Optimal Transportation of Vector-Valued Measures

Non-archimedean transportation problems and Kantorovich ultra-norms

We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field $\mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which…

General auction method for real-valued optimal transport

Optimal transportation theory is an area of mathematics with real-world applications in fields ranging from economics to optimal control to machine learning. We propose a new algorithm for solving discrete transport (network flow) problems, based on classical auction methods. Auction methods were originally developed as an alternative to the Hungarian method for the assignment problem, so the classic auction-based algorithms solve integer-valued optimal transport by converting such problems into assignment problems. The general transport auction method we propose works directly on real-valued …

On semi-discrete sub-partitions of vector-valued measures

We introduce a concept of optimal transport for vector-valued measures and its dual formulation. In this note we concentrate on the semi-discrete case and show some fundamental differences between the scalar and vector cases. A manifestation of this difference is the possibility of non-existence of optimal solution for the dual problem for feasible primer problems.

Interval Valued Vector Variational Inequalities and Vector Optimization Problems via Convexificators

In this paper, we consider interval-valued vector optimization problems $(IVOP)$ and derive their relationships to interval vector variational inequalities $(IVVI)$ of Minty and Stampacchia type in terms of convexificators and LU-efficient solution of $(IVOP)$ using LU-convexity assumption. Furthermore, we consider weak versions of $(IVVI)$ of Minty and Stampacchia type and find the relationships between weak versions of $(IVVI)$ of Minty and Stampacchia type and weakly LU-efficient solution of $(IVOP)$. The results presented in this paper extend and generalized some existing results in the li…

最优运输理论中的对偶定理与向量测度

Duality Theorems and Vector Measures in Optimal Transportation Theory

本文通过抽象化表述推广最优运输问题及其对偶定理，省略测度使用，从而涵盖多种最优运输问题及博弈论、线性规划等应用，并特别推广至向量测度情形，给出对偶定理及解的存在性条件。

Uniform estimates for a Modica-Mortola type approximation of branched transportation

Models for branched networks are often expressed as the minimization of an energy $M^α$ over vector measures concentrated on $1$-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation $M^α_\varepsilon$, introduced by Edouard Oudet and Filippo Santambrogio, which is defined over $H^1$ vector measures. These energies induce some pseudo-distances between $L^2$ functions obtained through the minimization problem $\min \{M^α_\varepsilon (u)\;:\;\nabla\cdot u=f^+-f^-\}$. We prove some uniform estimates on these pseudo-distances which allow us to estab…

Vector-Valued Multivariate Conditional Value-at-Risk

In this study, we propose a new definition of multivariate conditional value-at-risk (MCVaR) as a set of vectors for discrete probability spaces. We explore the properties of the vector-valued MCVaR (VMCVaR) and show the advantages of VMCVaR over the existing definitions given for continuous random variables when adapted to the discrete case.

集值映射的次正则性实体向量最优化的应用

On Subregularity Properties of Set-Valued Mappings. Applications to Solid Vector Optimization

本文将集值映射的在点映射分为两类，并通过几个例子分析了它们之间的关系。在这个理论之旅之后，我们使用次正则性来推导集值映射的隐式定理。最后，我们提出了一些应用研究的多目标最优化问题。