I want to compute the projection of a vector $\left( x\right) _{1\leq i,j\leq n}\in \lbrack 0,1]^{n\times n}$ on the following discrete set% $$ S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\text{ }% \sum_{j=1}^{n}x_{i,j}=1,\forall i\in \{1,...,n\}\right\} . $$

I think that we have to make it in two steps, the first we project $x$ on the set $$ \left\{ x\in \lbrack 0,1]^{n\times n}:\min (x_{i,j},x_{j,i})=0\right\} , $$ then we project on the second space $$ \left\{ x\in \{0,1\}^{n\times n}:\sum_{j=1}^{n}x_{i,j}=1,\forall i\in \{1,...,n\}\right\} , $$ by taking $\underset{i}{\max }x_{i,j}=1$ and the others zero for each $i$.

Is my projection optimal (not necessarily unique)?. Is there any algorithm to solve this problem?.

Many thanks for considering my request.

I want to compute the projection of a vector $\left( x\right) _{1\leq i,j\leq n}\in \lbrack 0,1]^{n\times n}$ on the following discrete set $$ S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\text{ }% \sum_{j=1}^{n}x_{i,j}=1,\forall i\in \{1,...,n\}\right\} . $$

I think that we have to make it in two steps, the first we project $x$ on the set $$ \left\{ x\in \lbrack 0,1]^{n\times n}:\min (x_{i,j},x_{j,i})=0\right\} , $$ then we project on the second space $$ \left\{ x\in \{0,1\}^{n\times n}:\sum_{j=1}^{n}x_{i,j}=1,\forall i\in \{1,...,n\}\right\} , $$ by taking $\underset{i}{\max }x_{i,j}=1$ and the others zero for each $i$.

Is my projection optimal (not necessarily unique)?. Is there any algorithm to solve this problem?.

Many thanks for considering my request.