arXiv subjects

Andreas Löhne

Publications and source records attributed to Andreas Löhne.

At least 19 recordsLinked to original sources

arXiv2026

Low-Rank Multi-Objective Linear Programming

When solving multi-objective programs (MOLPs), the number of objectives essentially determines the computing time. This can even lead to practically unsolvable problems. Consequently, it is of huge interest to reduce the number of objectives without losing information. There exist approaches for transforming vector linear programs (VLPs), i.e., the ordering cone is non-standard, into MOLPs in the literature. We here propose a method for solving MOLPs more efficiently by applying the transformation in reverse. In particular, we discuss MOLPs with linear dependent objective functions. The resulting VLPs have merely as many objectives as the rank of the objective matrices of the MOLPs. To achieve this, only a factorization of this matrix needs to be calculated. One factor then forms a new ordering cone, while the other remains as the objective matrix. Through multiple series of numerical experiments, we show that this approach indeed significantly reduces computing time, and therefore provides a technique for solving low-rank MOLPs in practice. As there are fewer objectives to consider, the approach additionally helps decision makers to get a better visualization as well as understanding of the actual problem. Moreover, we will point out that the equivalence between MOLPs and corresponding VLPs can be used to derive statements about the well-known concept of nonessential objectives.

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arXiv2026

A Solution Concept for Convex Vector Optimization Problems based on a User-defined Region of Interest

This work addresses arbitrary convex vector optimization problems, which constitute a general framework for multi-criteria decision-making in diverse real-world applications. Due to their complexity, such problems are typically tackled using polyhedral approximation. Existing solution concepts rely on additional assumptions, such as boundedness, polyhedrality of the ordering cone, or existence of interior points in the ordering cone, and typically focus on absolute error measures. We introduce a solution concept based on the homogenization of the upper image that employs relative error measures and avoids additional structural assumptions. Although minimality is not explicitly required, a form of approximate minimality is implicitly ensured. The concept is straightforward, requiring only a single precision parameter and, owing to relative errors, remains robust under scaling of the target functions. Homogenization also eliminates the need for the binary distinction between points far from the origin and directions which can lead to numerical difficulties. Furthermore, in practice decision-makers often identify a region where preferred solutions are expected. Our concept supports both a global overview of the upper image and a refined local perspective within such a user-defined region of interest (RoI).We present a decision-making procedure enabling iterative refinement of this region and the associated preferences.

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arXiv2025

Multi-objective stochastic linear programming with recourse and flexible decision making

Optimal inventory leads to stochastic optimization problems where deterministic delivery decisions have to be made in advance of stochastic demand realizations. Similarly, risk deposits have to be given before the random outcomes of investments are known. In this paper, multi-criteria versions of such stochastic recourse problems are studied. In addition to traditional concepts like Pareto-optimality, a decision maker for the multi-criteria problem may have a preference for greater flexibility in the second stage decision. This idea leads to a first stage optimization problem with a set-valued objective instead of a mere multi-criteria one. Under linearity assumptions, this problem becomes a polyhedral convex set optimization problem instead of a multi-objective linear program. Solution concepts for multi-objective/set-valued recourse problems are given as well as deterministic surrogates of the stochastic problem such as its deterministic equivalent, the so-called wait-and-see problem and the expected-value problem for the multi-objective case. Managerial decision making guidelines are obtained based on set optimization methods and the preference-for-flexibility approach: choose the deterministic first stage variable such that a maximum of flexibility is combined with a guarantee for Pareto minimal objective values. Two major examples illustrate the findings, a multi-objective newsvendor problem with an additional health/sustainability objective and a risk compensation problem where the availability of more than one asset for risk compensation, e.g., several currencies, leads to multiple objectives.

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arXiv2025

MOCVXPY: a CVXPY extension for multiobjective optimization

MOCVXPY is an open-source Python library for convex vector optimization. It is built on top of CVXPY, a domain-specific language for single-objective convex optimization. MOCVXPY enables practitioners to describe their convex vector optimization problem in an intuitive algebraic language, that closely follows the mathematical formulation. This work presents the main features of MOCVXPY, explains some background of the algorithms it employs to solve the optimization problems, and illustrates its functionality through examples and two real-world applications in finance and energy. MOCVXPY is available at https://github.com/salomonl/mocvxpy under the Apache 2.0 licence, with some documentation and examples.

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arXiv2025

Computing all Nash equilibria of low-rank bi-matrix games

We study constrained bi-matrix games, with a particular focus on low-rank games. Our main contribution is a framework that reduces low-rank games to smaller, equivalent constrained games, along with a necessary and sufficient condition for when such reductions exist. Building on this framework, we present three approaches for computing the set of extremal Nash equilibria, based on vertex enumeration, polyhedral calculus, and vector linear programming. Numerical case studies demonstrate the effectiveness of the proposed reduction and solution methods.

