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Frauke Liers

Publications and source records attributed to Frauke Liers.

At least 19 recordsLinked to original sources
arXiv2025

High-Quality Tomographic Image Reconstruction Integrating Neural Networks and Mathematical Optimization

In this work, we develop a novel technique for reconstructing images from projection-based nano- and microtomography. Our contribution focuses on enhancing reconstruction quality, particularly for specimen composed of homogeneous material phases connected by sharp edges. This is accomplished by training a neural network to identify edges within subpictures. The trained network is then integrated into a mathematical optimization model, to reduce artifacts from previous reconstructions. To this end, the optimization approach favors solutions according to the learned predictions, however may also determine alternative solutions if these are strongly supported by the raw data. Hence, our technique successfully incorporates knowledge about the homogeneity and presence of sharp edges in the sample and thereby eliminates blurriness. Our results on experimental datasets show significant enhancements in interface sharpness and material homogeneity compared to benchmark algorithms. Thus, our technique produces high-quality reconstructions, showcasing its potential for advancing tomographic imaging techniques.

cs.CV↗
arXiv2025

Feature Selection for Data-driven Explainable Optimization

Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while obtained solutions are globally optimal, they are frequently not comprehensible to humans, in particular when obtained by black-box routines. In contrast, explainability is a standard requirement for results in Artificial Intelligence, but it is rarely considered in optimization yet. There are only a few studies that aim to find solutions that are both of high quality and explainable. In recent work, explainability for optimization was defined in a data-driven manner: A solution is considered explainable if it closely resembles solutions that have been used in the past under similar circumstances. To this end, it is crucial to identify a preferably small subset of features from a presumably large set that can be used to measure instance similarity. In this work, we formally define the feature selection problem for explainable optimization and prove that its decision version is NP-complete. We introduce mathematical models for optimized feature selection. As their global solution requires significant computation time with modern mixed-integer linear solvers, we employ local heuristics. Our computational study using data that reflect real-world scenarios demonstrates that the problem can be solved practically efficiently for instances of reasonable size.

math.OC↗
arXiv2025

Scenario Reduction for Distributionally Robust Optimization

Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios or data points increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the proposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors.

math.OC↗
arXiv2024

Quantum Subroutines in Branch-Price-and-Cut for Vehicle Routing

Motivated by recent progress in quantum hardware and algorithms researchers have developed quantum heuristics for optimization problems, aiming for advantages over classical methods. To date, quantum hardware is still error-prone and limited in size such that quantum heuristics cannot be scaled to relevant problem sizes and are often outperformed by their classical counterparts. Moreover, if provably optimal solutions are desired, one has to resort to classical exact methods. As however quantum technologies may improve considerably in future, we demonstrate in this work how quantum heuristics with limited resources can be integrated in large-scale exact optimization algorithms for NP-hard problems. To this end, we consider vehicle routing as prototypical NP-hard problem. We model the pricing and separation subproblems arising in a branch-price-and-cut algorithm as quadratic unconstrained binary optimization problems. This allows to use established quantum heuristics like quantum annealing or the quantum approximate optimization algorithm for their solution. A key feature of our algorithm is that it profits not only from the best solution returned by the quantum heuristic but from all solutions below a certain cost threshold, thereby exploiting the inherent randomness is quantum algorithms. Moreover, we reduce the requirements on quantum hardware since the subproblems, which are solved via quantum heuristics, are smaller than the original problem. We provide an experimental study comparing quantum annealing to simulated annealing and to established classical algorithms in our framework. While our hybrid quantum-classical approach is still outperformed by purely classical methods, our results reveal that both pricing and separation may be well suited for quantum heuristics if quantum hardware improves.

quant-ph↗
arXiv2024

Optimized Noise Suppression for Quantum Circuits

Quantum computation promises to advance a wide range of computational tasks. However, current quantum hardware suffers from noise and is too small for error correction. Thus, accurately utilizing noisy quantum computers strongly relies on noise characterization, mitigation, and suppression. Crucially, these methods must also be efficient in terms of their classical and quantum overhead. Here, we efficiently characterize and mitigate crosstalk noise, which is a severe error source in, e.g., cross-resonance based superconducting quantum processors. For crosstalk characterization, we develop a simplified measurement experiment. Furthermore, we analyze the problem of optimal experiment scheduling and solve it for common hardware architectures. After characterization, we mitigate noise in quantum circuits by a noise-aware qubit routing algorithm. Our integer programming algorithm extends previous work on optimized qubit routing by swap insertion. We incorporate the measured crosstalk errors in addition to other, more easily accessible noise data in the objective function. Furthermore, we strengthen the underlying integer linear model by proving a convex hull result about an associated class of polytopes, which has applications beyond this work. We evaluate the proposed method by characterizing crosstalk noise for two chips with up to 127 qubits and leverage the resulting data to improve the approximation ratio of the Quantum Approximate Optimization Algorithm by up to 10 % compared to other established noise-aware routing methods. Our work clearly demonstrates the gains of including noise data when mapping abstract quantum circuits to hardware native ones.

