Variational principles for approximating thermodynamic potentials in the inverse Ising problem: How to go from double to single extremum?

I’m trying to wrap my head around section 2.2.6 (on variational principles) in the following paper (on the inverse Ising problem): https://arxiv.org/abs/1702.01522

Here the authors explain how to use variational principles to approximate the following thermodynamic potentials:
$$ F(boldsymbol{J}, vec{h}) = – ln{Z(boldsymbol{J}, vec{h})}, $$

$$ S(boldsymbol{chi}, vec{m}) = min_{boldsymbol{J}, vec{h}}left{{-sum_{i} h_i m_i -sum_{i < j} J_{ij} chi_{ij} – F(boldsymbol{J}, vec{h}) }right}, $$

$$G(boldsymbol{J}, vec{m}) = max_{vec{h}}left{ { sum_{i} h_i m_i + F(boldsymbol{J}, vec{h})} right}. $$

I understand that $F(boldsymbol{J}, vec{h})$ can be obtained by noting that for a distribution $q$,

$$D_{KL}(q||p) = langle H rangle_q + langle ln q rangle_q + ln Z(boldsymbol{J}, vec{h}),$$
so that, using the notation $U[q] equiv langle H rangle_q$, and $S[q] equiv langle ln q rangle_q$, we have:
$$F(boldsymbol{J}, vec{h}) = langle H rangle_q + langle ln q rangle_q – D_{KL}(q||p) longrightarrow F(boldsymbol{J}, vec{h}) = min_q{ U[q] – S[q]}.$$

However, the authors then go on to say that it is easy to see (using Lagrange multipliers) that the remaining two potentials can be expressed as:

$$G(boldsymbol{J}, vec{m}) = max_{vec{h}}left{ { sum_{i} h_i m_i + min_q{ U[q] – S[q]}} right} = min_{q in mathcal{G}}left{-sum_{i < j} J_{ij} langlesigma_i sigma_j rangle_q – S[q]right}, $$
and
$$S(boldsymbol{chi}, vec{m}) = min_{boldsymbol{J}, vec{h}}left{{-sum_{i} h_i m_i -sum_{i < j} J_{ij} chi_{ij} – min_q{ U[q] – S[q]} }right} = max_{q in mathcal{S}}{S[q]}.$$
Where $mathcal{G} equiv {q : langle sigma_i rangle_q = m_i},$ and $mathcal{S} equiv {q : langle sigma_i rangle_q = m_i & langle sigma_i sigma_j rangle_q = chi_{ij} }$.

My question is how do I, via Lagrange multipliers, go from these double extremum expressions to the single extremum expressions. Thanks in advance for any help I may recieve.