A convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming

J. B. Baillon, R. Cominetti

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20 Scopus citations

Abstract

We present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u(t) ∈ - ∂o(u(t)), where {ot: t ≥ 0} is a family of closed proper convex functions. The result is used to study the flow generated by the family ot(x) = f(x, r(t)), where f(x, r) := cTx + r ∑ exp[(Aix-bi)/r] is the exponential penalty approximation of the linear program min {cTx: Ax ≤ b}, and r(t) is a positive function tending to 0 when t → ∞. We prove that the trajectory u(t) converges to an optimal solution u of the linear program, and we give conditions for the convergence of an associated dual trajectory μ(t) toward an optimal solution of the dual program.

Original languageEnglish
Pages (from-to)263-273
Number of pages11
JournalJournal of Functional Analysis
Volume187
Issue number2
DOIs
StatePublished - 20 Dec 2001

Keywords

  • Evolution equations
  • Exponential penalty
  • Linear programming

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