0
votes
0answers
19 views

Hyperplane - soft and hard questions

This would be a rather long question. Apologies for that. Instead of asking three separate questions I've consolidated them in one. I am trying to learn hyperplanes, convex hulls, separations theorems ...
0
votes
2answers
56 views

Solution in general for a seemingly simple problem

Let $\mathbb{S}$ be a closed, bounded, convex set in $\mathbb{R}^N$. Let $\mathbb{x}=[x_1,\dots,x_N]$ be any arbitrary vector in $\mathbb{S}$. Then what can we comment on the problem \begin{align} ...
0
votes
1answer
58 views

Explain the convexity by looking the hessian matrix of a function

The hessian matrix of a function is given by, $$ H = \begin{bmatrix} a & b & c \\[0.3em] b & b & 0\\[0.3em] c & 0 & c \end{bmatrix} $$ where, ...
1
vote
0answers
49 views

Monotonically increasing maximum eigenvalue

Let a matrix $A \in \mathbb{R}^{n \times n}$ be the convex combination of two matrices as $A = qB + (1-q)C$. Define $B$ as unit anti-diagonal. Define $C_{i,j} = \delta_{i,i+1}$. Consider $A$ for ...
0
votes
0answers
26 views

Isomorphisms of Convex Cones

A convex cone $C$ is a subsets $C \subseteq V$ of a vector space which is closed under positive linear combinations, i.e. for $\lambda, \mu > 0$ and $u,v \in C$ it is $\lambda u + \mu v \in C$. An ...
1
vote
0answers
43 views

Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
0
votes
1answer
28 views

Finding the determinant of a $ k \times k$ matrix (Hessian matrix)

Given $H(x_{1}, x_{2}, x_{3}) = \begin{bmatrix} -2 & \frac{1}{2} & 0 \newline \frac{1}{2} & -2 & 0 \newline 0 & 0 & -4 \end{bmatrix}$, I want to find (I think) the leading ...
1
vote
0answers
25 views

Intersection of affine subspace of $\mathbb{R}^n$ with $[0, 1]^n$

Suppose I have an affine subspace $V \subseteq \mathbb{R}^n$, say given by a rank-$r$ system of $m$ equations in $n$ variables. I'm interested in two questions: Is there a straightforward way to ...
2
votes
0answers
96 views

Voronoi-Diagram - split concave polygon

I have a concave polygon c and a set of points p[]. What I need is to have a set of polygons splitted by the voronoi-diagram. My problem is the following: I already found a library to calculate the ...
2
votes
0answers
27 views

Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
0
votes
1answer
25 views

show this Polyhedral set is convex

show that polyhedral set is convex. A is a matrix ( m x n ) and b is element of Rm (i think)
1
vote
1answer
55 views

Normed space problem

I am currently dealing with the following problem: Imagine you have two points $x,y$ in a normed space $(X,||.||)$ and in a convex set $K \subset X$. Now you know that $B_{\varepsilon}(x) \subset ...
0
votes
1answer
22 views

Converting a Probability constraint to a Norm constraint

Let $\mathbf{z}$ be a $N\times 1$ complex vector. Let $\mathbf{u}$ be a $N\times 1$ random Gaussian vector whose entries are i.i.d with zero mean and $\sigma^2$ variance. Consider the following ...
1
vote
0answers
23 views

What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
1
vote
1answer
31 views

Need an analytic expression to find the index of the first positive element of an array

I have an array of length M. The elements of the array are either zero or positive real numbers. I need to derive a function/analytic expression (preferably linear/convex/concave) that finds the index ...
0
votes
1answer
48 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 ...
0
votes
0answers
70 views

Help prove a lemma similar to Fredholm alternative (linear algebra / convex optimization)

I was asked to help with a proof of a lemma similar to Fredholm alternative. It looks too similar so I think it may be wrong - could you please advise whether it is true and hint how to prove it? The ...
0
votes
0answers
37 views

convexity in oriented matroid theory

I would like to try to solve the following problems. If someone knows how to prove at least part (a), could you show me the proof? I am having a LOT of trouble understanding oriented matroid theory. ...
0
votes
1answer
48 views

Is this function convex or concave?

