Consider the following problem from textbook Convex Optimization Algorithm p.10:
\begin{equation} \begin{aligned} &{\text{min}} & & F(x)+G(y)\\ & \text{s.t.} & & x \in \mathcal{X},\ \ y \in \mathcal{Y} \\ & & & Ax+By=c \\ \end{aligned} \end{equation}
By partitioning, it can be written as
\begin{equation} \begin{aligned} &{\text{min}} & & F(X) +\underset{By=c-Ax,\ \ y\in \mathcal{Y}}{\text{inf}} G(y)\\ & \text{s.t.} & & x \in \mathcal{X} \\ \end{aligned} \end{equation}
It seems this trick just transform the original problem to a new one with only one variable $x$. The condition is that the oprinal problem should be seperable.
There are two constraints, one is $y \in \mathcal{Y}$ and a equality constraint. Can we say more about the second problem?
The go-to method for attacking your problem is the ADMM (Alternating Directions Method of Multipliers).
Your can learn ADMM from Stephen Boyd's ADMM Page. For a more in-depth theory, see
Eckstein, J., Bertsekas, D.P.: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 55 (1992)
Glowinski, R., Marroco, A.: Sur l’approximation, par ́elements finis d’ordre un, et la resolution, par p ́enalisation-dualit ́e d’une classe de problemes de Dirichlet non lin ́eaires. ESAIM: Mathe- matical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique 9 (1975)
Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Computers & Mathematics with Applications 2 (1976)
Hope this helps.