Reduction/transformation from MinCostSat to MaxSat
First thing to realize is that minimization can be rephrased as maximization of the negated literal (and vice a versa). So minimizing $x+y$ is the same as  maximizing $\lnot x + \lnot y$. So to get a partial maxsat problem you just add the unit clauses $\lnot x$, $\lnot y$ as soft clauses and the original clauses as hard. (partial MaxSAT is crucial here, weighted partial MaxSAT is needed if you have  coefficients in the minimization function)