arXiv subjects

Kathrin Klamroth

Publications and source records attributed to Kathrin Klamroth.

At least 19 recordsLinked to original sources

arXiv2026

Consistent Path Selection for Bi-objective Median Location Problems on Graphs

We consider single-facility median location problems on graphs where two conflicting cost values are associated with the edges. As an example, suppose that a decision maker wants to locate one new facility, e. g., a pizza delivery place, which uses bicycles for delivery. The two objective functions could then be the total traveling time and the total number of left turns, as the latter are very risky. Path choices then depend on the preferences of the decision maker, and in general, there may not exist a unique optimal path between a customer and a new facility location. In this paper, we consider the location decision and the routing decision in a coupled problem, i. e., we search for an optimal location and for consistent delivery paths simultaneously. We introduce the concept of consistent paths, where we assume that the choice of a path from the facility to a demand node implies certain preferences. All paths of a solution are consistent if the preferences of all paths do not contradict each other. We present an algorithm that computes a minimum complete set of efficient solutions with consistent path choices and illustrate the results at example instances in the city of Wuppertal in Germany.

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arXiv2026

A multi-objective perspective on block-structured integer programs with one soft coupling constraint

This paper presents a multi-objective perspective on block-structured integer programs featuring a single soft coupling constraint. By interpreting the coupling constraint as a second objective, we transform the coupled single-objective problem into an additively-separable bi-objective optimization problem. To avoid the expensive computation of the full Pareto front, we introduce an algorithm, which uses a binary search to isolate a region of interest around the soft constraint limit. This algorithm provides provable bounds on the single-objective optimum. We further enhance this algorithm, by exploiting the block-structure, using a novel $\lambda$-lookup mechanism to skip repeated sub-problem calculations. Finally, for scenarios requiring all non-dominated solutions within the region of interest, we propose a new approach, that works its way from the middle of the region of interest outwards. This algorithm shows quick convergence in terms of representation. Computational studies demonstrate that our methods dramatically reduce integer programming calls, thereby outperforming traditional dichotomic search. For large instances the method works as a strong heuristic providing bounds on the gap to an optimal solution, providing trade-off information in addition to the solution.

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arXiv2026

Adapting Polyhedral Dominance Cones to Ordinal Preference Structures

In combinatorial optimization, ordinal costs can be used to model the quality of elements whenever numerical values are not available. When considering, for example, routing problems for cyclists, the safety of a street can be ranked in ordered categories like safe (separate bike lane), medium safe (street with a bike lane) and unsafe (street without a bike lane). However, ordinal optimization may suggest unrealistic solutions with huge detours to avoid unsafe street segments. In this paper, we investigate how partial preference information regarding the relative quality of the ordinal categories can be used to improve the relevance of the computed solutions. By introducing preference weights which describe how much better a category is at least or at most, compared to the subsequent category, we enlarge the ordinal dominance cone. This leads to a smaller set of alternatives, i. e., of ordinally efficient solutions. We show that the corresponding weighted ordinal ordering cone is a polyhedral cone and provide descriptions via its extreme rays and via its facets. The latter implies a linear transformation to an associated multi-objective optimization problem. This paves the way for the application of standard multi-objective solution approaches. We demonstrate the usefulness of the weighted ordinal ordering cone by investigating a safest path problem with different preference weights. Moreover, we investigate the interrelation between the weighted ordering cone to standard dominance concepts of multi-objective optimization, like, e.g., Pareto dominance, lexicographic dominance and weighted sum dominance.

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arXiv2024

On the Computational Complexity of Multi-Objective Ordinal Unconstrained Combinatorial Optimization

Multi-objective unconstrained combinatorial optimization problems (MUCO) are in general hard to solve, i.e., the corresponding decision problem is NP-hard and the outcome set is intractable. In this paper we explore special cases of MUCO problems that are actually easy, i.e., solvable in polynomial time. More precisely, we show that MUCO problems with up to two ordinal objective functions plus one real-valued objective function are tractable, and that their complete nondominated set can be computed in polynomial time. For MUCO problems with one ordinal and a second ordinal or real-valued objective function we present an even more efficient algorithm that applies a greedy strategy multiple times.

cs.DM↗

arXiv2024

Event-based MILP models for ride pooling applications

Ridepooling services require efficient optimization algorithms to simultaneously plan routes and pool users in shared rides. We consider a static dial-a-ride problem (DARP) where a series of origin-destination requests have to be assigned to routes of a fleet of vehicles. Thereby, all requests have associated time windows for pick-up and delivery, and may be denied if they can not be serviced in reasonable time or at reasonable cost. Rather than using a spatial representation of the transportation network we suggest an event-based formulation of the problem, resulting in significantly improved computational times. While the corresponding MILP formulations require more variables than standard models, they have the advantage that capacity, pairing and precedence constraints are handled implicitly. The approach is tested and validated using a standard IP-solver on benchmark data from the literature. Moreover, the impact of, and the trade-off between, different optimization goals is evaluated on a case study in the city of Wuppertal (Germany).

