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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.

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Both solutions being equal is a property of strong duality. The uniqueness of the solution requires further assumptions, namely strict convexity of the primal problem and satisfaction of a constraint qualification, e.g., Slater's.

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