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First eliminate $x_1$ by solving an ordinary least squares, and then you need to essentially solve a problem of the form: $\min x_2^TMx_2$ s.t. $\|x_2\|=g$, for appropriate $M$. This problem is the fa …

A classical fact that is often used is that a continuous function on a compact set attains its minimum AND maximum More flexible is, however, the following Theorem (see e.g., Thm 1.9 in Rockafellar & …

After your edit, the problem becomes equivalent to the convex problem: $$\min x^TAx - b^Tx + C\|x\|_1$$ This is a very-well studied problem, and here are the keywords that will help you find algorit …

For your problem, where the relations between objects are specified via a distance matrix, the formulation of Correlation Clustering, seems to be more appropriate. Here you do not need to pick $k$ in …

I think what you are actually after is the elusive combinatorial interpretation of SDPs. While this is in general a rather tricky issue, a very nice piece of work that is a good starting point, and br …

There are several methods that one could apply to this problem. I don't have time to write out a full solution, but here's a quick idea. Replace $\Omega$ by $\hat\Omega = \Omega + \delta_+$, where $\d …

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 …

One "quick" idea for solving the associated QP is to try CVXOPT, which is a nice package for doing convex optimization. Since the Hessian of your QP is tridiagonal, implementing a customized solver th …

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 …

Building on Nesterov's work, in his Ph.D thesis, Peter Richtarik considers first-order methods with relative error of approximation guarantees. I haven't looked in too closely, but I am sure that a la …

This function is not differentiable (consider $A=0$). If you are interested in learning about its subdifferential (and more on subdifferential of spectral functions), please refer to the excellent pap …

I think in several cases "best self interest" can lead to overall poorer solutions than solutions with interaction. More formally, the self-interest maximizing version falls into the domain of tradi …

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, …

Here is a simple argument. First, define $r_t = \|x_t - x^\ast\|$, where $x^\ast$ denotes an optimal point. Since $f$ is convex, we have \begin{equation*} f(x^\ast) \ge f(x_t) + \langle f'(x_t), x …

Here is one way to solve your problem numerically using CVX under Matlab: % assuming we have defined v and m above cvx_begin sdp variable d(2*m,2*m) diagonal maximize det_rootn(v*d*v') su …