Lemma 4 in How easy is local search by Johnson, Papadimitriou, and Yannakakis, states:
If a PLS problem is NP-hard then NP = P
So assuming L is a PLS problem (polynomial local search problem) that is NP-hard, then every NP problem can be reduced to L.
Usually if $P_1$ is reducible to $P_2$ (being two decision problems) then there is a polynomial time function $f$ such that:
$I_1$ is a positive instance of $P_1$ $\qquad$ if and only if $\qquad$ $f(I_1)$ is a positive instance of $P_2$
However I'm not completely sure how would a reduction from a decision problem to a local search problem be.
I guess my question is: if $P$ is a decision problem which is reducible to the problem $L$, mentioned above, with the polynomial time function (or procedure) $g$ then complete the sentence:
$I$ is a positive instance of $P$ $\qquad$ if and only if $\qquad$ $g(I)$ is ....................