Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(μ), s(μ))}μ > 0 in linear programming at bothμ = 0 andμ = ∞. We establish the existence of a partition (B∞,N∞) of the index set { 1, ⋯,n } such thatxi(μ) → ∞ andsj(μ) → ∞ asμ → ∞ fori ∈ B∞, andj ∈ N∞, andxN∞ (μ),sB∞ (μ) converge to weighted analytic centers of certain polytopes. For allk ⩾ 1, we show that thekth order derivativesx(k) (μ) ands(k) (μ) converge whenμ → 0 andμ → ∞. Consequently, the derivatives of each order are bounded in the interval (0, ∞). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k ⩾ 2) converge to zero whenμ → ∞.