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

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 investigate the unsolved critical case of the implicit functions theorem, define the notion of generalized solution and prove existence and basic properties. Relevant examples and counterexamples are also indicated. The applications concern new necessary conditions (with less Lagrange multipliers), perturbations and algorithms in non convex optimization problems.