convxy
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Find a dual problem with one dual decision variable to the problem of finding the orthogonal projection of a given vector
2 votes

I've checked my answer and it works. Here is a Matlab code that I've written.

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Find the dual problem of $\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}$
1 votes

$$\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}=\min_{x,z_i}\{||z_1||+||z_2||+||z_3||,a_i\in\mathbb{R}^n\,z_i=x-a_i\}$$ and the Lagrangian is:\ $L(x,z,\lambda)= ||z_1||+||z_2||+||z_3||+\...

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Let $A\in\mathbb{R}^{m\times n},c\in\mathbb{R}^n$. Show that exactly one of the following two systems is feasible:
1 votes

Assume that A is feasible we'll show that B is not feasible. $A^Ty\geq c\iff x^TA^Ty\geq x^Tc\iff(Ax)^Ty\geq c^Tx>0\to (Ax)^Ty>0$ but $Ax\geq0$ therefore $y$ must satisfy that $y>0$ and B is ...

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