  The data-compatibility approach to constrained optimization, proposed here,
strives to a point that is "close enough" to the solution set and whose target
function value is "close enough" to the constrained minimum value. These
notions can replace analysis of asymptotic convergence to a solution point of
infinite sequences generated by specific algorithms. We consider a problem of
minimizing a convex function over the intersection of the fixed point sets of
nonexpansive mappings and demonstrate the data-compatibility of the Hybrid
Subgradient Method (HSM). A string-averaging HSM is obtained as a by-product
and relevance to the minimization over disjoint hard and soft constraints sets
is discussed.
