arXiv subjects

Martin Schmidt

Publications and source records attributed to Martin Schmidt.

At least 37 records · Page 2Linked to original sources

arXiv2024

On Coupling Constraints in Linear Bilevel Optimization

It is well-known that coupling constraints in linear bilevel optimization can lead to disconnected feasible sets, which is not possible without coupling constraints. However, there is no difference between linear bilevel problems with and without coupling constraints w.r.t. their complexity-theoretical hardness. In this note, we prove that, although there is a clear difference between these two classes of problems in terms of their feasible sets, the classes are equivalent on the level of optimal solutions. To this end, given a general linear bilevel problem with coupling constraints, we derive a respective problem without coupling constraints and prove that it has the same optimal solutions (when projected back to the original variable space).

math.OC↗

arXiv2023

On the Relation Between Affinely Adjustable Robust Linear Complementarity and Mixed-Integer Linear Feasibility Problems

We consider adjustable robust linear complementarity problems and extend the results of Biefel et al. (2022) towards convex and compact uncertainty sets. Moreover, for the case of polyhedral uncertainty sets, we prove that computing an adjustable robust solution of a given linear complementarity problem is equivalent to solving a properly chosen mixed-integer linear feasibility problem.

math.OC↗

arXiv2023

Mixed-Integer Quadratic Optimization and Iterative Clustering Techniques for Semi-Supervised Support Vector Machines

Among the most famous algorithms for solving classification problems are support vector machines (SVMs), which find a separating hyperplane for a set of labeled data points. In some applications, however, labels are only available for a subset of points. Furthermore, this subset can be non-representative, e.g., due to self-selection in a survey. Semi-supervised SVMs tackle the setting of labeled and unlabeled data and can often improve the reliability of the results. Moreover, additional information about the size of the classes can be available from undisclosed sources. We propose a mixed-integer quadratic optimization (MIQP) model that covers the setting of labeled and unlabeled data points as well as the overall number of points in each class. Since the MIQP's solution time rapidly grows as the number of variables increases, we introduce an iterative clustering approach to reduce the model's size. Moreover, we present an update rule for the required big-$M$ values, prove the correctness of the iterative clustering method as well as derive tailored dimension-reduction and warm-starting techniques. Our numerical results show that our approach leads to a similar accuracy and precision than the MIQP formulation but at much lower computational cost. Thus, we can solve solve larger problems. With respect to the original SVM formulation, we observe that our approach has even better accuracy and precision for biased samples.

math.OC↗

arXiv2023

Adaptive Nonlinear Optimization of District Heating Networks Based on Model and Discretization Catalogs

We propose an adaptive optimization algorithm for operating district heating networks in a stationary regime. The behavior of hot water flow in the pipe network is modeled using the incompressible Euler equations and a suitably chosen energy equation. By applying different simplifications to these equations, we derive a catalog of models. Our algorithm is based on this catalog and adaptively controls where in the network which model is used. Moreover, the granularity of the applied discretization is controlled in a similar adaptive manner. By doing so, we are able to obtain optimal solutions at low computational costs that satisfy a prescribed tolerance w.r.t. the most accurate modeling level. To adaptively control the switching between different levels and the adaptation of the discretization grids, we derive error measure formulas and a posteriori error measure estimators. Under reasonable assumptions we prove that the adaptive algorithm terminates after finitely many iterations. Our numerical results show that the algorithm is able to produce solutions for problem instances that have not been solvable before.

math.OC↗

arXiv2023

On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to $\varepsilon$-feasibility regarding its nonlinear constraints for an arbitrarily small but positive $\varepsilon$, the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater's constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of $\varepsilon$-feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.

math.OC↗

arXiv2023

A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt et al. (2018, 2021) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems.

math.OC↗

arXiv2023

A Brief Introduction to Robust Bilevel Optimization

Bilevel optimization is a powerful tool for modeling hierarchical decision making processes. However, the resulting problems are challenging to solve - both in theory and practice. Fortunately, there have been significant algorithmic advances in the field so that we can solve much larger and also more complicated problems today compared to what was possible to solve two decades ago. This results in more and more challenging bilevel problems that researchers try to solve today. In this article, we give a brief introduction to one of these more challenging classes of bilevel problems: bilevel optimization under uncertainty using robust optimization techniques. To this end, we briefly state different versions of uncertain bilevel problems that result from different levels of cooperation of the follower as well as on when the uncertainty is revealed. We highlight these concepts using an academic example and discuss recent results from the literature concerning complexity as well as solution approaches. Finally, we discuss that the sources of uncertainty in bilevel optimization are much richer than in single-level optimization and, to this end, introduce the concept of decision uncertainty.

