51 research outputs found
The Radius of Metric Subregularity
There is a basic paradigm, called here the radius of well-posedness, which
quantifies the "distance" from a given well-posed problem to the set of
ill-posed problems of the same kind. In variational analysis, well-posedness is
often understood as a regularity property, which is usually employed to measure
the effect of perturbations and approximations of a problem on its solutions.
In this paper we focus on evaluating the radius of the property of metric
subregularity which, in contrast to its siblings, metric regularity, strong
regularity and strong subregularity, exhibits a more complicated behavior under
various perturbations. We consider three kinds of perturbations: by Lipschitz
continuous functions, by semismooth functions, and by smooth functions,
obtaining different expressions/bounds for the radius of subregularity, which
involve generalized derivatives of set-valued mappings. We also obtain
different expressions when using either Frobenius or Euclidean norm to measure
the radius. As an application, we evaluate the radius of subregularity of a
general constraint system. Examples illustrate the theoretical findings.Comment: 20 page
Teaching mechanics with individual exercise assignments and automated correction
Solving exercise problems by yourself is a vital part of developing a
mechanical understanding. Yet, most mechanics lectures have more than 200
participants, so the workload for manually creating and correcting assignments
limits the number of exercises. The resulting example pool is usually much
smaller than the number of participants, making verifying whether students can
solve problems themselves considerably harder. At the same time, unreflected
copying of tasks already solved does not foster the understanding of the
subject and leads to a false self-assessment. We address these issues by
providing a scalable approach for creating, distributing, and correcting
exercise assignments for problems related to statics, strength of materials,
dynamics, and hydrostatics. The overall concept allows us to provide individual
exercise assignments for each student. A quantitative survey among students of
our recent statics lecture assesses the acceptance of our teaching tool. The
feedback indicates a clear added value for the lecture, which fosters
self-directed and reflective learning
Advances in structure elucidation of small molecules using mass spectrometry
The structural elucidation of small molecules using mass spectrometry plays an important role in modern life sciences and bioanalytical approaches. This review covers different soft and hard ionization techniques and figures of merit for modern mass spectrometers, such as mass resolving power, mass accuracy, isotopic abundance accuracy, accurate mass multiple-stage MS(n) capability, as well as hybrid mass spectrometric and orthogonal chromatographic approaches. The latter part discusses mass spectral data handling strategies, which includes background and noise subtraction, adduct formation and detection, charge state determination, accurate mass measurements, elemental composition determinations, and complex data-dependent setups with ion maps and ion trees. The importance of mass spectral library search algorithms for tandem mass spectra and multiple-stage MS(n) mass spectra as well as mass spectral tree libraries that combine multiple-stage mass spectra are outlined. The successive chapter discusses mass spectral fragmentation pathways, biotransformation reactions and drug metabolism studies, the mass spectral simulation and generation of in silico mass spectra, expert systems for mass spectral interpretation, and the use of computational chemistry to explain gas-phase phenomena. A single chapter discusses data handling for hyphenated approaches including mass spectral deconvolution for clean mass spectra, cheminformatics approaches and structure retention relationships, and retention index predictions for gas and liquid chromatography. The last section reviews the current state of electronic data sharing of mass spectra and discusses the importance of software development for the advancement of structure elucidation of small molecules
CONVERGENT SEMIDEFINITE PROGRAMMING RELAXATIONS FOR GLOBAL BILEVEL POLYNOMIAL OPTIMIZATION PROBLEMS *
International audienceIn this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper-and the lower-level problems are polynomi-als. We present methods for finding its global minimizers and global minimum using a sequence of semidefinite programming (SDP) relaxations and provide convergence results for the methods. Our scheme for problems with a convex lower-level problem involves solving a transformed equivalent single-level problem by a sequence of SDP relaxations, whereas our approach for general problems involving a nonconvex polynomial lower-level problem solves a sequence of approximation problems via another sequence of SDP relaxations
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