## 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.

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 |
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Pages (from-to) | 301 - 327 |

Journal | Journal of Global Optimization |

Volume | 83 |

DOIs | |

Publication status | Published - 2022 |

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

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