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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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