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Limiting behavior of weighted central paths in linear programming

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Abstract

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 thatx i(μ) → ∞ ands j (μ) → ∞ asμ → ∞ fori ∈ B ∞, andj ∈ N ∞, andx N∞ (μ),s B∞ (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 → ∞.

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Author notes

  1. Osman Güler

    Present address: Department of Mathematics and Statistics, University of Maryland Baltimore County, 21228, Baltimore, MD, USA

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  1. Faculty of Technical Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    Osman Güler

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  1. Osman Güler
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Güler, O. Limiting behavior of weighted central paths in linear programming. Mathematical Programming 65, 347–363 (1994). https://doi.org/10.1007/BF01581702

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  • DOI: https://doi.org/10.1007/BF01581702

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