Optimality criteria form the foundations of mathematical programming both theoretically and computationally. In general, these criteria can be classified as either necessary or sufficient. Of course, one would like to have the same criterion be both necessary and sufficient. However, this occurs only under somewhat ideal conditions which are rarely satisfied in practice. In the absence of conve...

Abstract In the paper ‘On Dirac–Frenkel Variational Principle on Tensor Banach Spaces’ , we provided a geometrical description of manifolds tensors in Tucker format with fixed multilinear (or Tucker) rank tensor spaces, that allowed to extend variational principle framework topological spaces. The purpose this note is these results more general formats. More precisely, provide new tree-based hi...

Let f be an integrable function on an infinite measure space (S,S , π). We show that if a regenerative sequence {Xn}n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫ S fdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and inte...

We prove that the notion of KT-invexity for continuous-time nonlinear optimization problem is a necessary and sufficient condition for global optimality of a KarushKuhn-Tucker point.