Interior point solutions of variational problems and global inverse function theorems

Abstract

SUMMARYVariational 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. We are also 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 analysing the nonlinear matrix Riccati differential equation. The main results on the existence of solutions to geometrically constrained well-connected pairs are then presented and illustrated by a geometric analysis of the existence of interior points for linear programming problems.

The Generalized Moment Problem with Complexity Constraint

“…In this paper our first contribution is to show that this problem is wellposed in the sense of Hadamard, i.e., that the solution not only exists and is unique but is also continuous (in fact, smooth, where appropriate) with respect to the initial conditions. We previously have demonstrated this under the hypothesis that P ⊂ C 2 [a, b] [11,12]. Recently, using the results obtained in [21], it is possible to prove this in the case P ⊂ C 1 [a, b].…”

“…Later, in Sections 2 and 3 we will extend the range of P. We have previously shown [11,12] that for each c ∈ C + and p ∈ P + there exists a unique q ∈ P + so that the generalized moment problem with the complexity constraint (1.6) is solvable. In this paper our first contribution is to show that this problem is wellposed in the sense of Hadamard, i.e., that the solution not only exists and is unique but is also continuous (in fact, smooth, where appropriate) with respect to the initial conditions.…”

The Circulant Rational Covariance Extension Problem: The Complete Solution

“…In fact, if , such representations are more parsimonious containing at most parameters. This is a proper subclass of (57), since, in view of (8) and (35), rational symbols (58) can also be written as pseudo-polynomials (57a). Note that the number of parameters in is equal to the number of given covariance data.…”

“…In the circulant rational covariance extension problem we consider Hermitian circulant matrices (7) which can be represented in form (8) where is the nonsingular cyclic shift matrix…”

A new algorithm for Kullback-Leibler approximation of spectral densities

“…[9]. This result implies that, under assumption (3), there exists (a unique)Λ in L + (Γ) satisfying (17).…”