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arXiv2024

A solution method for arbitrary polyhedral convex set optimization problems

We provide a solution method for the polyhedral convex set optimization problem, that is, the problem to minimize a set-valued mapping with polyhedral convex graph with respect to a set ordering relation which is generated by a polyhedral convex cone . The method is proven to be correct and finite without any further assumption to the problem.

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arXiv2024

A Decision-Making Method in Polyhedral Convex Set Optimization

Optimization problems with set-valued objective functions arise in contexts such as multi-stage optimization with vector-valued objectives. The aim is to identify an optimizer -- a feasible point with an optimal objective value -- based on an ordering relation on a family of sets. When faced with multiple optimizers, a decision maker must choose one. Visualizing the values associated with these optimizers could provide a solid basis for decision-making. However, these values are sets, making it challenging to visualize many of them. Therefore, we propose a method where an optimizer is selected by designing the respective outcome set through a trial-and-error process. In a polyhedral convex setting, we discuss an implementation and prove that an optimizer can be found using this method after a finite number of design steps. We motivate the problem setting and illustrate the process using an example: a two-stage bi-objective network flow problem.

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arXiv2024

The natural ordering cone of a polyhedral convex set-valued objective mapping

For a given polyhedral convex set-valued mapping we define a polyhedral convex cone which we call the natural ordering cone. We show that the solution behavior of a polyhedral convex set optimization problem can be characterized by this cone. Under appropriate assumptions the natural ordering cone is proven to be the smallest ordering cone which makes a polyhedral convex set optimization problem solvable.

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arXiv2024

Finite Representation of Quantile Sets for Multivariate Data via Vector Linear Programming

Empirical quantiles for finitely distributed univariate random variables can be obtained by solving a certain linear program. It is shown in this short note that multivariate empirical quantiles can be obtained in a very similar way by solving a vector linear program. This connection provides a new approach for computing Tukey depth regions and more general cone quantile sets.

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arXiv2024

Convex sets approximable as the sum of a compact set and a cone

The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced. These sets are called approximately Motzkin-decomposable and generalize the notion of Motzkin-decomposability, i.e. the representation of a set as the sum of a compact convex set and a closed convex cone. We characterize these sets in terms of their support functions and show that they coincide with self-bounded sets, i.e. sets contained in the sum of a compact convex set and a closed convex cone, if their recession cones are polyhedral but are more restrictive in general. In particular we prove that a set is approximately Motzkin-decomposable if and only if its support function has a closed domain relative to which it is continuous.

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arXiv2023

Existence of solutions for polyhedral convex set optimization problems

Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear programs and multiple objective linear programs provide proper subclasses. In this article we choose a solution concept for arbitrary polyhedral convex set optimization problems out of several alternatives, show existence of solutions and characterize the existence of solutions in different ways. Two known results are obtained as particular cases, both with proofs being easier than the original ones: The existence of solutions of bounded polyhedral convex set optimization problems and a characterization of the existence of solutions of vector linear programs.

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arXiv2023

Polyhedral approximation of spectrahedral shadows via homogenization

This article is concerned with the problem of approximating a not necessarily bounded spectrahedral shadow, a certain convex set, by polyhedra. By identifying the set with its homogenization the problem is reduced to the approximation of a closed convex cone. We introduce the notion of homogeneous δ-approximation of a convex set and show that it defines a meaningful concept in the sense that approximations converge to the original set if the approximation error δ diminishes. Moreover, we show that a homogeneous δ-approximation of the polar of a convex set is immediately available from an approximation of the set itself under mild conditions. Finally, we present an algorithm for the computation of homogeneous δ-approximations of spectrahedral shadows and demonstrate it on examples.

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arXiv2023

A polyhedral approximation algorithm for recession cones of spectrahedral shadows

The intersection of an affine subspace with the cone of positive semidefinite matrices is called a spectrahedron. An orthogonal projection thereof is called a spectrahedral shadow or projected spectrahedron. Spectrahedra and their projections can be seen as a generalization of polyhedra. This article is concerned with the problem of approximating the recession cones of spectrahedra and spectrahedral shadows via polyhedral cones. We present two iterative algorithms to compute outer and inner approximations to within an arbitrary prescribed accuracy. The first algorithm is tailored to spectrahedra and is derived from polyhedral approximation algorithms for compact convex sets and relies on the fact, that an algebraic description of the recession cone is available. The second algorithm is designed for projected spectrahedra and does not require an algebraic description of the recession cone, which is in general more difficult to obtain. We prove correctness and finiteness of both algorithms and provide numerical examples.

