Asymptotic expansion of penalty-gradient flows in linear programming

J. B. Baillon, R. Cominetti

Research output: Contribution to journalArticlepeer-review


We establish asymptotic expansions for nonautonomous gradient flows of the form u̇(t) = - ∇f(u(t), r(t)), where f(x, r) is a penalty approximation of a linear program and the penalty parameter r(t) tends to 0 as t→ ∞. Under appropriate conditions we show that every integral curve satisfies u(t) = u + r(t) d*0 + ṙ (t)r(t) w*0 + o(ṙ(t)r(t)) for suitable vectors u∞, d*0, and w* 0. We deduce an asymptotic expansion for a related dual trajectory, and we show that the primal-dual limit point is a pair of strictly complementary optimal solutions for the linear program.

Original languageEnglish
Pages (from-to)728-739
Number of pages12
JournalSIAM Journal on Optimization
Issue number2
StatePublished - 2009


  • Asymptotic behavior
  • Gradient flows
  • Linear programming
  • Penalty schemes


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