All Questions
12 questions with no upvoted or accepted answers
3
votes
0
answers
282
views
Continuum of Lagrange multipliers, duality gap, and minimax theorem
Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
- 781
3
votes
0
answers
122
views
Convex optimization upper bound for a non-linear optimization
Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
- 287
2
votes
0
answers
46
views
Notion of distance between linear programs
Consider the linear programming problem
\begin{align}
\max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1
\end{align}
where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a ...
- 1,421
2
votes
0
answers
64
views
Finding orthogonal basis with constraint
Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...
- 909
1
vote
0
answers
98
views
Solution of a simple optimization problem
Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...
- 287
1
vote
0
answers
42
views
Computation of sub-gradient for a concave envelope
Let $x_1<\cdots<x_n$ be $n$ points on real line and $g=(g_1,\cdots, g_n)\in\mathbb R^n$ be the scattered data. Let $u_g: [x_1,x_n]\to\mathbb R$ be the linear interpolation of $g_1,\cdots, g_n$, ...
- 345
1
vote
0
answers
81
views
Maximizing sum of homogeneous functions of order one over a polytope
Let $f_i: \mathbb{R}^n\rightarrow \mathbb{R}$ be
concave, increasing (i.e., if $x\geq y$ where the inequality is entry wise, we have $f_i(x)\geq f_i(y)$), and a
homogeneous function of order one for ...
- 393
0
votes
0
answers
55
views
Relationship of optimal solutions between the total function and the sub function
This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
- 1
0
votes
0
answers
124
views
The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
0
votes
0
answers
108
views
How to find a set given its support function
Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\...
- 53
0
votes
0
answers
68
views
Convex optimization under asymmetric loss in infinite dimensional space
The following problem is common in financial economics
$$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$
That is, given a random variable $y(\theta)$ ($\...
- 45
0
votes
0
answers
104
views
Big eigenvalues of a special stochastic matrix
Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq |\lambda_2|(M)....
- 105