Convex Projection and Convex Multi-Objective Optimization

TL;DR

This paper establishes a theoretical link between convex projection problems and multi-objective convex optimization, enabling approximate solutions to be computed efficiently with controlled error increases.

Contribution

It demonstrates that convex projection can be reformulated as a multi-objective convex optimization problem, extending known polyhedral results to the convex case with error tolerance adjustments.

Findings

Approximate solutions to convex projection relate to multi-objective problems with proportional error increases.

The connection between convex projection and multi-objective optimization is established and proven.

Limitations of generalizing polyhedral projection results to convex sets are discussed.

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 multi-objective 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 arXiv:1507.00228. In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) 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 multi-objective 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 a multiplier. These multipliers 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 arXiv:1507.00228, 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.