On M-stationarity conditions in MPECs and the associated qualification conditions

2017 ◽  
Vol 168 (1-2) ◽  
pp. 229-259 ◽  
Author(s):  
Lukáš Adam ◽  
René Henrion ◽  
Jiří Outrata
Keyword(s):  
Qualification Conditions ◽  
Stationarity Conditions

scholarly journals On implicit variables in optimization theory

2021 ◽  
Vol Volume 2 (Original research articles) ◽  
Author(s):  
Matúš Benko ◽  
Patrick Mehlitz
Keyword(s):  
Optimization Problems ◽  
Constraint Qualifications ◽  
Optimization Theory ◽  
Mathematical Optimization ◽  
General Setting ◽  
Necessary Optimality Conditions ◽  
Detailed Comparison ◽  
Mordukhovich Stationarity ◽  
Stationarity Conditions ◽  
The One

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory. Comment: 34 pages

scholarly journals Intervention Models in Functional Connectivity Identification Applied to fMRI

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
João Ricardo Sato ◽  
Daniel Yasumasa Takahashi ◽  
Ellison Fernando Cardoso ◽  
Maria da Graça Morais Martin ◽  
Edson Amaro Júnior ◽  
...  
Keyword(s):  
Structural Equation ◽  
Motor Task ◽  
Equation Modeling ◽  
Association Measure ◽  
Intervention Models ◽  
Simple Motor ◽  
Simple Motor Task ◽  
Stationarity Conditions ◽  
Neuroimaging Techniques ◽  
Illustrative Application

Recent advances in neuroimaging techniques have provided precise spatial localization of brain activation applied in several neuroscience subareas. The development of functional magnetic resonance imaging (fMRI), based on the BOLD signal, is one of the most popular techniques related to the detection of neuronal activation. However, understanding the interactions between several neuronal modules is also an important task, providing a better comprehension about brain dynamics. Nevertheless, most connectivity studies in fMRI are based on a simple correlation analysis, which is only an association measure and does not provide the direction of information flow between brain areas. Other proposed methods like structural equation modeling (SEM) seem to be attractive alternatives. However, this approach assumes prior information about the causality direction and stationarity conditions, which may not be satisfied in fMRI experiments. Generally, the fMRI experiments are related to an activation task; hence, the stimulus conditions should also be included in the model. In this paper, we suggest an intervention analysis, which includes stimulus condition, allowing a nonstationary modeling. Furthermore, an illustrative application to real fMRI dataset from a simple motor task is presented.

Non-linear threshold autoregressive models for non-linear random vibrations

1981 ◽  
Vol 18 (02) ◽  
pp. 443-451 ◽  
Author(s):  
Tohru Ozaki
Keyword(s):  
Real Data ◽  
Time Series Models ◽  
Random Vibrations ◽  
Restoring Force ◽  
Threshold Autoregressive ◽  
Threshold Autoregressive Model ◽  
Linear Threshold ◽  
Non Linear ◽  
Stationarity Conditions ◽  
Structural Aspects

Time series models for non-linear random vibrations are discussed from the viewpoint of the specification of the dynamics of the damping and restoring force of vibrations, and a non-linear threshold autoregressive model is introduced. Typical non-linear phenomena of vibrations are demonstrated using the models. Stationarity conditions and some structural aspects of the model are briefly discussed. Applications of the model in the statistical analysis of real data are also shown with numerical results.

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