legendre Transformation in Convex Analysis. It was defined like this: Let $\varphi$ be a continuous function on $\left[0,\infty\right)$ and convex on $\mathbb{R}_{&...
MoinsUnPuissanceN's user avatar
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0 answers
18 views

Quasiconvexity implies rank-one convexity proof

I'm struggling to understand the proof of Proposition 5.3 in Rindler's 'Calculus of Variations'. I cannot follow why $u_j$ is $0$ on the boundary of $Q_n$. The next line states that $u_j$ converges ...
Fluffy Alpaca's user avatar
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0 answers
76 views

Maximum and concavity of function

I have a function \begin{align} G(x_1,x_2,x_3)=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi} \frac{d\theta_1d\theta_2d\theta_3}{(2\pi)^3} p(\vec{x},\vec{\theta}) \log\left( p(\vec{x}, \vec{\theta})\right) \...
nervxxx's user avatar
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0 answers
8 views

Is a concave parametric curve along a concave surface guaranteed to be concave along another concave surface?

Take a parametrized probability distribution $\mathbf{p}(\theta)=( p_0(\theta),p_1(\theta),\cdots p_n(\theta))$ and two permutation-symmetric, everywhere-concave functions $S_1(\mathbf{p})$ and $S_2(\...
Quantum Mechanic's user avatar
1 vote
1 answer
44 views

Compute the conjugate of $\varphi (t) = \frac{1}{p}|t|^p$.

I'm a bit confused on how to actually compute the conjugate on a function like this based on the definition alone. The conjugate is given by $$\varphi^*(f) = \sup_{x\in \mathbb{R}} \{f(x)-\varphi(x)\}$...
H4z3's user avatar
  • 800
4 votes
0 answers
204 views

An arrangements of the hyperplanes.

Consider a finite set L of lines in the plane. They divide the plane into convex subsets of various dimensions, as is indicated in the following picture with 4 lines: The intersections of the lines, ...
D. S.'s user avatar
  • 282
2 votes
1 answer
66 views

Number of cells in simple arrangement

Consider a finite set L of lines in the plane. They divide the plane into convex subsets of various dimensions, as is indicated in the following picture with 4 lines: The intersections of the lines, ...
D. S.'s user avatar
  • 282
2 votes
0 answers
27 views

Gauss map is a diffeomorphism

Consider a smooth, convex function $f:\Bbb{R}^n\to \Bbb{R}$ such that the open subset $U=f^{-1}(-\infty, 0)$ has a smooth boundary $\partial U=f^{-1}(0)$. Consider the following smooth map $g: \...
dsjkhbjdskfhkjh jhdj's user avatar
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0 answers
21 views

Is squared geodesic distance about a point convex when defined on a convex ball centered at that point?

Let $\mathcal{M}$ be a complete Riemannian manifold. Let $x \in \mathcal{M}$, and let $r_x >0$ be the convexity radius at $x$. Let $B \subset \mathcal{M}$ be a geodesic ball centered at $x$ with ...
Spencer Kraisler's user avatar
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0 answers
28 views

Proximal point operator of maximum of functions

I want to know whether the proximal point operator of a function with a special structure can be expressed in terms of simpler proximal point operators. Setup: For a proper, convex and lower ...
AverageJoe's user avatar
0 votes
1 answer
25 views

Normal Cones on a Circular Arc

Let $C = \{(x, y) \in \mathbb{R}^2: x^2 + y^2 \leq 1, y \geq 0, y \leq x\}.$ I wish to determine the normal cone $N_C(x_0, y_0)$, for any $(x_0, y_0)$ on the boundary of $C$. There are six cases to ...
V. Elizabeth's user avatar
0 votes
1 answer
41 views

Concavity of Distance-to-Boundary Function

Let $B := \bar{B}_{R}(0) \subseteq \mathbb{R}^2$ be the closed ball of radius $R$. The distance-to-boundary function $d: \mathbb{R} \rightarrow \mathbb{R}$ gives the distance of any point $x_0$ to the ...
V. Elizabeth's user avatar
0 votes
1 answer
36 views

A question on strict convexity vs strong convexity

Is it possible to build a function $f: [0, +\infty)$ such that $f, f', f'' > 0$ and $x \mapsto x/ f(x)$ is strictly increasing? Of course, if $f, f'>0$ and $f''> \epsilon > 0$, this is not ...
Aristodog's user avatar
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1 vote
1 answer
63 views

Proving the two optimization are equivalent or not

Here are two optimization problems: $$ (P) : \inf_{x} [ f(x)|g(x) \leq 0 ] \text{ where } g(x) = \inf_{y} [ h(x,y)|y \in Y ] $$ $$ (Q) : \inf_{x,y} [f(x)|h(x, y) \leq 0, y \in Y ] $$ Are they ...
Albertvosky's user avatar
0 votes
1 answer
19 views

Upper and lower contour sets of a convex function?

