scholarly journals Minimization of Vectors of Curvilinear Functionals on Second-Order Jet Bundle: Dual Program Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Mihai Postolache
Keyword(s):  
Variational Problem ◽  
Necessary Conditions ◽  
Optimization Theory ◽  
General Setting ◽  
Second Order ◽  
Mechanical Work ◽  
Program Theory ◽  
Jet Bundle ◽  
Dual Program ◽  
Suitable Framework

A previous paper (2011), Pitea and Postolache, considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework offered by the second-order jet bundle, and initiated an optimization theory for this class of problems by introducing necessary conditions. As natural continuation of these results, the present work introduces a dual program theory, the general setting, and the theory which is new as a whole, containing our results.

scholarly journals Sufficient Efficiency Conditions for Vector Ratio Problem on the Second-Order Jet Bundle

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Ariana Pitea
Keyword(s):  
Optimality Conditions ◽  
Mechanical Engineering ◽  
Optimization Theory ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Second Order ◽  
Mechanical Work ◽  
Jet Bundle ◽  
Multiobjective Variational Problems ◽  
Efficiency Conditions

Motivated by its possible applications in mechanics and mechanical engineering, in our previous published work (Pitea and Postolache, 2011), we initiated an optimization theory for the second-order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary optimality conditions. As natural continuation of these results, the present work introduces a study of sufficient efficiency conditions.

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

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

Argumentation and Reflection in Plato’s Gigantomachia (Sophist 245e6–249d5)

2018 ◽  
Vol 100 (3) ◽  
pp. 241-285
Author(s):  
Daniel Vázquez
Keyword(s):  
Necessary Conditions ◽  
Second Order ◽  
Complex Activity ◽  
Causal Relations ◽  
First Order ◽  
Theoretical Assumptions ◽  
Argumentative Strategies ◽  
New Interpretation ◽  
Argumentative Structure ◽  
The Impact

Abstract This paper argues that Plato’s gigantomachia is simultaneously concerned with first-order arguments about metaphysics and epistemology and with second-order arguments that reflect on the impact of ethical components, argumentative strategies and theoretical assumptions in the conversation. This complex argumentative structure reveals, I suggest, an organic and systematic conception of philosophy where all the elements are interdependent. This interpretation has four consequences, two at the second-order level, and two concerning the first-order arguments. First, it shows that there are methodological and ethical requirements without which philosophy is impossible. Second, it shows that the text does not refute materialism but tries to reflect the necessary conditions to consider possible the existence of incorporeal beings. Third, it argues that the text assumes a conception of knowledge where knowing something is a complex activity composed of two causal relations. Finally, it offers a new interpretation of the overall conclusion of the passage.

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