reducing a decision problem to a local search problem
To compare search problems, polytime many-one reductions work as follows: $A$ is reducible to $B$ if there are polynomial-time computable functions $f$, $g$ such that for any $A$-instance $x$, if $y$ is a $B$-solution to $g(x)$, then $f(x,y)$ is an $A$-solution to $x$.

We can view decision problems as special cases of search problems with unique binary answers, but note that the reduction notions disagree. In particular, we cannot really see the difference between yes and no instances in the search problem view.