All Questions
8 questions
0
votes
1
answer
84
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Non linear optimization wrt to square matrix
Given vectors $\pmb{y}_0, \pmb{y}_1, \dots \pmb{y}_t \in \mathbb{R}^n$, let $F : \mathbb{R}^{n \times n} \to \mathbb{R}_0^+$ be defined by
$$F(W) := \left\| \pmb{y}_1 - W \pmb{y}_0 \right\|^2 + \left\|...
1
vote
1
answer
121
views
Projection to triangular matrices whose symmetrisation has positive eigenvalues
Let $\lambda>0$, and $\mathbb{R}^{n\times n}$ be the space of matrices equipped with Frobenius norm. Consider the set
$$
D=\{A\in \mathbb{R}^{n\times n}\mid \textrm{$A$ is lower triangular and ...
- 13.5k
1
vote
1
answer
488
views
Using KKT conditions to solve the following problem?
I'm trying to optimize the following equation:
$$min_y \frac{1}{2}||y-z||^2_2 \\ s.t \ (\mathbf{y} - \mathbf{\sigma})^T\mathbf{A}(\mathbf{y}-\mathbf{\sigma}) \leq 1$$
Note: $A ⪰ 0$
I've started out by ...
- 375
0
votes
0
answers
224
views
How to use barrier method for constraints like $X \succ 0$?
When reading about interior-point methods in Stephen Boyd & Lieven Vandenberghe's Convex Optimization, a question arose about how to use barrier method for the constraint $X$ is positive definite, ...
- 951
2
votes
0
answers
123
views
Is Proximal Gradient Method (PGM) suitable for solving matrix optimization problems?
Generally, the well-known Compressed Sensing (CS) task can be modeled by the following vector optimization problem with a pre-defined convex regularizer $\mathcal{R}(\cdot)$:
$$
\underset{\mathbf{\hat{...
- 648
0
votes
0
answers
106
views
Minimization of Frobenius norm with nuclear norm penalization
Define $\mathcal{M}_n$ and $\mathcal{S}_n$ as the space of $n\times n$ real matrices and $n\times n$ symmetric real matrices, respectively. I want to solve the problem
$$
\min_{A\in S_n}\frac{1}{2}\|...
- 1,426
0
votes
0
answers
174
views
Approximating Hessian with BFGS for a matrix variable
So as we know the approximation of the inverse of the hessian matrix using the BFGS method
is calculated with the following formulas :
$$q_{k+1} = (I-p_k s_k (y_k)^T)q_k(I- p_k y_k (s_k)^T) + p_k ...
- 49
3
votes
4
answers
1k
views
Linear Least Squares with Linear Equality Constraints - Iterative Solver
I am looking for iterative procedures for solution of the linear least squares problems with linear equality constraints.
The Problem:
$$ \arg \min_{x} \frac{1}{2} \left\| A x - b \right\|_{2}^{2}, \...
- 31