math.OC↗
arXiv2023

Improving reconstructions in nanotomography for homogeneous materials via mathematical optimization

Compressed sensing is an image reconstruction technique to achieve high-quality results from limited amount of data. In order to achieve this, it utilizes prior knowledge about the samples that shall be reconstructed. Focusing on image reconstruction in nanotomography, this work proposes enhancements by including additional problem-specific knowledge. In more detail, we propose further classes of algebraic inequalities that are added to the compressed sensing model. The first consists in a valid upper bound on the pixel brightness. It only exploits general information about the projections and is thus applicable to a broad range of reconstruction problems. The second class is applicable whenever the sample material is of roughly homogeneous composition. The model favors a constant density and penalizes deviations from it. The resulting mathematical optimization models are algorithmically tractable and can be solved to global optimality by state-of-the-art available implementations of interior point methods. In order to evaluate the novel models, obtained results are compared to existing image reconstruction methods, tested on simulated and experimental data sets. The experimental data comprise one 360{\deg} electron tomography tilt series of a macroporous zeolite particle and one absorption contrast nano X-ray computed tomography (nano-CT) data set of a copper microlattice structure. The enriched models are optimized quickly and show improved reconstruction quality, outperforming the existing models. Promisingly, our approach yields superior reconstruction results, particularly when information about the samples is available for a small number of tilt angles only

cond-mat.mtrl-sci↗
arXiv2023

A Framework for Data-Driven Explainability in Mathematical Optimization

Advancements in mathematical programming have made it possible to efficiently tackle large-scale real-world problems that were deemed intractable just a few decades ago. However, provably optimal solutions may not be accepted due to the perception of optimization software as a black box. Although well understood by scientists, this lacks easy accessibility for practitioners. Hence, we advocate for introducing the explainability of a solution as another evaluation criterion, next to its objective value, which enables us to find trade-off solutions between these two criteria. Explainability is attained by comparing against (not necessarily optimal) solutions that were implemented in similar situations in the past. Thus, solutions are preferred that exhibit similar features. Although we prove that already in simple cases the explainable model is NP-hard, we characterize relevant polynomially solvable cases such as the explainable shortest path problem. Our numerical experiments on both artificial as well as real-world road networks show the resulting Pareto front. It turns out that the cost of enforcing explainability can be very small.

math.OC↗
arXiv2023

Quality Control in Particle Precipitation via Robust Optimization

In this work, we propose a robust optimization approach to mitigate the impact of uncertainties in particle precipitation. Our model incorporates partial differential equations, more particular nonlinear and nonlocal population balance equations to describe particle synthesis. The goal of the optimization problem is to design products with desired size distributions. Recognizing the impact of uncertainties, we extend the model to hedge against them. We emphasize the importance of robust protection to ensure the production of high-quality particles. To solve the resulting robust problem, we enhance a novel adaptive bundle framework for nonlinear robust optimization that integrates the exact method of moments approach for solving the population balance equations. Computational experiments performed with the integrated algorithm focus on uncertainties in the total mass of the system as it greatly influence the quality of the resulting product. Using realistic parameter values for quantum dot synthesis, we demonstrate the efficiency of our integrated algorithm. Furthermore, we find that the unprotected process fails to achieve the desired particle characteristics, even for small uncertainties, which highlights the necessity of the robust process. The latter consistently outperforms the unprotected process in quality of the obtained product, in particular in perturbed scenarios.

math.OC↗
arXiv2023

A Digital Twin to overcome long-time challenges in Photovoltaics

The recent successes of emerging photovoltaics (PV) such as organic and perovskite solar cells are largely driven by innovations in material science. However, closing the gap to commercialization still requires significant innovation to match contradicting requirements such as performance, longevity and recyclability. The rate of innovation, as of today, is limited by a lack of design principles linking chemical motifs to functional microscopic structures, and by an incapacity to experimentally access microscopic structures from investigating macroscopic device properties. In this work, we envision a layout of a Digital Twin for PV materials aimed at removing both limitations. The layout combines machine learning approaches, as performed in materials acceleration platforms (MAPs), with mathematical models derived from the underlying physics and digital twin concepts from the engineering world. This layout will allow using high-throughput (HT) experimentation in MAPs to improve the parametrization of quantum chemical and solid-state models. In turn, the improved and generalized models can be used to obtain the crucial structural parameters from HT data. HT experimentation will thus yield a detailed understanding of generally valid structure-property relationships, enabling inverse molecular design, that is, predicting the optimal chemical structure and process conditions to build PV devices satisfying a multitude of requirements at the same time. After motivating our proposed layout of the digital twin with causal relationships in material science, we discuss the current state of the enabling technologies, already being able to yield insight from HT data today. We identify open challenges with respect to the multiscale nature of PV materials and the needed volume and diversity of data, and mention promising approaches to address these challenges.

