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As mentioned by another poster, the work of Nesterov and Nemirovski summarized in Interior-Point Polynomial Algorithms in Convex Programming showed that many convex optimization problems (including li …

The real issue here is the constraint $\sum_{i} x_{i}1_{x_{i}>a} < b $ whose left hand side has horrible discontinuities. Rather than using a solver designed for problems with continuous variable …

I'd suggest that you start by encoding your signal in terms of the symbols +1 and -1 rather than 0 and 1. If you have two signals x and y, then take elementwise products of x and y and sum to get a m …

As others have already indicated, there's a good chance that you don't actually need $(E-aX)^{-1}$, but rather $(E-aX)^{-1}v$ for some vector $v$. In that case, you might be better off using an itera …

Some books to start with for background reading would include: Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Springer, 2003. Y. Nesterov and A. Nemirovsky, Interior Poin …

Your problem has been studied extensively in the context of trust region methods for optimization, and there are a number of algorithms that have been developed. See for example: W. W. Hager, Mini …

Your notation is somewhat confusing, in that you apply the subscript $i$ to $w$, and have a vector $w_{i}$, but don't use $i$ in any meaningful way in your problem. I'm going to take the liberty of …

First, I'm assuming that your nonconvex feasible set $D$ is a subset of some larger convex set $C$ on which the objective function $f(x)$ is defined. It doesn't really make sense to talk about $f(x)$ …

Unfortunately, this problem can't be represented by an LP, since your feasible region is in general nonconvex, and the feasible region of an LP (being the intersection of a bunch of half spaces) is al …

Where do the different objective functions come from? Perhaps if we knew more about your problem we could make a helpful suggestion about how to approach it. If the objective functions are totall …

These kinds of feasible sets can often be written in terms of second order cone programming and/or semidefinite programming constraints. If that's the case, then optimizing over the feasible set is r …

As a practical matter, a lot depends on $n$ and the dimension of $x$. If the problem is small enough then you might not be in deep trouble. If the problem is large (e.g. $x$ might have thousands of …

Many (but not all) problems involving the eigenvalues of a graph are convex optimization problems that can be formulated as semidefinite programming problems. There are a number of "tricks" that you …

It's important that you understand the definition of a neighborhood used in this book. This is definition 1.3 on page 7. $N$ is not a set but rather a function that maps a solution to the problem to …

As in your previous question, this is a nonconvex optimization problem, so it won't be LP, SOCP, or SDP representable. You've only got a 21 dimensional problem, and the constraint functions have ea …