First let me present an observation. Suppose $f$ is a convex function. Suppose $f(x)<f(y)$. Then for any $t>0$, we must have $f(y+t(y-x))\geqslant f(y)$. The proof is as follows. Suppose instead ...
Ypbor's user avatar
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2 votes
0 answers
57 views

Intersect polytopes defined by half-planes

Is there a quick way to find this region or its convex hull? $N$ is large, $M=d=5$ or so $$ \bigcap_{n=1}^N \bigcup_{m=1}^M \{x \in \mathbb{R}^d :a_{nm}'x \geq 1\} $$ The slow way to do it (the only ...
Christian Chapman's user avatar
0 votes
0 answers
32 views

Understanding Feasibility, Constraints in the Lagrange Dual Function: A Query on Boyd's Least-Squares Example

I am confused by a basic problem with the Lagrange dual function. As Boyd states, $g(\lambda, \mu) = \underset{\scriptsize \text{$x \in D$}}{inf}L(x, \lambda, \mu) = \underset{\scriptsize \text{$x \in ...
LoveDance's user avatar
2 votes
0 answers
57 views

Covariances not below $\Sigma$

$\Sigma_0,\Sigma_1,\dots,\Sigma_K$ are real covariance matrices. I’m interested in the set of matrices $$\bigcap_{k=1}^K \left\{x: 0 \preceq x \preceq \Sigma_0, \ x\not\prec\Sigma_k\right\}.$$ I’m ...
Christian Chapman's user avatar
3 votes
0 answers
135 views

Improving Ellipsoid Theorem $t+\mathcal{E} \subseteq K \subset t + c\cdot n \mathcal{E}$

Setting: Definition (ellipsoid): An ellipsoid is $\mathcal{E} := \{y \in \mathbb{R}^n : ||A^{-1}(y-t)|| \leq 1\}$, i.e images under a given $A$ of full rank of the unit closed ball of $\mathbb{R}^n$. ...
jacopoburelli's user avatar
0 votes
1 answer
68 views

Checking if a function is convex

I like to find out if this function convex. $$f(x)=x_1^2+x_2e^{x_1}-x_1x_2+x_2^2$$ So I checked the definition: A function $f$ is convex on a convex set $S$ if it satisfies $f(\alpha x + (1 -\alpha)y)...
Superunknown's user avatar
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0 votes
0 answers
29 views

Inequality with two concave functions

is this inequality holds true $[\frac{\frac{f_2(x_0,y_0)}{y_0}}{\frac{f_2(x,y)}{y}}-\frac{f_1(x_0,y_0)}{f_1(x,y_0)}]\times [\frac{f_1(x,y_0)}{f_1(x_0,y_0)}-\frac{f_2(x,y)}{f_2(x_0,y_0)}]\leq 0 \; \...
Sayed Sayari's user avatar
0 votes
1 answer
58 views

Is the entropy of a distribution that follows the exponential family differential equation always concave in the natural parameter?

Are there any proofs, conjectures, counterexamples, or other helpful references related to the titular question? To clarify, let the entropy of a random variable $X$ distributed according to a ...
nlupugla's user avatar
4 votes
1 answer
131 views

Can a convex function with a unique support be non-differentiable?

Let $\mathrm{M}$ be an open convex set of a normed space and $f:\mathrm{M} \to \mathbf{R}$ a continuous convex function. Definition. A support of $f$ at $v \in \mathrm{M}$ is an affine function $x \...
William M.'s user avatar
  • 7,706
0 votes
1 answer
43 views

This function has no saddle points: correctness of this reasoning

I would like to know if my reasoning is correct. I have the function $f(x, y) = e^{3x}(1+25x^2+25y^2)$ and I have to study the stationary points. After computing the gradient I found $$\begin{cases} 3 ...
Heidegger's user avatar
  • 3,482
4 votes
1 answer
58 views

Which condition implies that $f(s)/s$ is increasing?

I'm trying to find a class of functions $f$ that has this property: $f(s)/s$ is increasing in some interval $(0, \lambda)$. The function $f(s)= s^p$ with $p>1$ has this property, for example. My ...
Ilovemath's user avatar
  • 3,004
1 vote
1 answer
48 views

Curvature measure for convex functions in $\mathbb R^d$

TLDR: convex functions in high dimensions are much weirder than in 1D: do you please have insights or references to share? Let $c(x)$ be a convex function on the real line. There is an obvious ...
Guillaume Dehaene's user avatar
0 votes
1 answer
24 views

Orthogonal Projection onto a Polyhedron (Matrix Inequality)

How to efficiently solve: $$\begin{align*} \arg \min_{\boldsymbol{X}} \quad & \frac{1}{2} {\left\| \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} \\ \text{subject to} \quad & \begin{aligned}...
Royi's user avatar
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