arXiv subjects

Gabriela Kováčová

Publications and source records attributed to Gabriela Kováčová.

8 recordsLinked to original sources

arXiv2025

Non-conservative optimal transport

Motivated by optimal re-balancing of a portfolio, we formalize an optimal transport problem in which the transported mass is scaled by a mass-change factor depending on the source and destination. This allows direct modeling of the creation or destruction of mass. We discuss applications and position the framework alongside unbalanced, entropic, and unnormalized optimal transport. The existence of optimal transport plans and strong duality are established. The existence of optimal maps are deduced in two central regimes, i.e., perturbative mass-change and quadratic mass-loss. For $\ell_p$ costs we derive the analogue of the Benamou-Brenier dynamic formulation.

q-fin.PM↗

arXiv2024

Approximations of unbounded convex projections and unbounded convex sets

We consider the problem of projecting a convex set onto a subspace, or equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex sets by polyhedrons. The existing literature on convex projections provides methods for bounded convex sets only, in this paper we propose a method that can handle both bounded and unbounded problems. The algorithms we propose build on the ideas of inner and outer approximation. In particular, we adapt the recently proposed methods for solving unbounded convex vector optimization problems to handle also the class of projection problems.

math.OC↗

arXiv2024

Vector-valued robust stochastic control

We study a dynamic stochastic control problem subject to Knightian uncertainty with multi-objective (vector-valued) criteria. Assuming the preferences across expected multi-loss vectors are represented by a given, yet general, preorder, we address the model uncertainty by adopting a robust or minimax perspective, minimizing expected loss across the worst-case model. For loss functions taking real (or scalar) values, there is no ambiguity in interpreting supremum and infimum. In contrast to the scalar case, major challenges for multi-loss control problems include properly defining and interpreting the notions of supremum and infimum, and addressing the non-uniqueness of these suprema and infima. To deal with these, we employ the notion of an ideal point vector-valued supremum for the robust part of the problem, while we view the control part as a multi-objective (or vector) optimization problem. Using a set-valued framework, we derive both a weak and strong version of the dynamic programming principle (DPP) or Bellman equations by taking the value function as the collection of all worst expected losses across all feasible actions. The weak version of Bellman's principle is proved under minimal assumptions. To establish a stronger version of DPP, we introduce the rectangularity property with respect to a general preorder. We also further study a particular, but important, case of component-wise partial order of vectors, for which we additionally derive DPP under a different set-valued notion for the value function, the so-called upper image of the multi-objective problem. Finally, we provide illustrative examples motivated by financial problems. These results will serve as a foundation for addressing time-inconsistent problems subject to model uncertainty through the lens of a set-valued framework, as well as for studying multi-portfolio allocation problems under model uncertainty.

math.OC↗

arXiv2023

Computing the recession cone of a convex upper image via convex projection

It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase (Wagner et al., 2023). In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic and linear vector optimization problems and provide some numerical examples.

math.OC↗

arXiv2023

Algorithms to solve unbounded convex vector optimization problems

This paper is concerned with solution algorithms for general convex vector optimization problems (CVOPs). So far, solution concepts and approximation algorithms for solving CVOPs exist only for bounded problems [Ararat et al. 2022, Doerfler et al. 2021, Loehne et al. 2014]. They provide a polyhedral inner and outer approximation of the upper image that have a Hausdorff distance of at most $\varepsilon$. However, it is well known (see [Ulus, 2018]), that for some unbounded problems such polyhedral approximations do not exist. In this paper, we will propose a generalized solution concept, called an $(\varepsilon,δ)$--solution, that allows also to consider unbounded CVOPs. It is based on additionally bounding the recession cones of the inner and outer polyhedral approximations of the upper image in a meaningful way. An algorithm is proposed that computes such $δ$--outer and $δ$--inner approximations of the recession cone of the upper image. In combination with the results of [Loehne et al. 2014] this provides a primal and a dual algorithm that allow to compute $(\varepsilon,δ)$--solutions of (potentially unbounded) CVOPs. Numerical examples are provided.

math.OC↗

arXiv2021

Convex Projection and Convex Multi-Objective Optimization

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.

math.OC↗

arXiv2020

Acceptability maximization

The aim of this paper is to study the optimal investment problem by using coherent acceptability indices (CAIs) as a tool to measure the portfolio performance. We call this problem the acceptability maximization. First, we study the one-period (static) case, and propose a numerical algorithm that approximates the original problem by a sequence of risk minimization problems. The results are applied to several important CAIs, such as the gain-to-loss ratio, the risk-adjusted return on capital and the tail-value-at-risk based CAI. In the second part of the paper we investigate the acceptability maximization in a discrete time dynamic setup. Using robust representations of CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we establish an intriguing dichotomy: if the corresponding family of DCRMs is recursive (i.e. strongly time consistent) and assuming some recursive structure of the market model, then the acceptability maximization problem reduces to just a one period problem and the maximal acceptability is constant across all states and times. On the other hand, if the family of DCRMs is not recursive, which is often the case, then the acceptability maximization problem ordinarily is a time-inconsistent stochastic control problem, similar to the classical mean-variance criteria. To overcome this form of time-inconsistency, we adapt to our setup the set-valued Bellman's principle recently proposed in \cite{KovacovaRudloff2019} applied to two particular dynamic CAIs - the dynamic risk-adjusted return on capital and the dynamic gain-to-loss ratio. The obtained theoretical results are illustrated via numerical examples that include, in particular, the computation of the intermediate mean-risk efficient frontiers.

q-fin.MF↗

arXiv2020

Time consistency of the mean-risk problem

Choosing a portfolio of risky assets over time that maximizes the expected return at the same time as it minimizes portfolio risk is a classical problem in Mathematical Finance and is referred to as the dynamic Markowitz problem (when the risk is measured by variance) or more generally, the dynamic mean-risk problem. In most of the literature, the mean-risk problem is scalarized and it is well known that this scalarized problem does not satisfy the (scalar) Bellman's principle. Thus, the classical dynamic programming methods are not applicable. For the purpose of this paper we focus on the discrete time setup, and we will use a time consistent dynamic convex risk measure to evaluate the risk of a portfolio. We will show that when we do not scalarize the problem, but leave it in its original form as a vector optimization problem, the upper images, whose boundary contains the efficient frontier, recurse backwards in time under very mild assumptions. Thus, the dynamic mean-risk problem does satisfy a Bellman's principle, but a more general one, that seems more appropriate for a vector optimization problem: a set-valued Bellman's principle. We will present conditions under which this recursion can be exploited directly to compute a solution in the spirit of dynamic programming. Numerical examples illustrate the proposed method. The obtained results open the door for a new branch in mathematics: dynamic multivariate programming.

q-fin.MF↗