Convex approximation sets for multiobjective optimization problems are a
well-studied relaxation of the common notion of approximation sets. Instead of
approximating each image of a feasible solution by the image of some solution
in the approximation set up to a multiplicative factor in each component, a
convex approximation set only requires this multiplicative approximation to be
achieved by some convex combination of finitely many images of solutions in the
set. This makes convex approximation sets efficiently computable for a wide
range of multiobjective problems - even for many problems for which (classic)
approximations sets are hard to compute.
In this article, we propose a polynomial-time algorithm to compute convex
approximation sets that builds upon an exact or approximate algorithm for the
weighted sum scalarization and is, therefore, applicable to a large variety of
multiobjective optimization problems. The provided convex approximation quality
is arbitrarily close to the approximation quality of the underlying algorithm
for the weighted sum scalarization. In essence, our algorithm can be
interpreted as an approximate variant of the dual variant of Benson's Outer
Approximation Algorithm. Thus, in contrast to existing convex approximation
algorithms from the literature, information on solutions obtained during the
approximation process is utilized to significantly reduce both the practical
running time and the cardinality of the returned solution sets while still
guaranteeing the same worst-case approximation quality. We underpin these
advantages by the first comparison of all existing convex approximation
algorithms on several instances of the triobjective knapsack problem and the
triobjective symmetric metric traveling salesman problem