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Yes, the same approach can be used for complex matrices, with the constraint becoming $\left[ \begin{array}{rr} W_{1} & X \\ X^{H} & W_{2} \\ \end{array} \right] \succeq 0 $ Here, the constraint sa …

No. The problem with this simple minded approach is that the sequence of constraints that you add might go on forever without adding a critical constraint. Consider the following example problem. …

In general, your problem can become infeasible under arbitrarily small perturbations $\Delta b$. For example, consider the problem $\min X_{11}+X_{22}$ subject to $X_{11}=0$ $X_{22}=0$ $X \suc …

You need to start by by being explicit about the "equivalence" between the problems. Do you mean that "For every $s$, there is a $t$ such that if $x^{ * }$ is an optimal solution to the first problem …

Here's a solution for the specific case where $f(x)=\| x \|_{1}$ and $g(x)=\| y-Ax \|_{2}$. The same appraoch applies to $g(x)=\| A^{T}(y-Ax) \|_{\infty}$. There are two cases. If $g(0) \leq s$, …

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 …

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 …