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relative interior
The relative interior of the base polyhedron and the core
      
In this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi-relative interior, which, in all the concrete problems considered, is nonempty.
      
An interior point of a triangle is calledCP-point if its orthogonal projection on the line containing each side lies in the relative interior of that side.
      
We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions.
      
Our result depended on a constraint qualification involving the notion ofquasi relative interior.
      
We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption.
      
We study the problem of finding a point in the relative interior of the optimal face of a linear program.
      
Furthermore, we give a strong polynomial algorithm for constructing such a B-T Simplex Tableau when a solution in the relative interior of the optimal face is known.
      
It is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function.
      
It is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function.
      
The difficulties arising from the structure of the constraint set are overcome by means of generalized constraint qualification assumptions based on the concept of quasi relative interior of a convex set.
      
Separating sets with relative interior in Fréchet spaces
      
A separation theorem, valid in infinite dimensional spaces, and involving the relative interior of the sets to be separated, will be extended to Fréchet spaces.
      
We note thatP has disjointk-nuclei if and only if there exists a hyperplane inEd which bisects the (relative) interior of everyk-face ofP, and that this is possible only if [d+2/2]≤k≤d-1.
      
If in addition (Df(?t(x0))+lI)n-1 takes k into the relative interior of K(t) for all t>amp;gt;0 then is in the relative interior of K(t) for all t>amp;gt;0.
      
In order to obtain our result, we'll make use of the new concept of quasi relative interior.
      
Lagrange multipliers useful in characterizations of solutions to spectral estimation problems are proved to exist in the absence of Slater's condition provided a new constraint involving the quasi-relative interior holds.
      
We prove that the limit of the sequence generated by the method lies in the relative interior of the solution set, and furthermore is the closest optimal solution to the initial point, in the sense of the Bregman distance.
      
This bound implies that the undiscounted value of a competitive Markov decision process is continuous in the relative interior of the space of transition rules.
      
Although there is no empty circle that encloses e, the depicted circumcircle of e encloses no vertex that is visible from the relative interior of e.
      
 

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