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Asymptotic convergence of the steepest descent method for the exponential penalty in linear programming. (English) Zbl 0837.90082
The author studies the asymptotic behavior of a non-autonomous nonlinear evolution equation of the form \[ \begin{cases} \dot u(t) & = -\nabla_x f(u(t), r(t))\\ u(t_0) & = u_0\end{cases}\tag{E} \] where \(f(x, r)\) is convex and differentiable with respect to the \(x\) variable for every field \(r> 0\), and \(r(t)\) is a positive real differentiable function decreasing to 0 as \(t\to \infty\).
The function \(f(x, r)\) is the exponential penalty function associated with the linear program (P) \(\min\{c' x: Ax\leq b\}\). The author shows the existence of global solutions for (E) and asymptotic convergence towards an optimal solution of (P).

MSC:
90C05 Linear programming
90C25 Convex programming
49M30 Other numerical methods in calculus of variations (MSC2010)
90C31 Sensitivity, stability, parametric optimization
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