zbMATH — the first resource for mathematics

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).

90C05 Linear programming
90C25 Convex programming
49M30 Other numerical methods in calculus of variations (MSC2010)
90C31 Sensitivity, stability, parametric optimization
PDF BibTeX Cite
Full Text: EMIS EuDML