Linear programming, complexity theory and elementary functional analysis

We propose analyzing interior-point methods using notions of problem-instance size which are direct generalizations of the condition number of a matrix. The notions pertain to linear programming quite generally; the underlying vector spaces are not required to be finite-dimensional and, more importantly, the cones defining nonnegativity are not required to be polyhedral. Thus, for example, the notions are appropriate in the context of semi-definite programming. We prove various theorems to demonstrate how the notions can be used in analyzing interior-point methods. These theorems assume little more than that the interiors of the cones (defining nonnegativity) are the domains of self-concordant barrier functions.