arXiv subjects

Karl-Heinz Küfer

Publications and source records attributed to Karl-Heinz Küfer.

13 recordsLinked to original sources

arXiv2026

Immunity to Increasing Condition Numbers of Linear Superiorization versus Linear Programming

Given a family of linear constraints and a linear objective function one can consider whether to apply a Linear Programming (LP) algorithm or use a Linear Superiorization (LinSup) algorithm on this data. In the LP methodology one aims at finding an optimal point, i.e., a point that fulfills the constraints and has the minimal value of the objective function over these constraints. The Linear Superiorization approach considers the same data as in linear programming problems but instead of attempting to solve with linear programming methods it employs perturbation resilient feasibility-seeking algorithms that steer the iterations toward a feasible point with reduced (not necessarily minimal) objective function value. This aim of the superiorization method (SM) is less demanding than aiming to reach full-fledged constrained optimality and it places more importance on reaching feasibility than on reaching optimality. Previous studies (e.g., [1]) compared LP and LinSup in terms of their respective outputs and the resources they use. Here, we investigate classical LP approaches and LinSup in terms of their sensitivity to condition numbers of the system of linear constraints. Condition numbers are a measure for the impact of deviations in the input data on the output of a problem and, in particular, they describe the factor of error propagation when given wrong or erroneous data. Therefore, the ability of LP and LinSup to cope with increased condition numbers, thus with illposed problems, is an important matter to consider which was not studied until now. We investigate experimentally the advantages and disadvantages of both LP and LinSup on exemplary sets of data of problems of linear programming with multiple condition numbers and different problem dimensions.

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arXiv2026

An adaptive discretization algorithm for locally optimal experimental design with constraints

We develop a novel iterative algorithm for locally optimal experimental design under constraints, like budget or performance constraints. It is an adaptive discretization algorithm. In every iteration, a discretized version of the constrained-design problem is solved and then the discretization is adaptively refined by adding an approximate violator of a suitable sufficient $\eps$-optimality condition for the current design. We prove that with $\eps = 0$, our algorithm converges to an optimal design and that with $\eps > 0$, our algorithm finitely terminates at an $\eps$-optimal design. Compared to the existing algorithms on constrained experimental design, our algorithm comes with considerably less computational effort because the nonlinear subproblems in our algorithm have a smaller dimension and have to be solved only approximately and only in selected iterations (typically the last few). Additionally, our algorithm covers a considerably larger class of constraints. We demonstrate the good convergence properties of the algorithm on experimental design problems from chemical engineering that feature time and yield constraints.

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arXiv2026

Multi-Criteria Inverse Robustness in Radiotherapy Planning Using Semidefinite Programming

Radiotherapy planning naturally leads to a multi-criteria optimization problem which is subject to different sources of uncertainty. In order to find the desired treatment plan, a decision maker must balance these objectives as well as the level of robustness towards uncertainty against each other. This paper showcases a quantitative approach to do so, which combines the theoretical model with the ability to deal with practical challenges. To this end, the uncertainty, which can be expressed via the so-called dose-influence matrix, is modelled using interval matrices. We use inverse robustness to introduce an additional objective, which aims to maximize the volume of the uncertainty set. A multi-criteria approach allows to handle the uncertainty while keeping appropriate values of the other objective functions. We solve the resulting quadratically constrained quadratic optimization problem (QCQP) by first relaxing it to a convex semidefinite problem (SDP) and then reconstructing optimal solutions of the QCQP from solutions of the SDP.

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arXiv2024

An Adaptive Three-Stage Algorithm For Solving Adjustable Min-Max-Regret Problems

This work uniquely combines an affine linear decision rule known from adjustable robustness with min-max-regret robustness. By doing so, the advantages of both concepts can be obtained with an adjustable solution that is not over-conservative. This combination results in a bilevel optimization problem. For solving this problem, a three-stage algorithm which uses adaptive discretization of the uncertainty set via two criteria is presented and its convergence is proven. The algorithm is applicable for an example of optimizing a robust pump operation plan for a drinking water supply system facing uncertain demand. The algorithm shows a notable ability to scale, presenting an opportunity to solve larger instances that might challenge existing optimization approaches.

