The author of a optimization paper states this:
maximizing the minimum value of a function over the index set is equivalent to minimizing a parameter subject to the negative of all functional values over the index set being smaller than or equal to $\underset{\textbf{u}}{\text{max}}\ \underset{t\in \Omega}{\text{min}} \ f(t,\textbf{u})$, that is is equivalent to $\underset{\textbf{u}}{\text{min}\ v }$ subject to $-f(t,\textbf{u}) \leq v ,\forall t\in \Omega$
Could anyone help me out why this transformation problme is valid?
Another trasnformation problem: $\text{min}\ J = \underset{a}{\text{max}}\left (\underset{b}{\text{max}}\ y(t) - \underset{b}{\text{min}}\ y(t)\right ) $
Its transformation is equal to : $$\text{min}\ J = \alpha_1v_1 + \alpha_2v_2 $$
$$s.t.\ y(t) \leq v_1 \ \forall t\text{ satisfy a and b}$$ $$\ \ \ \ -y(t) \leq v_2 \ \forall t\text{ satisfy a and b}$$
I do not know how this transformation forms. Why two different scalars? Could you help?