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arXiv2024

Transforming the Challenge of Constructing Low-Discrepancy Point Sets into a Permutation Selection Problem

Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear how low the discrepancy of point sets can go; in other words, how regularly distributed can points be in a given space. Recent insights using optimization and machine learning techniques have led to substantial improvements in the construction of low-discrepancy point sets, resulting in configurations of much lower discrepancy values than previously known. Building on the optimal constructions, we present a simple way to obtain $L_{\infty}$-optimized placement of points that follow the same relative order as an (arbitrary) input set. Applying this approach to point sets in dimensions 2 and 3 for up to 400 and 50 points, respectively, we obtain point sets whose $L_{\infty}$ star discrepancies are up to 25% smaller than those of the current-best sets, and around 50% better than classical constructions such as the Fibonacci set.

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arXiv2024

Solving the Multiobjective Quasi-Clique Problem

Given a simple undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $\gamma$ $(0 < \gamma \leq 1)$. Finding a maximum quasi-clique has been addressed from two different perspectives: $i)$ maximizing vertex cardinality for a given edge density; and $ii)$ maximizing edge density for a given vertex cardinality. However, when no a priori preference information about cardinality and density is available, a more natural approach is to consider the problem from a multiobjective perspective. We introduce the Multiobjective Quasi-clique Problem (MOQC), which aims to find a quasi-clique by simultaneously maximizing both vertex cardinality and edge density. To efficiently address this problem, we explore the relationship among MOQC, its single-objective counterpart problems, and a biobjective optimization problem, along with several properties of the MOQC problem and quasi-cliques. We propose a baseline approach using $\varepsilon$-constraint scalarization and introduce a Two-phase strategy, which applies a dichotomic search based on weighted sum scalarization in the first phase and an $\varepsilon$-constraint methodology in the second phase. Additionally, we present a Three-phase strategy that combines the dichotomic search used in Two-phase with a vertex-degree-based local search employing novel sufficient conditions to assess quasi-clique efficiency, followed by an $\varepsilon$-constraint in a final stage. Experimental results on real-world sparse graphs indicate that the integrated use of dichotomic search and local search, together with mechanisms to assess quasi-clique efficiency, makes the Three-phase strategy an effective approach for solving the MOQC problem in terms of running time and ability to produce new efficient quasi-cliques.

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arXiv2024

Ensuring connectedness for the Maximum Quasi-clique and Densest $k$-subgraph problems

Given an undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $γ$ $(0 < γ\leq 1)$. Two optimization problems can be defined for quasi-cliques: the Maximum Quasi-Clique (MQC) Problem, which finds a quasi-clique with maximum vertex cardinality, and the Densest $k$-Subgraph (DKS) Problem, which finds the densest subgraph given a fixed cardinality constraint. Most existing approaches to solve both problems often disregard the requirement of connectedness, which may lead to solutions containing isolated components that are meaningless for many real-life applications. To address this issue, we propose two flow-based connectedness constraints to be integrated into known Mixed-Integer Linear Programming (MILP) formulations for either MQC or DKS problems. We compare the performance of MILP formulations enhanced with our connectedness constraints in terms of both running time and number of solved instances against existing approaches that ensure quasi-clique connectedness. Experimental results demonstrate that our constraints are quite competitive, making them valuable for practical applications requiring connectedness.