math.OC↗

arXiv2023

NV center magnetometry up to 130 GPa as if at ambient pressure

Engineering a layer of nitrogen-vacancy (NV) centers on the tip of a diamond anvil creates a multipurpose quantum sensors array for high pressure measurements, especially for probing magnetic and superconducting properties of materials. Expanding this concept above 100 GPa appears to be a substantial challenge. We observe that deviatoric stress on the anvil tip sets a limit at 40-50 GPa for practical magnetic measurements based on optically detected magnetic resonance (ODMR) of NV centers under pressure. We show that this limit can be circumvented up to at least 130 GPa by machining a micropillar on the anvil tip to create a quasi-hydrostatic stress environment for the NV centers. This is quantified using the pressure dependence of the diamond Raman shift, the NV ODMR dependence on applied magnetic field, and NV photoluminescence spectral shift. This paves the way for direct and reliable detection of the Meissner effect in superconductors above 100 GPa, such as super-hydrides.

quant-ph↗

arXiv2022

Efficient and all-carbon electrical readout of a NV based quantum sensor

The spin readout of an ensemble of nitrogen-vacancy (NV) centers in diamond can be realized by a photoconductive detection that is a complementary method to the optical detection of the NV electron spin magnetic resonance. Here, we implement the photoconductive detection through graphitic planar electrodes that collect the photocurrent. Graphitic electrodes are patterned using a xenon Focused-Ion Beam on an Optical-Grade quality diamond crystal containing a nitrogen concentration of ~1 ppm and a NV concentration of a few ppb. Resistance and current-voltage characteristics of the NV-doped diamond junction are investigated tuning the 532 nm pump beam intensity. The junction has an ohmic behavior and under a strong bias field, we observe velocity saturation of the optically-induced carriers in the diamond junction. We perform the photoconductive detection in continuous-wave regime of the magnetic resonance of the NV centers ensemble for a magnetic field applied along the <100> and the <111> direction with a magnitude above 100 mT. This technique enables the realization of all-carbon diamond quantum sensors integrating graphitic microstructures for the electrical readout.

quant-ph↗

arXiv2022

Optical properties of SiV and GeV color centers in nanodiamonds under hydrostatic pressures up to 180 GPa

We investigate the optical properties of silicon-vacancy (SiV) and germanium-vacancy (GeV) color centers in nanodiamonds under hydrostatic pressure up to 180 GPa. The nanodiamonds were synthetized by Si or Ge-doped plasma assisted chemical vapor deposition and, for our experiment, pressurized in a diamond anvil cell. Under hydrostatic pressure we observe blue-shifts of the SiV and GeV zero-phonon lines by 17 THz (70 meV) and 78 THz (320 meV), respectively. These measured pressure induced shifts are in good agreement with ab initio calculations that take into account the lattice compression based on the equation of state of diamond and that are extended to the case of the tin-vacancy (SnV) center. This work provides guidance on the use of group-IV-vacancy centers as quantum sensors under extreme pressures that will exploit their specific optical and spin properties induced by their intrinsic inversion-symmetric structure.

quant-ph↗

arXiv2022

Proximal-like algorithms for equilibrium seeking in mixed-integer Nash equilibrium problems

We consider potential games with mixed-integer variables, for which we propose two distributed, proximal-like equilibrium seeking algorithms. Specifically, we focus on two scenarios: i) the underlying game is generalized ordinal and the agents update through iterations by choosing an exact optimal strategy; ii) the game admits an exact potential and the agents adopt approximated optimal responses. By exploiting the properties of integer-compatible regularization functions used as penalty terms, we show that both algorithms converge to either an exact or an $ε$-approximate equilibrium. We corroborate our findings on a numerical instance of a Cournot oligopoly model.

math.OC↗

arXiv2022

Thermometry of an optically levitated nanodiamond

Using the spin properties of nitrogen-vacancy (NV) centers in levitated diamond, we characterize the absorption of single nanodiamonds. We first calibrate the thermometry response of the NV centers embedded in our nanodiamonds. Then, using this calibration, we estimate the absorption cross-section of single levitated nanodiamonds. We show that this absorption is extrinsic and dominated by volumic effects. Our work opens the way to diamond materials optimization for levitation quantum experiments. It also demonstrates optical levitation as a unique platform to characterize material thermal properties at the nanoparticle level.

