New perspective on some classical results in analysis and optimization

Abstract

The paper illustrates connections between classical results of Analysis and Optimization. The focus is on new elementary proofs of Implicit Function Theorem, Lusternik's Theorem, and optimality conditions for equality constrained optimization problems. The proofs are based on Fermat's Theorem and the Weierstrass Theorem and do not use the contraction mapping principle or other advanced results of Real Analysis, so they can be used in any introductory course on Optimization or Real Analysis without the requirement of the advanced background in analysis. The paper also presents a simple proof of Implicit Function Theorem in normed linear spaces.

Keywords:

• Implicit Function Theorem
• Lusternik's Theorem
• Fermat's Theorem
• the Weierstrass Theorem
• optimality conditions

2010 Mathematics Subject Classifications:

• 26B10
• 47J07
• 58C15
• 90C30
• 90C46

Acknowledgments

The authors thank the anonymous reviewers for their careful reading of our manuscript and for their constructive comments that helped us greatly improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Professor Tret'yakov and Academician Evtushenko were supported by the Russian Foundation for Basic Research (project No. 17-07-00510) and the RAS Presidium Program (program 27).

Notes on contributors

Olga Brezhneva

Olga Brezhneva is an Associate Professor in the Department of Mathematics at Miami University in Oxford, Ohio. She received her Master's Degree in Applied Mathematics from Moscow State University and her Ph.D. in Mathematics from the Russian Academy of Sciences in 2000. Before coming to Miami University in 2004, she was a postdoctoral associate at the Institute for Mathematics and its Applications at the University of Minnesota. Dr. Brezhneva has publications in the fields of optimization, differential equations, and numerical analysis.

Yuri G. Evtushenko

Yuri G. Evtushenko received the B.Sc. degree in Applied Mathematics from Moscow Institute of Physics and Technology (MIPT) in 1962 and the Doctoral Degree in Physics and Mathematics in 1966. He joined the Computing Center of the USSR Academy of Sciences in 1967. In 1989, he was elected the director of the Computing Center and held the position until 2018. His research interests include theory, algorithms and applications of computational optimization. For many years, Academician Evtushenko heads one of the leading scientific schools of the Russian Federation in the direction of “Methods for solving complex optimization problems.” The school activities have been supported by grants from the Russian President since 1996. Academician Evtushenko was awarded the Prize of the Government of the Russian Federation for the creation of a set of manuals and monographs in the area of “Life Safety for Educational Institutions of Higher Professional Education.” He was awarded the Order of the Badge of Honor (1981), a medal to the Order of Merit for the Fatherland (1999), and other medals.

Alexey A. Tret'yakov

Alexey A. Tret'yakov graduated from Moscow State University and received the Doctor of Science degree in Mathematics in 1988. Professor Tret'yakov worked at the Moscow State University, Russian Academy of Sciences, Polish Academy of Sciences, and at the Siedlce University in Poland. Professor Tret'yakov has numerous publications, including several monographs.