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arXiv2024

Constructing Optimal $L_{\infty}$ Star Discrepancy Sets

The $L_{\infty}$ star discrepancy is a very well-studied measure used to quantify the uniformity of a point set distribution. Constructing optimal point sets for this measure is seen as a very hard problem in the discrepancy community. Indeed, optimal point sets are, up to now, known only for $n\leq 6$ in dimension 2 and $n \leq 2$ for higher dimensions. We introduce in this paper mathematical programming formulations to construct point sets with as low $L_{\infty}$ star discrepancy as possible. Firstly, we present two models to construct optimal sets and show that there always exist optimal sets with the property that no two points share a coordinate. Then, we provide possible extensions of our models to other measures, such as the extreme and periodic discrepancies. For the $L_{\infty}$ star discrepancy, we are able to compute optimal point sets for up to 21 points in dimension 2 and for up to 8 points in dimension 3. For $d=2$ and $n\ge 7$ points, these point sets have around a 50% lower discrepancy than the current best point sets, and show a very different structure.

cs.CG↗

arXiv2023

Ridepooling and public bus services: A comparative case-study

This case-study aims at a comparison of the service quality of time-tabled buses as compared to on-demand ridepooling cabs in the late evening hours in the city of Wuppertal, Germany. To evaluate the service quality of ridepooling as compared to bus services, and to simulate bus rides during the evening hours, transport requests are generated using a predictive simulation. To this end, a framework in the programming language R is created, which automatically combines generalized linear models for count regression to model the demand at each bus stop. Furthermore, we use classification models for the prediction of trip destinations. To solve the resulting dynamic dial-a-ride problem, a rolling-horizon algorithm based on the iterative solution of Mixed-Integer Linear Programming Models (MILP) is used. A feasible-path heuristic is used to enhance the performance of the algorithm in presence of high request densities. This allows an estimation of the number of cabs needed depending on the weekday to realize the same or a better general service quality as the bus system.

cs.DM↗

arXiv2023

A new perspective on dynamic network flow problems via port-Hamiltonian systems

We suggest a global perspective on dynamic network flow problems that takes advantage of the similarities to port-Hamiltonian dynamics. Dynamic minimum cost flow problems are formulated as open-loop optimal control problems for general port-Hamiltonian systems with possibly state-dependent system matrices. We prove well-posedness of these systems and characterize optimal controls by the first-order optimality system, which is the starting point for the derivation of an adjoint-based gradient descent algorithm. Our theoretical analysis is complemented by a proof of concept, where we apply the proposed algorithm to static minimum cost flow problems and dynamic minimum cost flow problems on a simple directed acyclic graph. We present numerical results to validate the approach.

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arXiv2023

A Tight Formulation for the Dial-a-Ride Problem

Ridepooling services play an increasingly important role in modern transportation systems. With soaring demand and growing fleet sizes, the underlying route planning problems become increasingly challenging. In this context, we consider the dial-a-ride problem (DARP): Given a set of transportation requests with pick-up and delivery locations, passenger numbers, time windows, and maximum ride times, an optimal routing for a fleet of vehicles, including an optimized passenger assignment, needs to be determined. We present tight mixed-integer linear programming (MILP) formulations for the DARP by combining two state-of-the-art models into novel location-augmented-event-based formulations. Strong valid inequalities and lower and upper bounding techniques are derived to further improve the formulations. We then demonstrate the theoretical and computational superiority of the new model: First, the formulation is tight in the sense that, if time windows shrink to a single point in time, the linear programming relaxation yields integer (and hence optimal) solutions. Second, extensive numerical experiments on benchmark instances show that computational times are on average reduced by 49.7% compared to state-of-the-art event-based approaches.

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arXiv2023

On the online path extension problem -- Location and routing problems in board games

We consider an online version of a longest path problem in an undirected and planar graph that is motivated by a location and routing problem occurring in the board game "Turn & Taxis". Path extensions have to be selected based on only partial knowledge on the order in which nodes become available in later iterations. Besides board games, online path extension problems have applications in disaster relief management when infrastructure has to be rebuilt after natural disasters. For example, flooding may affect large parts of a road network, and parts of the network may become available only iteratively and decisions may have to be made without the possibility of planning ahead. We suggest and analyse selection criteria that identify promising nodes (locations) for path extensions. We introduce the concept of tentacles of paths as an indicator for the future extendability. Different initialization and extension heuristics are suggested on compared to an ideal solution that is obtained by an integer linear programming formulation assuming complete knowledge, i.e., assuming that the complete sequence in which nodes become available is known beforehand. All algorithms are tested and evaluated on the original "Turn & Taxis" graph, and on an extended version of the "Turn & Taxis" graph, with different parameter settings. The numerical results confirm that the number of tentacles is a useful criterion when selecting path extensions, leading to near-optimal paths at relatively low computational costs.

