Calculus of Tangents and Beyond

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.23671/VNC.2018.4.9165 Calculus of Tangents and Beyond Kusraev, A. G. , Kutateladze, S. S. Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 4. Abstract: Optimization is the choice of what is most preferable. Geometry and local analysis of nonsmooth objects are needed for variational analysis which embraces optimization. These involved admissible directions and tangents as the limiting positions of the former. The calculus of tangents is one of the main techniques of optimization. Calculus reduces forecast to numbers, which is scalarization in modern parlance. Spontaneous solutions are often labile and rarely optimal. Thus, optimization as well as calculus of tangents deals with inequality, scalarization and stability. The purpose of this article is to give an overview of the modern approach to this range of questions based on non-standard models of set theory. A model of a mathematical theory is usually called nonstandard if the membership within the model has interpretation different from that of set theory. In the recent decades much research is done into the nonstandard methods located at the junctions of analysis and logic. This area requires the study of some new opportunities of modeling that open broad vistas for consideration and solution of various theoretical and applied problems. Keywords: Hadamard cone, Bouligand cone, Clarke cone, general position, operator inequality, Boolean valued analysis, nonstandard analysis Language: English Download the full text For citation: Kusraev A. G., Kutateladze S. S. Calculus of Tangents and Beyond // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], 2017, vol. 19, no. 1, pp. 27-34. DOI 10.23671/VNC.2018.4.9165 + References 1. Clarke F. Nonsmooth Analysis in Systems and Control Theory. Encyclopedia of Complexity and Control Theory. Berlin: Springer-Verlag, 2009. pp. 6271-6184. 2. Kusraev A. G. and Kutateladze S. S. Subdifferential Calculus: Theory and Applications. Moscow: Nauka, 2007. 3. Bell J. L. Set Theory: Boolean Valued Models and Independence Proofs. Oxford: Clarendon Press, 2005. 4. Kusraev A. G. and Kutateladze S. S. Introduction to Boolean Valued Analysis. Moscow: Nauka, 2005. 5. Kanovei V. and Reeken M. Nonstandard Analysis: Axiomatically. Berlin: Springer-Verlag, 2004. 6. Gordon E. I., Kusraev A. G., and Kutateladze S. S. Infinitesimal Analysis: Selected Topics. Moscow: Nauka, 2011. 7. Ariew R. G. W. Leibniz and Samuel Clarke Correspondence. Indianopolis: Hackett Publ. Comp., 2000. 8. Ekeland I. The Best of All Possible Worlds: Mathematics and Destiny. Chicago-London: The Univer. of Chicago Press, 2006. 9. Kusraev A. G. and Kutateladze S. S. Boolean Methods in Positivity. J. Appl. Indust. Math., 2008, vol. 2, no. 1, pp. 81-99. 10. Kutateladze S. S. Mathematics and Economics of Leonid Kantorovich. Sib. Math. J., 2012, vol. 53, no. 1, pp. 1-12. 11. Kutateladze S. S. Harpedonaptae and Abstract Convexity. J. Appl. Indust. Math., 2008, vol. 2, no. 1, pp. 215-221. 12. Kutateladze S. S. Boolean Trends in Linear Inequalities, J. Appl. Indust. Math., 2010, vol. 4, no. 3, pp. 340-348. 13. Kutateladze S. S. Nonstandard tools of nonsmooth analysis, J. Appl. Indust. Math., 2012, vol. 6, no. 3, pp. 332-338. 14. Scott D. Boolean Models and Nonstandard Analysis, Applications of Model Theory to Algebra, Analysis, and Probability. Holt, Rinehart, and Winston, 1969. P. 87-92. 15. Kutateladze S. S. Leibnizian, Robinsonian, and Boolean Valued Monads. J. Appl. Indust. Math., 2011, vol. 5, no. 3, pp. 365-373. ← Contents of issue


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