Assume we meet this problem: $$ \min_x f(x)\\ s.t. g(x)=0, h(x)\leq 0 $$ Let's assume the solution is $(x^*,\nu^*,\lambda^*)$
Its dual problem is $$ \max_{\nu,\lambda} l(\nu,\lambda)\\ s.t. \lambda\geq 0 $$ where $l(v,\lambda)=\min_xf(x)+\nu g(x)+\lambda h(x)$. The solution is $\nu^{**},\lambda^{**}$.
Can we prove $\nu^*=\nu^{**}$ and $\lambda^*=\lambda^{**}$ with strong duality and also $(x^*,\nu^*,\lambda^*)$ being unique, or is it wrong? Of course, if you can prove this with only strong duality, it would be better.
Thank you for finishing reading this. I appreciate it.