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arXiv2023

Modeling Minimum Cost Network Flows With Port-Hamiltonian Systems

We give a short overview of advantages and drawbacks of the classical formulation of minimum cost network flow problems and solution techniques, to motivate a reformulation of classical static minimum cost network flow problems as optimal control problems constrained by port-Hamiltonian systems (pHS). The first-order optimality system for the port-Hamiltonian system-constrained optimal control problem is formally derived. Then we propose a gradient-based algorithm to find optimal controls. The port-Hamiltonian system formulation naturally conserves flow and supports a wide array of further modeling options as, for example, node reservoirs, flow dependent costs, leaking pipes (dissipation) and coupled sub-networks (ports). They thus provide a versatile alternative to state-of-the art approaches towards dynamic network flow problems, which are often based on computationally costly time-expanded networks. We argue that this opens the door for a plethora of modeling options and solution approaches for network flow problems.

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arXiv2023

Non-convex shape optimization by dissipative Hamiltonian flows

Shape optimization with constraints given by partial differential equations (PDE) is a highly developed field of optimization theory. The elegant adjoint formalism allows to compute shape gradients at the computational cost of a further PDE solve. Thus, gradient descent methods can be applied to shape optimization problems. However, gradient descent methods that can be understood as approximation to gradient flows get stuck in local minima, if the optimization problem is non-convex. In machine learning, the optimization in high dimensional non-convex energy landscapes has been successfully tackled by momentum methods, which can be understood as passing from gradient flow to dissipative Hamiltonian flows. In this paper, we adopt this strategy for non-convex shape optimization. In particular, we provide a mechanical shape optimization problem that is motivated by optimal reliability considering also material cost and the necessity to avoid certain obstructions in installation space. We then show how this problem can be solved effectively by port Hamiltonian shape flows.

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arXiv2023

PINN Training using Biobjective Optimization: The Trade-off between Data Loss and Residual Loss

Physics informed neural networks (PINNs) have proven to be an efficient tool to represent problems for which measured data are available and for which the dynamics in the data are expected to follow some physical laws. In this paper, we suggest a multiobjective perspective on the training of PINNs by treating the data loss and the residual loss as two individual objective functions in a truly biobjective optimization approach. As a showcase example, we consider COVID-19 predictions in Germany and built an extended susceptibles-infected-recovered (SIR) model with additionally considered leaky-vaccinated and hospitalized populations (SVIHR model) to model the transition rates and to predict future infections. SIR-type models are expressed by systems of ordinary differential equations (ODEs). We investigate the suitability of the generated PINN for COVID-19 predictions and compare the resulting predicted curves with those obtained by applying the method of non-standard finite differences to the system of ODEs and initial data. The approach is applicable to various systems of ODEs that define dynamical regimes. Those regimes do not need to be SIR-type models, and the corresponding underlying data sets do not have to be associated with COVID-19.

cs.LG↗

arXiv2022

Consensus-Based Optimization for Multi-Objective Problems: A Multi-Swarm Approach

We propose a multi-swarm approach to approximate the Pareto front of general multi-objective optimization problems that is based on the Consensus-based Optimization method (CBO). The algorithm is motivated step by step beginning with a simple extension of CBO based on fixed scalarization weights. To overcome the issue of choosing the weights we propose an adaptive weight strategy in the second modelling step. The modelling process is concluded with the incorporation of a penalty strategy that avoids clusters along the Pareto front and a diffusion term that prevents collapsing swarms. Altogether the proposed $K$-swarm CBO algorithm is tailored for a diverse approximation of the Pareto front and, simultaneously, the efficient set of general non-convex multi-objective problems. The feasibility of the approach is justified by analytic results, including convergence proofs, and a performance comparison to the well-known non-dominated sorting genetic algorithm (NSGA2) and the recently proposed one-swarm approach for multi-objective problems involving Consensus-based Optimization.

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arXiv2022

Ordinal Optimization Through Multi-objective Reformulation

We analyze combinatorial optimization problems with ordinal, i.e., non-additive, objective functions that assign categories (like good, medium and bad) rather than cost coefficients to the elements of feasible solutions. We review different optimality concepts for ordinal optimization problems and discuss their similarities and differences. We then focus on two prevalent optimality concepts that are shown to be equivalent. Our main result is a bijective linear transformation that transforms ordinal optimization problems to associated standard multi-objective optimization problems with binary cost coefficients. Since this transformation preserves all properties of the underlying problem, problem-specific solution methods remain applicable. A prominent example is dynamic programming and Bellman's principle of optimality, that can be applied, e.g., to ordinal shortest path and ordinal knapsack problems. We extend our results to multi-objective optimization problems that combine ordinal and real-valued objective functions.

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