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arXiv2023

On unbounded polyhedral convex set optimization problems

A polyhedral convex set optimization problem is given by a set-valued objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean space whose graph is a convex polyhedron. This problem can be seen as the most elementary subclass of set optimization problems, comparable to linear programming in the framework of optimization with scalar-valued objective function. Polyhedral convex set optimization generalizes both scalar and multi-objective (or vector) linear programming. In contrast to scalar linear programming but likewise to multi-objective linear programming, unbounded problems can indeed have minimizers and provide a rich class of problem instances. In this paper we extend the concept of finite infimizers from multi-objective linear programming to not necessarily bounded polyhedral convex set optimization problems. We show that finite infimizers can be obtained from finite infimizers of a reformulation of the polyhedral convex set optimization problem into a vector linear program. We also discuss two natural extensions of solution concepts based on the complete lattice approach. Surprisingly, the attempt to generalize the solution procedure for bounded polyhedral convex set optimization problems introduced in [A. Löhne and C. Schrage. An algorithm to solve polyhedral convex set optimization problems. {\em Optimization} 62(1):131--141, 2013.] to the case of not necessarily bounded problems uncovers some problems, which will be discussed.

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arXiv2023

Approximate Vertex Enumeration

The problem to compute the vertices of a polytope given by affine inequalities is called vertex enumeration. The inverse problem, which is equivalent by polarity, is called the convex hull problem. We introduce `approximate vertex enumeration' as the problem to compute the vertices of a polytope which is close to the original polytope given by affine inequalities. In contrast to exact vertex enumerations, both polytopes are not required to be combinatorially equivalent. Two algorithms for this problem are introduced. The first one is an approximate variant of Motzkin's double description method. Only under certain strong conditions, which are not acceptable for practical reasons, we were able to prove correctness of this method for polytopes of arbitrary dimension. The second method, called shortcut algorithm, is based on constructing a plane graph and is restricted to polytopes of dimension 2 and 3. We prove correctness of the shortcut algorithm. As a consequence, we also obtain correctness of the approximate double description method, only for dimension 2 and 3 but without any restricting conditions as still required for higher dimensions. We show that for dimension 2 and 3 both algorithm remain correct if imprecise arithmetic is used and the computational error caused by imprecision is not too high. Both algorithms were implemented. The numerical examples motivate the approximate vertex enumeration problem by showing that the approximate problem is often easier to solve than the exact vertex enumeration problem. It remains open whether or not the approximate double description method (without any restricting condition) is correct for polytopes of dimension 4 and higher.

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arXiv2021

On the approximation error for approximating convex bodies using multiobjective optimization

A polyhedral approximation of a convex body can be calculated by solving approximately an associated multiobjective convex program (MOCP). An MOCP can be solved approximately by Benson type algorithms, which compute outer and inner polyhedral approximations of the problem's upper image. Polyhedral approximations of a convex body can be obtained from polyhedral approximations of the upper image of the associated MOCP. We provide error bounds in terms of the Hausdorff distance for the polyhedral approximations of a convex body in dependence of the stopping criterion of the primal and dual Benson type algorithms which are applied to the associated MOCP.

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arXiv2020

A Benson-Type Algorithm for Bounded Convex Vector Optimization Problems with Vertex Selection

We present an algorithm for approximately solving bounded convex vector optimization problems. The algorithm provides both an outer and an inner polyhedral approximation of the upper image. It is a modification of the primal algorithm presented by Löhne, Rudloff, and Ulus in 2014. There, vertices of an already known outer approximation are successively cut off to improve the approximation error. We propose a new and efficient selection rule for deciding which vertex to cut off. Numerical examples are provided which illustrate that this method may solve fewer scalar problems overall and therefore may be faster while achieving the same approximation quality.

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arXiv2020

Solving polyhedral d.c. optimization problems via concave minimization

The problem of minimizing the difference of two convex functions is called polyhedral d.c. optimization problem if at least one of the two component functions is polyhedral. We characterize the existence of global optimal solutions of polyhedral d.c. optimization problems. This result is used to show that, whenever the existence of an optimal solution can be certified, polyhedral d.c. optimization problems can be solved by certain concave minimization algorithms. No further assumptions are necessary in case of the first component being polyhedral and just some mild assumptions to the first component are required for the case where the second component is polyhedral. In case of both component functions being polyhedral, we obtain a primal and dual existence test and a primal and dual solution procedure. Numerical examples are discussed.

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