  We revisit the decoupling approach widely used (often intuitively) in
nonlinear analysis and optimization and initially formalized about a quarter of
a century ago by Borwein & Zhu, Borwein & Ioffe and Lassonde. It allows one to
streamline proofs of necessary optimality conditions and calculus relations,
unify and simplify the respective statements, clarify and in many cases weaken
the assumptions. In this paper we study weaker concepts of quasiuniform
infimum, quasiuniform lower semicontinuity and quasiuniform minimum, putting
them into the context of the general theory developed by the aforementioned
authors. On the way, we unify the terminology and notation and fill in some
gaps in the general theory. We establish rather general primal and dual
necessary conditions characterizing quasiuniform $\varepsilon$-minima of the
sum of two functions. The obtained fuzzy multiplier rules are formulated in
general Banach spaces in terms of Clarke subdifferentials and in Asplund spaces
in terms of Fr\'echet subdifferentials. The mentioned fuzzy multiplier rules
naturally lead to certain fuzzy subdifferential calculus results. An
application from sparse optimal control illustrates applicability of the
obtained findings.