cond-mat.mtrl-sci↗
arXiv2023

Data-driven Distributionally Robust Optimization over Time

Stochastic Optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. As the latter is often unknown, Distributionally Robust Optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy which converges at a rate of $\mathcal{O}(\log T / \sqrt{T})$, where $T$ is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations.

math.OC↗
arXiv2023

Gamma counterparts for robust nonlinear combinatorial and discrete optimization

Gamma uncertainty sets have been introduced for adjusting the degree of conservatism of robust counterparts of (discrete) linear programs. The contribution of this paper is a generalization of this approach to (mixed integer) nonlinear optimization programs. We focus on the cases in which the uncertainty is linear or concave but also derive formulations for the general case. By applying reformulation techniques that have been established for nonlinear inequalities under uncertainty, we derive equivalent formulations of the robust counterpart that are not subject to uncertainty. The computational tractability depends on the structure of the functions under uncertainty and the geometry of its uncertainty set. We present cases where the robust counterpart of a nonlinear combinatorial program is solvable with a polynomial number of oracle calls for the underlying nominal program. Furthermore, we present robust counterparts for practical examples, namely for (discrete) linear, quadratic and piecewise linear settings. Keywords: Budget Uncertainty, Discrete Optimization, Combinatorial Optimization,

math.OC↗
arXiv2023

Enhancing Quantum Algorithms for Quadratic Unconstrained Binary Optimization via Integer Programming

To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing realizes, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem or, equivalently, for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of QAOA, a parameterized quantum algorithm, by deriving optimal parameters for special instances which motivates a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.

quant-ph↗
arXiv2023

A Safe Approximation Based on Mixed-Integer Optimization for Non-Convex Distributional Robustness Governed by Univariate Indicator Functions

In this work, we present an algorithmically tractable safe approximation of distributionally robust optimization (DRO) problems that contain univariate indicator functions. The latter appear in different applications, but render the model nonlinear and nonconvex. The considered ambiguity sets can exploit moment information. Typically, reformulation approaches using duality theory need to make strong assumptions on the structure of the underlying constraints, such as convexity in the decisions or concavity in the uncertainty which cannot be assumed in our setting. We nevertheless present an equivalent semi-infinite reformulation that is subsequently approximated by a discretized counterpart. Under mild assumptions, the latter provides a safe approximation that is formulated as a tractable mixed-integer linear problem, which can be solved by available standard software. Obtained solutions are guaranteed to be feasible for the original distributionally robust problem. Although we show that in general convergence to the true DRO problem cannot be expected, we furthermore prove that the approximation of the adversarial problem indeed converges to its true value for increasingly fine discretization. On the practical side, the approach is made concrete for a challenging, fundamental task in material design, namely in particle separation. Computational results for a realistic setting show that the safe approximation yields robust solutions of high-quality and can be computed within short time.

math.OC↗
arXiv2022

Improving Quantum Computation by Optimized Qubit Routing

In this work we propose a high-quality decomposition approach for qubit routing by swap insertion. This optimization problem arises in the context of compiling quantum algorithms onto specific quantum hardware. Our approach decomposes the routing problem into an allocation subproblem and a set of token swapping problems. This allows us to tackle the allocation part and the token swapping part separately. Extracting the allocation part from the qubit routing model of Nannicini et al. (arXiv:2106.06446), we formulate the allocation subproblem as a binary program. Herein, we employ a cost function that is a lower bound on the overall routing problem objective. We strengthen the linear relaxation by novel valid inequalities. For the token swapping part we develop an exact branch-and-bound algorithm. In this context, we improve upon known lower bounds on the token swapping problem. Furthermore, we enhance an existing approximation algorithm. We present numerical results for the integrated allocation and token swapping problem. Obtained solutions may not be globally optimal due to the decomposition and the usage of an approximation algorithm. However, the solutions are obtained fast and are typically close to optimal. In addition, there is a significant reduction in the number of gates and output circuit depth when compared to state-of-the-art heuristics. Reducing these figures is crucial for minimizing noise when running quantum algorithms on near-term hardware. As a consequence, using the novel decomposition approach leads to compiled algorithms with improved quality. Indeed, when compiled with the novel routing procedure and executed on real hardware, our experimental results for quantum approximate optimization algorithms show an significant increase in solution quality in comparison to standard routing methods.