I have a function, $f(x_{0},x_{1},......x_{n})=\sum^{n-1}_{i=0}A_{i}r^{-(x_{i}+x_{i+1})}-B$ $A_{i} >0$ for all $i$ and $B>0$ and $1 \leq r \leq 2$ so above function is convex or concave? ...
1
vote
1answer
41 views

Does a discrete set of points in $\mathbb{R}^{n}$ define a locally finite collection of hyperplanes?

Let $v_{1},v_{2},...$ be a discrete set of non-zero vectors in $\mathbb{R}^{n}$. By discrete, I mean that any $v_{i}$ is surrounded by an $\epsilon$-ball not containing any other point $v_{j}$. ...
0
votes
1answer
32 views

Effect of Moving within the Feasible Region

$f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a concave function with local maximum at $\mathbf{x}^*$ in a convex, closed feasible set $\mathcal{F}\subset\mathbb{R}^n$. Now consider a suboptimal point ...
0
votes
0answers
80 views

Proof of corollary of Farkas' lemma

I tried to prove the following lemma of Farkas' lemma: Given the system $Ax<b$, $A\in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^m$, the system is infeasible iff there exists $\lambda\in ...
2
votes
0answers
50 views

Maximizing the smallest eigenvalue of a linear combination of matrices

I have an engineering back ground. Due to work, I came across this problem \begin{align} &\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\ ...
0
votes
1answer
43 views

Subtle question concerning intersection of convex sets

I am attempting to convince myself that if $$\{S_{\alpha}: \alpha \in \mathcal{A}\}$$ is any collection of convex sets, then $$\cap_{\alpha \in \mathcal{A}}S_{\alpha}$$ is convex. This is my proof so ...
2
votes
1answer
54 views

Find the real vector $x$ which satisfies all this?

I got this after applying KKT conditions to an optimization problem. Let $\mathbf{h}$ be a given $N\times 1$ real vector. Let $\alpha$ be a real constant. We need to find $\lambda$ and the $N\times ...
0
votes
1answer
255 views

Convexity of sum and intersection of convex sets

Let $A_i$ be a subset of $\Bbb{R}^m$ which is convex for $i=1,...,n$. How can I prove that the sum of $A_i$ is also convex? I know how to prove it with two sets: Let $x = a_1 + b_1$ and $y = a_2 ...
0
votes
0answers
46 views

Existence of strictly positive solution

I have a linear system \begin{equation} \left[\begin{array}{c|c} A & \\ \hline I & I \\ \end{array}\right] \left[\begin{array}{c} x_0\\ x_1 \\ \end{array}\right] = \left[\begin{array}{c} ...
1
vote
0answers
36 views

Convex polyhedron is union of simplices

Given a convex polyhedron $P$, how can we prove that every point $x \in P$ is in some simplex whose vertices are vertices of $P$? One proof is to inductively build a triangulation of $P$. If $P$ is ...
0
votes
0answers
50 views

Linear map preserves closedness of a convex set with property on recession cone

Let $\mathbf{E},\mathbf{Y}$ be two euclidean space, $C$ be a non-empty closed convex set in $\mathbf{E}$. The map $A:\mathbf{E}\rightarrow\mathbf{Y}$ is linear, and $N(A)\cap 0^+(C)$ is a linear ...
1
vote
1answer
50 views

Interpreting a theorem about convex sets

I'm studying about linear programming and I bumped into following theorem (I have added my questions into the image): So in 1st rectangle how did we obtain the resulting equation when $\alpha ...
1
vote
1answer
67 views

Help with a proof of a theorem about convex sets

I'm studying the following theorem: Theorem 1. Let $C$ be a convex set and let $\textbf{y}$ be a point exterior to the closure of $C$. Then there is a vector $\textbf{a}$ such that ...
3
votes
1answer
119 views

optimizing a logdet function with respect to a scalar and the Hessian matrix

Given a logdet function $\mathcal{L}(\gamma)$, $$ \mathcal{L}(\gamma) = \log\vert \mathbf{I} + \gamma\mathbf{S} \vert - \mathbf{q}^T(\gamma^{-1}\mathbf{I} + \mathbf{S})^{-1} \mathbf{q}, $$ where ...
0
votes
0answers
166 views

Gradient of a cost function with respect to a matrix that pertains a special structure

Suppose we have a typical logdet function $\mathcal{L}$ $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q}, $$ where ...
2
votes
2answers
172 views

Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
5
votes
1answer
172 views

Characterisation of linearly separable points of a hypercube

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane. E.g. for a cube, the following 4 points (red) are not linearly ...
1
vote
1answer
16 views

Let $Ax = b$ be a system of hyper-planes that form a bounded convex $D$. Can $D$ be partitioned into union of adjacent simplices?

