Assume both $f(x)$ and $g(x)$ are non-negative and non-convex scalar functions. Further assume that for any $x_1,\, x_2$ it holds $f(x_1) \le f(x_2)$ if and only if $g(x_1) \ge g(x_2)$.
Can we say that for any $\alpha \ge 0$ in which $\{x: g(x)\le \alpha \}$ is not empty, there exist a $0\le\lambda\le 1$ such that \begin{equation} \min_x f(x) \ \ \ \mathrm{s.t.} \ \ g(x)\le \alpha \end{equation}
and \begin{equation} \min_x \big\{(1-\lambda) f(x) + \lambda g(x)\big\} \end{equation}
have same solution?