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convex projection and convex multi-objective optimization

JOURNAL OF GLOBAL optimization 2022年第2期83卷 301-327页

作者： Kovacova, Gabriela Rudloffl, Birgit Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria

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 Lohne and Weissing (Math Methods Oper Res 84(2):411-426, 2016). 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 Lohne 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.

关键词： convex projection convex vector optimization convex multi-objective optimization

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Deep learning the efficient frontier of convex vector optimization problems

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JOURNAL OF GLOBAL optimization 2024年第2期90卷 429-458页

作者： Feinstein, Zachary Rudloff, Birgit Stevens Inst Technol Sch Business Hoboken NJ 07030 USA Vienna Univ Econ & Business Inst Stat & Math A-1020 Vienna Austria

In this paper, we design a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater's condition. The proposed machine learning methodology provides both an inner and outer approximation of the weakly efficient frontier, as well as an upper bound to the error at each approximated efficient point. In numerical case studies we demonstrate that the proposed algorithm is effectively able to approximate the true weakly efficient frontier of CVOPs. This remains true even for large problems (i.e., many objectives, variables, and constraints) and thus overcoming the curse of dimensionality.

关键词： convex vector optimization convex multi-objective optimization Machine learning Deep learning Neural networks Efficient frontier

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