All Questions
Tagged with convex-optimization convex-geometry
70 questions
2
votes
1
answer
92
views
Does approximately null gradient imply approximately global minimum for convex functions?
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a non-negative and differentiable convex function which vanishes in a non-empty convex set $\Omega$ - possibly unbounded. Usually, when one ...
- 225
0
votes
0
answers
35
views
Describing the boundary of the feasible direction cone to a convex open subset of $\mathbb{R}^n$ at a boundary point: connection via subdifferential?
Let $U\subset \mathbb{R}^n$ be a convex, open set with nonempty boundary. Let $x_0\in \partial{U}.$ We can describe $U$ locally near $x_0$ as a super level set of a suitable continuous concave ...
- 1,467
0
votes
0
answers
69
views
Degree of reflectional symmetry of (unbounded) convex polyhedra in Euclidean spaces
Let $U \subset \mathbb{R}^m$ be an open domain. I'm trying to come up with a measure of its degree of reflectional symmetry and I have a question. The post in two-part, where in PART I I introduce the ...
- 1,467
1
vote
0
answers
143
views
Integer points inside the high-dimensional ball (asymptotics)
Let $N(\alpha, n)$ denote the number of integer points inside the origin-centered ball of radius $\alpha \sqrt n$ in $n$ dimensions, where $\alpha \in (0,\infty)$ is some fixed constant. Precisely:
$$...
- 435
1
vote
0
answers
45
views
Inequality Involving Concave Monotonic Function
Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
0
votes
1
answer
150
views
When are infimal convolutions contractions?
Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution
$$
\...
- 364
1
vote
0
answers
79
views
Touring a sequence of convex polygons with minimal energy
There is a known problem of touring a sequence of $n$ polygons: given a starting point $s$, an ending point $t$ and a sequence of polygons $P_1,\dots,P_k$ with a total of $n$ vertices, find points $...
- 177
6
votes
0
answers
48
views
Strengthening the Kovner-Besicovich theorem: Does every unit-area convex set in the plane contain a centrally symmetric hexagon of area $2/3$?
The Kovner-Besicovich theorem states that every convex set $S$ in the plane contains a centrally symmetric subset $C$ of at least $2/3$ the area of $S$, and that this bound is sharp for triangular $S$....
- 3,068
0
votes
0
answers
40
views
Iterating partially-unconstrained optimization with projection
Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex.
I want to optimize
$$
\tag{(A)...
- 5,405
1
vote
0
answers
95
views
Distance between two convex sets
Setting
If $A$ an $B$ are two symmetric matrices, we denote by $A >B$ when the matrice $A-B$ is definite positive.
In $\left(\mathbb{R}^{*}_{+} \right)^4$, consider the convex set $$ \Lambda = \...
- 125
1
vote
0
answers
32
views
Variants of cutting plane method for convex optimization
The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into ...
- 139
2
votes
1
answer
159
views
Conic hull of a rectangle
I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i ...
- 275
1
vote
1
answer
98
views
If $|P|<\infty$ and $C=P\cap\partial(\textrm{Conv}(P))$, then $P\subset\textrm{Conv}(C)$?
That is, if $P$ is a finite set, and $C$ is the set of points in $P$ which lie on the boundary of the convex hull of $P$, then is $P$ contained in the convex hull of $C$?
It seems true intuitively. In ...
2
votes
1
answer
594
views
Tangent cone of a closed convex cone
Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by)
$$
T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...
- 163
4
votes
2
answers
353
views
An upper bound of gradient norm for convex functions near minimizer
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ ...
- 311
1
vote
0
answers
61
views
Fitting a convex polytope with 𝑛 facets between two nested spheres
This is related to a research problem that is interested in approximation of spheres by convex polytopes.
Let $C_r$ and $C_R$ be two spheres in $\mathbb R^d$ of radius $r$ and $R$, respectively, where ...
1
vote
0
answers
121
views
John and Lowner ellipsoid
I am looking at a proof to show that Lowner ellipsoids are unique for centrally symmetric convex body $K$. I want to show basically that
$$
\DeclareMathOperator{\Vol}{Vol}\DeclareMathOperator{\Low}{...
- 11
2
votes
1
answer
119
views
Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$
Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set
$$
\alpha := \sup_{(x,y) \in C} ax + b y.
$$
Question. ...
