All Questions
8 questions
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
1
vote
0
answers
44
views
In convex optimization we know that the optimum solution is on which hyper plane
We have a standard linear program, I mean a set of inequalities $c_i^Tx\leq b_i$ where $i\in \{1,\ldots ,k\}$ and we want to find $max\{c^Ty| y\in \{\cap \{x|c_i^Tx\leq b_i\}\}$. I put some condition ...
- 19
2
votes
0
answers
385
views
(Quasi) convexity of separately convex homogeneous functions
Consider a function $f:\mathbb{R}^n_{\geq 0}\rightarrow \mathbb{R}$ that is separately convex, i.e. such that $\frac{d^2f}{dx_i^2}\geq 0$ for all $i\in \{1,\dots n\}$. Assume also that $f$ is ...
- 389
3
votes
1
answer
452
views
How to find extreme points of a set related to Minkowski's Theorem?
Let $S^{n-1}$ be the unit sphere in $\mathbb{R}^n$. For $m>n$, we can define $\Lambda$ to be the set
$$\{(\lambda_1, ..., \lambda_m):\sum_{i=1}^m \lambda_i=1, \lambda_i\ge0, and \mbox{ there exist}...
- 1,350
1
vote
0
answers
168
views
Projecting on a a special polyhedron
Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron
$$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$
Notice that $\mathcal P_X$ is symmetric about the origin.
...
- 6,853
1
vote
0
answers
77
views
Projecting on a convex compact polytope with special form
Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ...
- 6,853
1
vote
0
answers
80
views
A version of isotone projection cones
We write $a \succeq b$, where both $a, b \in \mathbb{R}^n$, as a shorthand for $a_i \ge b_i$ for all $1 \le i \le n$. Let $C$ be a closed convex cone in the first orthant of $\mathbb{R}^n$ and denote ...
- 773
1
vote
0
answers
126
views
Convex Optimization related problem
Suppose two non-negative convex functions $f$ and $g$ be given.
We want to solve the following optimization
$$\max_{g\leq\epsilon}f.$$
Now suppose that both $f$ and $g$ can be upper-bounded by a ...
- 1,109