Interior Point Solutions of Variational Problems and Global Inverse Function Theorems

Variational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. In this paper, we are interested in "well-connected" pairs of such problems which are not necessarily related by critical point considerations. We also study constrained problems of the kind which arise in mathematical programming as well as constraints of a geometric nature where a solution is sought on a leaf of a foliation. In these cases we are interested in interior minimizing points for the variational problem and in the well-posedness (in the sense of Hadamard) of solvability of the related systems of equations. We first prove a general result which implies the existence of interior points and which also leads to the development of certain generalization of the Hadamard-type global inverse function theorem, along the theme that uniqueness quite often implies existence. This result is illustrated by proving the non-existence of shock waves for certain initial data for the vector Burgers' equation. The global inverse function theorem is also illustrated by a derivation of the existence of positive definite solutions of matrix Riccati equations without first analyzing the nonlinear matrix Riccati di#erential equation. The main results on the existence of solutions to geometrically constrained well-connected pairs are then presented and illustrated by a universal solution to the generalized moment problem, with a nonclassical complexity constraint, given by a a geometrically derived minimization problem for a strictly convex nonlinear functional. A more general result is then illustrated by a geometric analysis of the existence of interior points for linear programming pro..