quant-ph↗
arXiv2022

Robust static and dynamic maximum flows

We study the robust maximum flow problem and the robust maximum flow over time problem where a given number of arcs $\Gamma$ may fail or may be delayed. Two prominent models have been introduced for these problems: either one assigns flow to arcs fulfilling weak flow conservation in any scenario, or one assigns flow to paths where an arc failure or delay affects a whole path. We provide a unifying framework by presenting novel general models, in which we assign flow to subpaths. These models contain the known models as special cases and unify their advantages in order to obtain less conservative robust solutions. We give a thorough analysis with respect to complexity of the general models. In particular, we show that the general models are essentially NP-hard, whereas, e.g. in the static case with $\Gamma = 1$ an optimal solution can be computed in polynomial time. Further, we answer the open question about the complexity of the dynamic path model for $\Gamma = 1$. We also compare the solution quality of the different models. In detail, we show that the general models have better robust optimal values than the known models and we prove bounds on these gaps.

math.OC↗
arXiv2021

Robust Market Equilibria under Uncertain Cost

This work studies equilibrium problems under uncertainty where firms maximize their profits in a robust way when selling their output. Robust optimization plays an increasingly important role when best guaranteed objective values are to be determined, independently of the specific distributional assumptions regarding uncertainty. In particular, solutions are to be determined that are feasible regardless of how the uncertainty manifests itself within some predefined uncertainty set. Our mathematical analysis adopts the robust optimization perspective in the context of equilibrium problems. First, we present structural insights for a single-stage, nonadjustable robust setting. We then go one step further and study the more complex two-stage or adjustable case where a part of the variables can adjust to the realization of the uncertainty. We compare equilibrium outcomes with the corresponding centralized robust optimization problem where thesum of all profits are maximized. As we find, the market equilibrium for the perfectly competitive firms differs from the solution of the robust central planner, which is in stark contrast to classical results regarding the efficiency of market equilibria with perfectly competitive firms. For the different scenarios considered, we furthermore are able to determine the resulting price of anarchy. In the case of non-adjustable robustness, for fixed demand in every time step the price of anarchy is bounded whereas it is unbounded if the buyers are modeled by elastic demand functions. For the two-stage adjustable setting, we show how to compute subsidies for the firms that lead to robust welfareoptimal equilibria.

math.OC↗
arXiv2020

Affinely Adjustable Robust Linear Complementarity Problems

Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite their close relation to optimization, the protection of LCPs against uncertainties -- especially in the sense of robust optimization -- is still in its infancy. During the last years, robust LCPs have only been studied using the notions of strict and $\Gamma$-robustness. Unfortunately, both concepts lead to the problem that the existence of robust solutions cannot be guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the latter, a part of the LCP solution is allowed to adjust via a function that is affine in the uncertainty. We show that this notion of robustness allows to establish strong characterizations of solutions for the cases of uncertain matrix and vector, separately, from which existence results can be derived. Our main results are valid for the case of an uncertain LCP vector. Here, we additionally provide sufficient conditions on the LCP matrix for the uniqueness of a solution. Moreover, based on characterizations of the affinely adjustable robust solutions, we derive a mixed-integer programming formulation that allows to solve the corresponding robust counterpart. If, in addition, the certain LCP matrix is positive semidefinite, we prove polynomial-time solvability and uniqueness of robust solutions. If the LCP matrix is uncertain, characterizations of solutions are developed for every nominal matrix, i.e., these characterizations are, in particular, independent of the definiteness of the nominal matrix. Robust solutions are also shown to be unique for positive definite LCP matrix but both uniqueness and mixed-integer programming formulations still remain open problems if the nominal LCP matrix is not psd.

math.OC↗
arXiv2018

Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations

In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robust feasibility problems with an empty first stage. This is equivalent to deciding whether the uncertainty set is contained within the projection of the feasible region onto the uncertainty-space. Moreover, the considered sets are assumed to be described by polynomials. For answering this question, two very general approaches using methods from polynomial optimization are presented - one for showing feasibility and one for showing infeasibility. The developed methods are approximated through sum of squares polynomials and solved using semidefinite programs. Deciding robust feasibility and infeasibility is important for gas network operations, which is a nonconvex feasibility problem where the feasible set is described by a composition of polynomials with the absolute value function. Concerning the gas network problem, different topologies are considered. It is shown that a tree structured network can be decided exactly using linear programming. Furthermore, a method is presented to reduce a tree network with one additional arc to a single cycle network. In this case, the problem can be decided by eliminating the absolute value functions and solving the resulting linearly many polynomial optimization problems. Lastly, the effectivity of the methods is tested on a variety of small cyclic networks. It turns out that for instances where robust feasibility or infeasibility can be decided successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically is sufficient.

math.OC↗