Let $Ax = b$ be linear system that forms $q$ bounded region $D$. If the columns of $A$ are independent, can $D$ be written as a union of adjacent simplices?
3
votes
1answer
63 views

Does convex and radially open imply open?

I want to show that a convex set $A$ is radially open iff $A\cap W$ is open in W, for every finite dimensional linear subspace. Here the 'openness' we are talking about is from any normed space. ...
2
votes
0answers
49 views

Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
1
vote
1answer
42 views

Does the Border (Boundary) Points of a convex shape in the positive quadrant make a convex function?

Let $\mathbb{S}$ be a convex body in 2-D with some non-zero intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis such that $(c,y)\in ...
1
vote
0answers
45 views

Solution of a Quadratic Optimization Problem

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem, \begin{align} ...
1
vote
1answer
39 views

Robust feasibility with halfspace?

Consider a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, such that for all $x \in \mathbb{R}^n$ we have $$ a_1^\top x + b_1 \leq f(x) \leq a_2^\top x + b_2 $$ for some given $a_1, a_2 ...
1
vote
1answer
77 views

About Balanced-Convex Hull of a Set

Definition. $\newcommand{\bco}{\operatorname{bco}}$Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, deonoted $\bco A$, is the intersection of all balanced-convex ...
1
vote
2answers
85 views

Does $\inf_{(a,b)\in A} ax+by=\inf_{(a,b)\in B} ax+by$ imply $A=B$?

Let $A,B\subseteq\Bbb [0,1]\times [0,1]$, and for all $(x,y)\in \Bbb R^2$ with $x^2+y^2=1$, $$\inf_{(a,b)\in A} ax+by=\inf_{(a,b)\in B} ax+by$$ Can we deduce $\overline A=\overline B$.
1
vote
1answer
95 views

Trace Constraints and rank-one positive semi-definite matrices.

Let $C_1$,$C_2$...$C_N$ be $M\times M$ hermitian matrices and $c$ be a given positive constant. Let $W$ be a positive (possibly semi) definite matrix such that \begin{align} \text{trace}\{WC_1\}\geq ...
1
vote
1answer
78 views

Convexity of product of elements from two convex set

Given two convex set $X\subseteq \mathbb{R}^N$,$Y\subseteq \mathbb{R}^{N\times N}$ Given a $x\in X$, is the set $\{z|z=yx,\forall y \in Y\}$ convex? If no, by adding what can force it to be convex? ...
2
votes
1answer
139 views

Hessian of a function that takes matrix arguments

I have a function that that takes a matrix and returns a scalar, $f : \mathbb{R}^{m\times n} \rightarrow \mathbb{R}$. I know how to calculate the derivative of this function with respect to the matrix ...
4
votes
1answer
112 views

Is this matrix function convex or non-convex?

Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
0
votes
2answers
131 views

Convexity of $\log \det(I+(P-\text{tr}(X))\cdot X)$

where $P-\text{tr}(X)>0$ and $X$ is a diagonal matrix with diagonal elements are $(x_1,x_2,\dots x_m)$
0
votes
2answers
89 views

An eigen problem

$K$ is a symmetric positive semidefefinit matrix. $K1 = 0$ (i.e. The sum of elements in each row is $0$. Or in other words matrix $K$ is centered. From this we conclude the smallest eigenvalue of $K$ ...
4
votes
1answer
138 views

Convexity of a Given Function

Is the following function convex or concave? $$f(\mathbf{x}) = \frac{1}{N}\sum_{i=1}^N\sqrt{1+\frac{x_i^2}{\left(\frac{1}{N}\sum_{i=1}^{N}x_i\right)^2}},$$ $\mathbf{x} = [x_1,x_2,...,x_N]^T, x_i \ge ...