A Hamiltonian approach to implicit systems, generalized solutions and applications in optimization

TL;DR

This paper presents a Hamiltonian-based method for solving implicit systems and generalized solutions, with applications in nonconvex optimization and algorithm development.

Contribution

It introduces a constructive Hamiltonian approach for local solutions of implicit systems and extends implicit function solutions to generalized solutions.

Findings

Provides a Hamiltonian method for implicit systems

Extends implicit function solutions to generalized solutions

Applies to nonconvex optimization problems

Abstract

We introduce a constructive method that provides the local solution of general implicit systems in arbitrary dimension via Hamiltonian type equations. A variant of this approach constructs parametrizations of the manifold, extending the usual implicit functions solution. We also discuss the critical case of the implicit functions theorem, define the notion of generalized solution and prove existence and properties. Examples are also indicated. The applications concern necessary conditions and algorithms in nonconvex optimization problems and their perturbations.