An old problem and new tools

Abstract

The aim of the paper is to demonstrate the efficiency of application of the Theory of Exact Penalties and Nonsmooth Optimization to solving variational problems. As an example we discuss the main (or so-called simplest) problem of the Calculus of Variations. It is shown that this approach allows one not only to get the main known results (e.g., the Euler and Weierstrass necessary conditions) but also to gain a deeper understanding of the intristic nature of the Euler condition, to derive new extremality conditions and to construct new direct numerical methods for solving variational problems based on the notions of subgradient and hypogradient of the exact penalty function.

Keywords:

• Calculus of Variations
• Exact penalties
• Minimax
• Nonsmooth analysis
• Steepest descent method
• Subdifferential
• Hypodifferential

Acknowledgements

The author is grateful to the anonymous referee for his useful remarks. The work was supported by the Russian Foundation for Fundamental Studies (RFFI) under Grant No. 03-01-00668.