Prove that a saddle point for the Lagrangian function - then it's a global min for constraint problem I have to prove: If $(x,u,v)$ is a saddle point for the Lagrange function then $x$ is a global minimum point for the corresponding constrained problem.
I tried to find hints on the web -- without success. 
I started with the defintion of a saddle-point:
$$L(x,u,v) \le L(x,u_1,v_1) \le L(x_1,u_1,v_1) \text{ for all } x,u,v$$
 A: I assuming that you have the problem $\min \{ f(x) | g(x) \le 0, h(x) = 0 \}$, and $L(x,u,v) = f(x) + u^T g(x) + v^T h(x)$, with appropriate
dimensions.
The saddle point condition is that there are $(x^*,u^*, v^*)$ with
$u^* \ge 0$ such that 
$L(x^*,u,v) \le L(x^*,u^*, v^*) \le L(x,u^*, v^*)$ for all $x, u \ge 0, v$.
First we note that $x^*$ is feasible for the original problem. To see
this, note that if $h_k(x^*) \neq 0$, then
$L(x^*,u, v+t e_k) = L(x^*,u, v)+t h_k(x^*)$, and hence by suitable
choice of $t$ the value $L(x^*,u, v+t e_k)$ is unbounded above, which
contradicts the saddle point assumption. In the same way we
verify that $g(x^*) \le 0$ (note that $u$ is restricted to be non
negative, $u \ge 0$).
Next we note that $L(x^*,u^*, v^*) = f(x^*)$. It is clear that
$(v^*)^T h(x^*) = 0$ since $x^*$ is feasible. If $(u^*)^T g(x^*) \neq 0$, then we must have $(u^*)^T g(x^*) < 0$ (again, since $x^*$ is feasible and $u^* \ge 0$),
in which case $L(x^*,u^*, v^*) < L(x^*,0, v^*)$, which contradicts
the saddle point assumption.
Hence we have $f(x^*) \le L(x,u^*, v^*)$ for all $x$. Hence for
feasible $x$, we have $g(x) \le 0$ and $h(x) = 0$, in which case
$f(x^*) \le f(x) + (u^*)^T g(x)+ (v^*)^T h(x) \le f(x)$, and so
we see that $x^*$ is a global minimiser.
Aside: The saddle point assumption is, in general,
 a fairly strong assumption.
A: For more details on this subject See the following URLs: https://mpra.ub.uni-muenchen.de/50598/1/MPRA_paper_50598.pdf
     and: https://pdfs.semanticscholar.org/9039/26a0558775ed243d39e4879894d6c34b5ea6.pdf
and:https://www.researchgate.net/publication/265939872_Mathematical_Optimization_and_Economic_Analysis