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arXiv2024

Multicriteria Adjustable Regret Robust Optimization for Building Energy Supply Design

Optimizing a building's energy supply design is a task with multiple competing criteria, where not only monetary but also, for example, an environmental objective shall be taken into account. Moreover, when deciding which storages and heating and cooling units to purchase (here-and-now-decisions), there is uncertainty about future developments of prices for energy, e.g. electricity and gas. This can be accounted for later by operating the units accordingly (wait-and-see-decisions), once the uncertainty revealed itself. Therefore, the problem can be modeled as an adjustable robust optimization problem. We combine adjustable robustness and multicriteria optimization for the case of building energy supply design and solve the resulting problem using a column and constraint generation algorithm in combination with an $\varepsilon$-constraint approach. In the multicriteria adjustable robust problem, we simultaneously minimize worst-case cost regret and carbon emissions. We take into account future price uncertainties and consider the results in the light of information gap decision theory to find a trade-off between security against price fluctuations and over-conservatism. We present the model, a solution strategy and discuss different application scenarios for a case study building.

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arXiv2024

Multicriteria Adjustable Robustness

Multicriteria adjustable robust optimization (MARO) problems arise in a wide variety of practical settings, for example, in the design of a building's energy supply. However, no general approaches, neither for the characterization of solutions to this problem class, nor potential solution methods, are available in the literature so far. We give different definitions for efficient solutions to MARO problems and look at three computational concepts to deal with the problems. These computational concepts can also be understood as additional solution definitions. We assess the advantages and disadvantages of the different computational approaches and analyze their connections to our initial definitions of MARO-efficiency. We observe that an $\varepsilon$-constraint inspired first-scalarize-then-robustify computational approach is beneficial because it provides an efficient set that is easy to understand for decision makers and provides tight bounds on the worst-case evaluation for a particular efficient solution. In contrast, a weighted sum first-scalarize-then-robustify approach keeps the problem structure more simple but is only beneficial if the desired trade-off between objectives is already known because the efficient set might look ambiguous. Further, we demonstrate that a first-robustify procedure only gives bad bounds and can be too optimistic as well as too pessimistic.

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arXiv2022

Hidden Symmetries and Model Reduction in Markov Decision Processes: Explained and Applied to the Multi-period Newsvendor Problem

Symmetry breaking is a common approach for model reduction of Markov decision processes (MDPs). This approach only uses directly accessible symmetries such as geometric symmetries. For some MDPs, it is possible to transform them equivalently such that symmetries become accessible -- we call this type of symmetries hidden symmetries. For these MDPs, hidden symmetries allow substantially better model reduction compared to directly accessible symmetries. The main idea is to reveal a hidden symmetry by altering the reward structure and then exploit the revealed symmetry by forming a quotient MDP. The quotient MDP is the reduced MDP, since it is sufficient to solve the quotient MDP instead of the original one. In this paper, we introduce hidden symmetries and the associated concept of model reduction. We demonstrate this concept on the multi-period newsvendor problem, which is the newsvendor problem considered over an infinite number of days. In this way, we show that hidden symmetries can reduce problems that directly accessible symmetries cannot, and present a basic idea of revealing hidden symmetries in multi-period problems. The presented approach can be extended to more sophisticated problems.

