All Questions
17 questions with no upvoted or accepted answers
7
votes
0
answers
217
views
Characterizing matrices with rank constraint
Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...
- 13.9k
3
votes
0
answers
122
views
Convex optimization upper bound for a non-linear optimization
Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
- 287
2
votes
0
answers
178
views
Can we get the exact solution of large-scale quadratic programming problems (quadratic objective, linear inequality constraints) using KKT condition?
Crossposted at Computational Science SE
Consider a quadratic programming problem with the following format:
$$
\text{min} Q(x) = c^Tx+\frac{1}{2}x^TDx \\
$$
$$
\text{s.t.} Ax\leq b, \\
x\geq 0
$$
...
- 21
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
210
views
projection of a matrix to the the space such that the diagonal elements are the greatest
Suppose there is a symmetric matrix $A$ in $\mathcal{S}^n$. I would like to compute the nearest symmetric matrix $X \in \mathcal{S}^n$ such that $X_{ij} \le X_{ii}$, $i ,j \in \{1,...,n \}$. In other ...
- 263
1
vote
0
answers
204
views
Matrix relative condition number
I've been working on some distributed optimization problems and faced a bit of a challenge with the following question.
Given $A_1, A_2, .., A_m \in M_n({\mathbb{R})} $ symmetric positive definite ...
- 11
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
150
views
Minimax optimization of diagonal entries of function of matrix
Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and ...
- 287
1
vote
0
answers
138
views
Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?
Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...
- 13.9k
1
vote
0
answers
483
views
minimize norm of matrix product
I have the matrix Product $PAP^H$ and I need to minimize $\|(PAP^H)^{-1}\|^2$ (over $P$ and Frobenius norm).
$A$ is a positive definite Hermitian matrix and $P$ has the structure
$$P=\left[\begin{...
- 129
1
vote
0
answers
100
views
Changing a nonlinear equality constraint into some conic inequality plus rank constraint
If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
- 33
0
votes
0
answers
72
views
Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 43
0
votes
0
answers
156
views
Optimal solution of complex optimization problem
Let $Q(x)=a(x)e^{jb(x)}$ be a complex function of $x$. We want to approximate this function with $R(x)=\alpha e^{jx\beta}$ such that
\begin{align}
\text{arg}\min_{\alpha,\beta} \int_{-\frac{A}{2}}^{\...
- 287
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
121
views
Closed form solution to an equation
Let $X \in \mathbb{R}^{n \times d}, w \in \mathbb{R}^d, y \in \{\pm 1
\}^{n}, \alpha \in (0, 1)$. Consider the equation
$$ X^{\top}(Xw-y)+\alpha \|w\|_{2}X^{\top}\operatorname{sign}(Xw-y)+\alpha\frac{...
0
votes
0
answers
80
views
Finding gradient of an optimization
I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...
- 161
0
votes
1
answer
153
views
Difference of two optimization problem's optimal value
Let we have two following optimization problems:
\begin{align}
\text{(P1)}\quad \alpha_1 = \max_{x_1,\ldots,x_M} &\quad \sum_{m=1}^{M}\log(1+f_m(x_1,\ldots,x_M))\\
\textrm{s.t.} &\quad \...
- 287