Lagrange and Wolf Dualities in Nonholonomic Optimization

TL;DR

This paper explores duality principles in nonholonomic optimization problems, utilizing differential geometry, Pfaff equations, and Riemannian geometry to provide new insights and solution methods for constrained optimization.

Contribution

It introduces novel duality concepts for nonholonomic programs using geometric and differential tools, linking optimization with advanced geometric theories.

Findings

Surprising duality results derived from Vranceanu theory

Application of Darboux's theorem to express Pfaff forms

Development of an original Riemannian geometry framework for constrained optimization

Abstract

This article deals with optimizing problems classified by the kinds of restrictions as required in differential geometry and in mechanics: holonomic and nonholonomic. The central issue relates to dual nonholonomic programs (what they mean and how they are solved?) when the nonholonomic constraints are given by Pfaff equations. The original results are surprising and include aspects derived from the Vranceanu theory of nonholonomic manifolds, from the geometric distributions theory and from Darboux's theorem on canonical coordinates in which we can express a Pfaff form. On these ways we get also an original Riemannian geometry attached to a given constrained optimization problem.