For linear optimization problems, we consider a curvature integral of the central path which was first introduced by Sonnevend et al. (Math Progr 52:527–553, 1991). Our main result states that in order to establish an upper bound for the total Sonnevend curvature of the central path of a polytope, it is sufficient to consider only the case where the number of inequalities is twice as many as the number of variables. This also implies that the worst cases of linear optimization problems for certain path-following interior-point algorithms can be reconstructed for the aforementioned case. As a by-product, our construction yields an asymptotically worst-case lower bound in the order of magnitude of the number of inequalities for Sonnevend’s curvature. Our research is motivated by the work of Deza et al. (Electron Notes Discrete Math 31:221–225, 2008) for the geometric curvature of the central path, which is analogous to the Klee–Walkup result for the diameter of a polytope.
Curvature Central path Polytopes Diameter Complexity Interior-point methods Linear optimization
Mathematics Subject Classification
52B05 65K05 68Q25 90C05 90C51 90C60
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Research was supported by a Start-up grant of Lehigh University. It was also supported by TAMOP-4.2.2.A-11/1KONV-2012-0012: Basic research for the development of hybrid and electric vehicles. The TAMOP Project was supported by the European Union and co-financed by the European Regional Development Fund.
Sonnevend, G., Stoer, J., Zhao, G.: On the complexity of following the central path of linear programs by linear extrapolation II. Math. Progr. 52, 527–553 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
Zhao, G., Stoer, J.: Estimating the complexity of a class of path-following methods for solving linear programs by curvature integrals. Appl. Math. Optim. 27, 85–103 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
Monteiro, R.D., Tsuchiya, T.: A strong bound on the integral of the central path curvature and its relationship with the iteration-complexity of primal-dual path-following LP algorithms. Math. Progr. 115, 105–149 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
Vavasis, S.A., Ye, Y.: A primal-dual interior point method whose running time depends only on the constraint matrix. Math. Progr. 74, 79–120 (1996)MathSciNetzbMATHGoogle Scholar
Roos, C., Terlaky, T., Vial, J.P.: Interior Point Methods for Linear Optimization. Springer, Berlin (2005)zbMATHGoogle Scholar