  This paper introduces a new computational methodology for determining
a-posteriori multi-objective error estimates for finite-element approximations,
and for constructing corresponding (quasi-)optimal adaptive refinements of
finite-element spaces. As opposed to the classical goal-oriented approaches,
which consider only a single objective functional, the presented methodology
applies to general closed convex subsets of the dual space and constructs a
worst-case error estimate of the finite-element approximation error. This
worst-case multi-objective error estimate conforms to a dual-weighted residual,
in which the dual solution is associated with an approximate supporting
functional of the objective set at the approximation error. We regard both
standard approximation errors and data-incompatibility errors associated with
incompatibility of boundary data with the trace of the finite-element space.
Numerical experiments are presented to demonstrate the efficacy of applying the
proposed worst-case multi-objective error in adaptive refinement procedures.