quant-ph↗

arXiv2022

Hot Brownian motion of optically levitated nanodiamonds

The Brownian motion of a particle hotter than its environment is an iconic out-of-equilibrium system. Its study provides valuable insights into nanoscale thermal effects. Notably, it supplies an excellent diagnosis of thermal effects in optically levitated particles, a promising platform for force sensing and quantum physics tests. Thus, understanding the relevant parameters in this effect is critical. In this context, we test the role of particles' shape and material, using optically levitated nanodiamonds hosting NV centers to measure the particles' internal temperature and center-of-mass dynamics. We present a model to assess the nanodiamond internal temperature from its dynamics, adaptable to other particles. We also demonstrate that other mechanisms affect the nanodiamond dynamics and its stability in the trap. Finally, our work, by showing levitating nanodiamonds as an excellent tool for studying nano-thermal effects, opens prospects for increasing the trapping stability of optically levitated particles.

quant-ph↗

arXiv2021

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 $Γ$-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↗

arXiv2020

Combined synchrotron X-ray diffraction and NV diamond magnetic microscopy measurements at high pressure

We report the possibility to simultaneously perform wide-field nitrogen-vacancy (NV) diamond magnetic microscopy and synchrotron X-ray diffraction (XRD) measurements at high pressure. NV color centers are created on the culet of a diamond anvil which is integrated in a diamond anvil cell for static compression of the sample. The optically detected spin resonance of the NV centers is used to map the stray magnetic field produced by the sample magnetization. Using this combined scheme, the magnetic and structural behaviors can be simultaneously measured. As a proof-of-principle, we record the correlated α-Fe to ε-Fe structural and magnetic transitions of iron that occur here between 15 and 20 GPa at 300 K.

cond-mat.mes-hall↗

arXiv2020

Nonlinear Optimization of District Heating Networks

We develop a complementarity-constrained nonlinear optimization model for the time-dependent control of district heating networks. The main physical aspects of water and heat flow in these networks are governed by nonlinear and hyperbolic 1d partial differential equations. In addition, a pooling-type mixing model is required at the nodes of the network to treat the mixing of different water temperatures. This mixing model can be recast using suitable complementarity constraints. The resulting problem is a mathematical program with complementarity constraints subject to nonlinear partial differential equations describing the physics. In order to obtain a tractable problem, we apply suitable discretizations in space and time, resulting in a finite-dimensional optimization problem with complementarity constraints for which we develop a suitable reformulation with improved constraint regularity. Moreover, we propose an instantaneous control approach for the discretized problem, discuss practically relevant penalty formulations, and present preprocessing techniques that are used to simplify the mixing model at the nodes of the network. Finally, we use all these techniques to solve realistic instances. Our numerical results show the applicability of our techniques in practice.

math.OC↗

arXiv2019

Port-Hamiltonian modeling of district heating networks

This paper provides a first contribution to port-Hamiltonian modeling of district heating networks. By introducing a model hierarchy of flow equations on the network, this work aims at a thermodynamically consistent port-Hamiltonian embedding of the partial differential-algebraic systems. We show that a spatially discretized network model describing the advection of the internal energy density with respect to an underlying incompressible stationary Euler-type hydrodynamics can be considered as a parameter-dependent finite-dimensional port-Hamiltonian system. Moreover, we present an infinite-dimensional port-Hamiltonian formulation for a compressible instationary thermodynamic fluid flow in a pipe. Based on these first promising results, we raise open questions and point out research perspectives concerning structure-preserving discretization, model reduction, and optimization.

math.OC↗

arXiv2019

Functorial Approach to Graph and Hypergraph Theory

We provide a new approach to categorical graph and hypergraph theory by using categorical syntax and semantics. For each monoid $M$ and action on a set $X$, there is an associated presheaf topos of $(X,M)$-graphs where each object can be interpreted as a generalized uniform hypergraph where each edge has cardinality $\#X$ incident vertices (including multiplicity) and where the monoid informs what type of cohesivity the edges possess. One distinguishing feature of $(X,M)$-graphs is the presence of unfixed edges. We prove that unfixed edges are a necessary feature of a category of graphs or uniform hypergraphs if one wants exponentials and effective equivalence relations to exist in the category. The main advantage of separating syntax (the $(X,M)$-graph theories) from semantics (the categories of $(X,M)$-graphs) is the ability to interpret the theory in any cocomplete category. This interpetation functor then yields a nerve-realization adjunction and allows us to transfer structure between the category of $(X,M)$-graphs and the receptive cocomplete category.

math.CO↗