\begin{align} f_{0}\left(x^{*}\right) &= \text{inf}_{x} \left(f_{0}\left(x\right) + \sum_{i=1}^{m} \lambda_{i}f_{i}\left(x\right) + \sum_{i=1}^{p} v_{i}h_{i}\left(x\right)\right) \\ &\leq f_{0}\left(x\right) + \dots \\ &\leq f_{0}\left(x\right) \end{align}