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This is very possibly a homework exercise, so I won't provide a complete solution. The computation of the gradient of $\| y-f(\beta)-J\delta \|^{2}$ simply isn't correct. If $y$ is a vector of siz …

Since you haven't assumed anything about the distribution of points in S, any algorithm for this problem must examine all $n$ points, and thus take $\Omega(n)$ time. The straight forward algorithm th …

If $M$ is a symmetric and positive definite matrix of size $n$ by $n$, then the range of $M$ is $R^{n}$. Unless $A^{T}$ also has $R^{n}$ as its range, you can't make this happen. You'd have a much …

I'll assume in my answer that you're using the convention that the primal problem is: $\max tr(CX) $ subject to $tr(A_{i}X)=b_{i}\;\; i=1, 2, ..., m$ $X \succeq 0$ where $X$ is an $n$ by $n$ sym …

You should look into the topic of "parameteric linear programming". In certain situations where the problem data vary linearly with parameters, you can derive formulas for the optimal solution and op …

Do you really want to solve this problem, or do you just want to get a good approximation to the function? I would bet that your overall goal is actually to get a fast and reasonably accurate approxi …

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 …

There are a number of widely used primal--dual interior point codes for SDP, including SeDuMi, SDPA, SDPT3, and CSDP. Of these, SeDuMi approaches this problem by using the self dual embedding, while …

No, that's not quite the generalization that you'll get when you extend the Schur complement theorem for positive definite matrices to negative definite matrices. Try using the fact that a matrix $X$ …

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 …

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 $f(x)$ …

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 …

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 …

You haven't said so explicitly, but it sounds as though your function evaluations may also be noisy in that the function value is the result of a Monte Carlo simulation that incorporates random number …

It might help to understand what background you already have. Have you taken any courses in ordinary differential equations? partial differential equations? real analysis? What mathematics courses h …