A Convergence Result for Nonautonomous Subgradient Evolution Equations and Its Application to the Steepest Descent Exponential Penalty Trajectory in Linear Programming

• Published in: Journal of functional analysis Vol. 187; no. 2; pp. 263 - 273
• Main Authors: Baillon, J.B., Cominetti, R.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 20.12.2001
• Subjects: evolution equations; exponential penalty; linear programming; exponential penalty; linear programming; evolution equations
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1006/jfan.2001.3828

We present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u(t)∈−∂ϕt(u(t)), where {ϕt:t⩾0} is a family of closed proper convex functions. The result is used to study the flow generated by the family ϕt(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.