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Augmented self-concordant barriers and nonlinear optimization problems with finite complexity

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Abstract.

In this paper we study special barrier functions for convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratic form. We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center of the augmented barrier, a problem that has some interesting applications. We show that for some special classes of quadratic forms and some convex cones, the computation of the analytic center requires an amount of operations independent of the particular data set. We argue that these problems form a class that is endowed with a property which we call finite polynomial complexity.

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References

  1. Goffin, J.-L., Vial, J.-Ph.: Multiple cuts in the analytic center cutting plane method. SIAM Journal on Optimization 11, 266–288 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Horn, R.A., Johnson, Ch.R.: Matrix Analysis, Cambridge University Press, Cambridge, 1985

  3. Nesterov, Yu., Nemirovsky, A.: Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia, 1994

  4. Nesterov, Yu., Vial, J.-Ph.: Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities. SIAM Journal on Optimization 9, 707–728 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. CORE Lecture Series, 1999

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Authors and Affiliations

  1. CORE, Catholic University of Louvain, 34 voie du Roman Pays, 1348, Louvain-la-Neuve, Belgium

    Yu. Nesterov

  2. HEC, Université de Genève, 40 Bd du Pont d’Arve, CH-1211, Geneva, Switzerland

    J.-Ph. Vial

Authors
  1. Yu. Nesterov
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  2. J.-Ph. Vial
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Corresponding author

Correspondence to Yu. Nesterov.

Additional information

This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming.

This author acknowledges the support of the grant 1214-057093-99 of the Swiss National Science Foundation.

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Nesterov, Y., Vial, JP. Augmented self-concordant barriers and nonlinear optimization problems with finite complexity. Math. Program., Ser. A 99, 149–174 (2004). https://doi.org/10.1007/s10107-003-0392-8

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  • DOI: https://doi.org/10.1007/s10107-003-0392-8

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