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arXiv2022

Computing T-optimal designs via nested semi-infinite programming and twofold adaptive discretization

Modeling real processes often results in several suitable models. In order to be able to distinguish, or discriminate, which model best represents a phenomenon, one is interested, e.g., in so-called T-optimal designs. These consist of the (design) points from a generally continuous design space at which the models deviate most from each other, under the condition that they are best fitted to those points. Thus, the T-criterion represents a bi-level optimization problem, which can be transferred into a semi-infinite one, but whose solution is very unstable or time consuming for non-linear models and non-convex lower- and upper-level problems. If one considers only a finite number of possible design points, a numerically well tractable linear semi-infinite optimization problem arises. Since this is only an approximation of the original model discrimination problem, we propose an algorithm which alternately and adaptively refines discretizations of the parameter as well as of the design space and, thus, solves a sequence of LSIPs. We prove convergence of our method and its subroutine and show on the basis of discrimination tasks from process engineering that our approach is stable and can outperform the known methods.

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arXiv2020

A Ray Tracing Technique for the Navigation on a Non-convex Pareto Front

A new interactive approach to navigate on approximations of in general non-convex but connected Pareto fronts is introduced. Given a finite number of precalculated representative Pareto-efficient solutions, an adapted Delaunay triangulation is generated. Based on interpolation and ray tracing techniques, real time navigation in the vicinity of Pareto-optimal solutions is made possible.

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arXiv2019

An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence

Semi-infinite programming can be used to model a large variety of complex optimization problems. The simple description of such problems comes at a price: semi-infinite problems are often harder to solve than finite nonlinear problems. In this paper we combine a classical adaptive discretization method developed by Blankenship and Falk and techniques regarding a semi-infinite optimization problem as a bi-level optimization problem. We develop a new adaptive discretization method which combines the advantages of both techniques and exhibits a quadratic rate of convergence. We further show that a limit of the iterates is a stationary point, if the iterates are stationary points of the approximate problems.

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arXiv2018

A generalized projection-based scheme for solving convex constrained optimization problems

In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility problems by iteratively constraining the objective function from above until the feasibility problem is inconsistent. For each of the feasibility problems one may apply any of the existing projection methods for solving it. In particular, the scheme allows the use of subgradient projections and does not require exact projections onto the constraints sets as in existing similar methods. We also apply the newly introduced concept of superiorization to optimization formulation and compare its performance to our scheme. We provide some numerical results for convex quadratic test problems as well as for real-life optimization problems coming from medical treatment planning.

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arXiv2016

Speedup of lexicographic optimization by superiorization and its applications to cancer radiotherapy treatment

Multicriteria optimization problems occur in many real life applications, for example in cancer radiotherapy treatment and in particular in intensity modulated radiation therapy (IMRT). In this work we focus on optimization problems with multiple objectives that are ranked according to their importance. We solve these problems numerically by combining lexicographic optimization with our recently proposed level set scheme, which yields a sequence of auxiliary convex feasibility problems; solved here via projection methods. The projection enables us to combine the newly introduced superiorization methodology with multicriteria optimization methods to speed up computation while guaranteeing convergence of the optimization. We demonstrate our scheme with a simple 2D academic example (used in the literature) and also present results from calculations on four real head neck cases in IMRT (Radiation Oncology of the Ludwig-Maximilians University, Munich, Germany) for two different choices of superiorization parameter sets suited to yield fast convergence for each case individually or robust behavior for all four cases.

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arXiv2014

Molecular modelling and simulation of the surface tension of real quadrupolar fluids

Molecular modelling and simulation of the surface tension of fluids with force fields is discussed. 29 real fluids are studied, including nitrogen, oxygen, carbon dioxide, carbon monoxide, fluorine, chlorine, bromine, iodine, ethane, ethylene, acetylene, propyne, propylene, propadiene, carbon disulfide, sulfur hexafluoride, and many refrigerants. The fluids are represented by two-centre Lennard-Jones plus point quadrupole models from the literature. These models were adjusted only to experimental data of the vapour pressure and saturated liquid density so that the results for the surface tension are predictions. The deviations between the predictions and experimental data for the surface tension are of the order of 20 percent. The surface tension is usually overestimated by the models. For further improvements, data on the surface tension can be included in the model development. A suitable strategy for this is multi-criteria optimization based on Pareto sets. This is demonstrated using the model for carbon dioxide as an example.

physics.comp-ph↗