Search Results

My preliminary experiments show that the answer is no. Here is why. In the paper Constructing copositive matrices from interior matrices, the following matrix (from Horn's quadratic form) is mentione …

Define, $$A(\alpha) = \sum_i \alpha_i A_i.$$ In your case (ignoring constraints for the moment), you have $$\min\quad\mbox{trace}(A(\alpha)^{-1}) = \sum_i e_i^TA(\alpha)^{-1}e_i,$$ which makes i …

Since we are dealing with real number computation, we cannot use the traditional Turing machine for complexity analysis. There will always be some $\epsilon$s lurking in there. That said, when analyz …

Let $p^*$ and $q^*$ be the conjugate exponents. Some (slightly laborious) algebra shows that the dual-norm is $\|A\|_{p^*,q^*}$. The conjugate function is the indicator function for the (unit) dual-no …

I looked at the problem again and saw that it can be simplified nicely. In summary: If $\alpha, \beta > 0$, the solution is independent of $\alpha$ and $\beta$ The solution can be easily computed ( …

Here is a partial solution to the first question in the original post. Let's look at the equation \begin{equation}\label{1}\tag{1} \sum\nolimits_{i=1}^m (X+ \Theta_i)^{-1} = Q. \end{equation} Lemma …

Not an answer, but some comments and a suggestion. The place I have previously seen $(1-a_ib_j)^{-1}$ is in Theorem~7.12.1 in Enumerative Combinatorics volume 2, by R. Stanley, where he mentions Cauc …

One approach is to solve the optimization problem: \begin{equation*} \min_x\quad \|Ax-y\|_\infty. \end{equation*} This is a nonsmooth optimization problem, but is amenable to a variety of scalable opt …

Have a look at the following slides (several pointers are in there) Optimization on the Stiefel manifold The point is that you can directly remain on the manifold while optimizing, so no explicit "c …

The said claim follows from the following general result on elementary symmetric polynomials, denoted $e_k$ below. $\newcommand{\vx}{\mathbf{x}}\newcommand{\vy}{\mathbf{y}}$ Theorem A (S. 2018). $ …

The paper cited in my answer here provides a detailed proof of the two-block case of alternating minimization (block coordinate descent). In particular, as mentioned in my comment, the convergence fol …

Your problem is a special case of a Fractional Linear Program, so as such following the recipe provided on Wikipedia you should be able to solve it by using a reformulation to an equivalent linear pro …

As far as I know, there is no "analytic" solution to your problem. Fortunately, this problem happens to be a special case of optimizing (over $\mathbb{C}^n$) a quadratic function subject to two quadra …

Your minimization problem is equivalent to \begin{equation*} \min_{R^TR=I}\quad\prod_{i=1}^p r_i^T\Sigma r_i, \end{equation*} and it can be shown (using Hadamard's determinant inequality and some more …

In practice, you would not want to run a vanilla prox-gradient method that requires knowledge of the Lipschitz constant. Instead, you'd use a method that combines line-search (these notes give a nice, …