# Convex Projection and Convex Vector Optimization

Publication: Scientific journalJournal articlepeer-review

## Abstract

In this paper we consider a problem, called convex projection, of projecting a convex set
onto a subspace. We will show that to a convex projection one can assign a particular multiobjective
convex optimization problem, such that the solution to that problem also solves the
convex projection (and vice versa), which is analogous to the result in the polyhedral convex
case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In
practice, however, one can only compute approximate solutions in the (bounded or selfbounded)
convex case, which solve the problem up to a given error tolerance. We will show
that for approximate solutions a similar connection can be proven, but the tolerance level needs
to be adjusted. That is, an approximate solution of the convex projection solves the multiobjective
problem only with an increased error. Similarly, an approximate solution of the
multi-objective problem solves the convex projection with an increased error. In both cases the
tolerance is increased proportionally to amultiplier. Thesemultipliers are deduced and shown
to be sharp. These results allow to compute approximate solutions to a convex projection
problem by computing approximate solutions to the corresponding multi-objective convex
optimization problem, for which algorithms exist in the bounded case. For completeness, we
will also investigate the potential generalization of the following result to the convex case. In
Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for
the polyhedral case, how to construct a polyhedral projection associated to any given vector
linear program and how to relate their solutions. This in turn yields an equivalence between
polyhedral projection, multi-objective linear programming and vector linear programming.
We will show that only some parts of this result can be generalized to the convex case, and
discuss the limitations.
Original language English 301 - 327 Journal of Global Optimization 83 https://doi.org/10.1007/s10898-021-01111-1 Published - 2022

## Austrian Classification of Fields of Science and Technology (ÖFOS)

• 101024 Probability theory
• 101007 Financial mathematics
• 502009 Corporate finance