grandes-ecoles 2016 Q20

grandes-ecoles · France · x-ens-maths__psi Proof Deduction or Consequence from Prior Results

For $A \in M _ { n , m } ( \mathbb { R } )$ and $b \in \mathbb { R } ^ { n }$ as in Farkas' lemma, we set $$B = \left( \begin{array} { c c c c } 0 & 0 & A & - b \\ 0 & 0 & - A & b \\ - { } ^ { t } A & { } ^ { t } A & 0 & 0 \\ { } ^ { t } b & - { } ^ { t } b & 0 & 0 \end{array} \right)$$ Let, by Tucker's theorem, $y = { } ^ { t } \left( z _ { 1 } , z _ { 2 } , x , t \right) \geq 0$ such that $B y \geq 0$ and $y + B y > 0$.
If $t > 0$ show that $A x = t b$ and conclude.

For $A \in M _ { n , m } ( \mathbb { R } )$ and $b \in \mathbb { R } ^ { n }$ as in Farkas' lemma, we set
$$B = \left( \begin{array} { c c c c } 
0 & 0 & A & - b \\
0 & 0 & - A & b \\
- { } ^ { t } A & { } ^ { t } A & 0 & 0 \\
{ } ^ { t } b & - { } ^ { t } b & 0 & 0
\end{array} \right)$$
Let, by Tucker's theorem, $y = { } ^ { t } \left( z _ { 1 } , z _ { 2 } , x , t \right) \geq 0$ such that $B y \geq 0$ and $y + B y > 0$.

If $t > 0$ show that $A x = t b$ and conclude.

Paper Questions

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