- 6,853
0
votes
1
answer
67
views
Analytic formula for minimizer of $f(x) := \sqrt{(x-a)^\top S(x-a)}+ r \|x\|_2$
Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the ...
- 6,853
5
votes
1
answer
355
views
Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?
Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is ...
- 225
0
votes
1
answer
101
views
Estimation via projecting onto a convex body
Assume that $\theta$ is in a convex body $K \in \mathbb{R}^n$ and we observe $y = \theta + z$, where $z$ is a noise term (following, say, the normal distribution). Consider an estimator of $\theta$ by ...
- 773
0
votes
0
answers
118
views
Weak derivative of projection onto probabilist's simplex
Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...
- 5,405
4
votes
1
answer
194
views
How to solve this minimax matrix optimization problem?
Recently, I want to know how well can a $\ell_1$ ball be approximated by the image of a $\ell_2$ ball under a linear transform. I formulate this problem as the following optimization problem.
\begin{...
- 550
1
vote
0
answers
52
views
When the summands of a positive definite matrix are positive definite
Let $A,B$ be two real symmetric matrices. Let $C = A+B$ be a positive-definite matrix. Write $C>0$ for $C$ being positive-definite. Suppose that $A>0 \implies C>0$ and $B > 0 \implies C>...
- 71
1
vote
1
answer
337
views
High-dimensional polytopes
I have two questions regarding polytopes in high dimensions, $\mathbb{R}^d$ where $d > 3$, that I could not find resources for on the web for. Suppose I have a polytope that is non-convex:
How can ...
- 43
2
votes
1
answer
345
views
Can the subdifferential become unbounded at interior points?
Consider $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ a lower-semicontinuous, proper, closed and convex. My question is, can the subdifferential of $\partial f$ be unbounded in the interior of $\text{...
- 179
2
votes
1
answer
189
views
Analytic expression for the Moreau envelope of $x \mapsto \|Ax\|$
Given an $m \times n$ matrix $A$ and a vector $c \in \mathbb R^n$, define $\eta(A,c) \ge 0$,
$$
\eta(A,c) := \sup_{u \in \mathbb R^n} u^\top c - \frac{1}{2}\|u\|^2 - \|Au\|.
$$
Note that $\eta(A,c) = \...
- 6,853
1
vote
0
answers
45
views
Analytic lower-bound for minimal value of $\|x\|^2$ such that $\|Cx-b\|^2 \le c^2$ (a hyperellipsoid)
Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position.
Question 1. ...
- 6,853
1
vote
0
answers
172
views
continuity of linear programming
I have the following conjecture:
Given a closed convex set $S \subseteq \mathbb{R}^n$ and one of its exposed face $F=\{x \in S \mid \pi x = \pi_0\}$, where $\pi x =\pi_0$ is the supporting hyperplane ...
- 37
4
votes
0
answers
140
views
A convex function is "usually" subdifferentiable
Let $X$ be a locally convex topological vector space, and let $f:X\to\mathbb R\cup\{\infty\}$ be a proper, convex, lower semicontinuous function, whose effective domain $D:=f^{-1}(\mathbb R)$ is ...
- 537
3
votes
2
answers
207
views
Generalizing Pythagorean Theorem: equations defining edges of a (convex) $n$-gon with $n-2$ prescribed angles?
Let $n\geq 3$ be an integer and $0<\alpha_1, \dots ,\alpha_{n-2}<1$. Let's say a tuple of positive numbers $(e_1,\dots, e_n)$ is nice if there is a convex $n$-gon $A_1\dots A_n$ such that $\hat ...
- 30.5k
5
votes
1
answer
226
views
Sufficient condition for geodesic convexity/connectedness
Let $(\Sigma,g)$ be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the ...
- 1,371
0
votes
0
answers
44
views
Let $A,B,C$ be centrally-symmetric convex bodies. What is this function $G(x,y) := \sup_{b \in B}\inf_{a \in A} a^T x - b^T y + \|a-b\|_C$?
Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\...
- 6,853
4
votes
0
answers
229
views
How to find the dimension of the polar cone of a convex cone generated by some given vectors
Suppose we have access to a generating set $\{v_1, ..., v_k\}\subseteq\mathbb{R}^n$ of the convex cone $C=cone(v_1, ..., v_k)$, where $cone(\cdot)$ is the conical hull (i.e. nonnegative span) of ...
- 461
1
vote
0
answers
68
views
Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
- 13.9k
5
votes
0
answers
135
views
Conv A = Dual B
I have two cones $A$ and $B$ in a Euclidean space.
