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What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$? I am interested in appr …

I assume that $\theta(y) = 1$ if $y\geq 0$, and $\theta(y)=0$ else. In that case you can "linearize" your constraint 5 as follows: Add new variables $y_t$ for $t \in T$. Replace constraint 5 with th …

This looks like a convex optimization problem. Fix a convex set $D \subseteq \mathbb{R}^N$. Fix parameters $t_1<t_2$. Let $x=(x_1, \ldots, x_N)$. Define the set $\mathcal{A}$ and function $g:\math …

Here is some insight into Jean's answer, showing what that final problem means. Suppose you transform the problem to finding a vector $(y,t) \in \mathbb{R}^{n+1}$ to solve: \begin{align*} &\mbox{Max …

Unfortunately, no. Here is an example for $n=1$ (1-dimension). For parameters $m>0$, $b\in\mathbb{R}$ define: $$f(x) = (m/2)(x-b)^2 $$ For any $b \in \mathbb{R}$, this function $f$ is strongly conve …