I want to show that $\mathrm{Conv}\,A=\mathrm{Dual}\,B$; that is, $a\in\mathrm{Conv}\,A$ if and only if $\langle a,b\rangle\ge 0$ for any $b\in B$.
...
- 45k
4
votes
1
answer
255
views
Inequality $\frac{C}{d_{max}} \le \pi $ relating perimeter and diameter of planar convex body
Let $C$ is a perimeter of a convex hull (plane geometry) and $d_{max}$ is the largest distance of two arbitrary points in the convex hull. I am looking for a proof that:
$$\frac{C}{d_{max}} \le \pi ...
- 1,776
1
vote
0
answers
99
views
Finding a point on a convex set
Given a compact bounded convex set $\mathcal C\subseteq\mathbb R^n$ given by $t$ hyperplane inequalities I want to find a point $u\in\mathcal C$ such that for all $v\in\mathcal C$
a convex relation $...
- 1,826
1
vote
1
answer
2k
views
Euclidean projection onto certain convex set
Consider the closed convex set
$$
C = \{ x \in R^n : \alpha \| x \| + \langle c, x \rangle \leq 0 \},
$$
for constants $\alpha > 0$, $c \in R^n$.
My question is whether the Euclidean projection ...
- 303
0
votes
1
answer
117
views
Closed form solutions for maximal subsets of convex polytopes
I'm looking for any known exact results about inscribing simple convex bodies inside a convex polytope. The most famous is the Löwner-John ellipsoid, but as far as I understood in general there is no ...
- 445
3
votes
1
answer
258
views
Polygon of convex arcs
Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...
- 422
3
votes
0
answers
203
views
Level sets of strongly convex and smooth functions
Let $f: \mathbb{R}^N \to \mathbb{R}$ be a $\alpha$-strongly convex and $\beta$-strongly smooth function, i.e.,
$$ f(x) + \langle\nabla f(x), y- x\rangle + \frac{\alpha}{2}\|y-x\|^2
\leq f(y) \leq f(x) ...
- 31
2
votes
1
answer
233
views
Why the convexity condition on the definition of a face of a convex set?
A face of a closed convex set $X\subseteq\mathbb{R}^n$ is defined to be a set $F\subseteq X$ such that:
$F$ is convex.
Every line segment from $X$ whose interior meets $F$ is contained in $F$.
Is ...
- 75
1
vote
1
answer
266
views
Relative interior of a normal cone at a face of a convex polytope?
Suppose $A$ is a nonempty convex polytope in $\mathbb{R}^n$. Suppose $F$ is a face of $A$.
Consider the normal cone of $A$ at $F$:
$C_A(F)=\{v\in\mathbb{R}^n:v\cdot x\geq v\cdot y\ \forall\ x\in ...
- 159
1
vote
0
answers
53
views
Projecting two convex polyhedra onto their intersection
Suppose we are given two convex polyhedra $\mathcal{C}_1, \mathcal{C}_2 \subset \mathbb{R}^n$ with non-empty intersection $\mathcal{C}_1 \cap \mathcal{C}_2 \neq \emptyset$.
For the orthogonal ...
- 337
11
votes
2
answers
559
views
Convex hull of the Stiefel manifold with non-negativity constraints
Consider the Stiefel manifold
$$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$
where $I_k$ is the $k$-dimensional identity matrix. It is well known that
$$\mathrm{conv} \left( ...
- 1,991
-1
votes
1
answer
137
views
Does a half plane contain intersection of some other half planes? [closed]
I'm doing research in Optimization and I have found this obstacle in the way.
If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
- 19
5
votes
0
answers
151
views
Dimensions of faces of convex hull of convex bodies
Let $K_1,\ldots,K_m\subset\mathbb{R}^n$ with $m\geq n$ some convex bodies (i.e. compact with nonempty interior). I am interested in sufficient criteria for the convex hull $K=\textrm{conv}(K_1,\ldots,...
- 3,031
9
votes
1
answer
749
views
property of convex functions
I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the ...
- 1,211
9
votes
0
answers
1k
views
Existence of barycenter
Let $(X,d)$ be a metric space. A barycenter of a Borel probability measure $\mu$ on $X$ is a minimizer of the function
\begin{equation}
\begin{split}
f \colon X & \to \mathbb{R}\\
x &\mapsto \...
- 1,347