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Dans cet article, nous proposons une nouvelle méthode de purification pour les problèmes de complémentarité linéaire, monotones. Cette méthode associe à chaque itéré de la suite, générée par une méthode de points intérieurs, une base non nécessairement réalisable. Nous montrons que, sous les hypothèses de complémentarité stricte et de non dégénérescence, la suite des bases converge en un nombre fini d’itérations vers une base optimale qui donne une solution exacte du problème. Le procédé adopté...
Dans cet article, nous proposons une nouvelle méthode de purification pour les problèmes de complémentarité linéaire, monotones. Cette méthode associe à chaque itéré de la suite, générée par une méthode de points intérieurs, une base non nécessairement réalisable. Nous montrons que, sous les hypothèses de complémentarité stricte et de non dégénérescence, la suite des bases converge en un nombre fini d'itérations vers une base optimale qui donne une solution exacte du problème. Le procédé adopté...
* This work was supported by National Science Foundation grant DMS 9404431.In this paper we prove that the Newton method applied to the
generalized equation y ∈ f(x) + F(x) with a C^1 function f and a set-valued map
F acting in Banach spaces, is locally convergent uniformly in the parameter y if
and only if the map (f +F)^(−1) is Aubin continuous at the reference point. We also
show that the Aubin continuity actually implies uniform Q-quadratic convergence
provided that the derivative of f is Lipschitz...
We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality of a numerical approximation obtained by the finite element method is compared with the exact solution and the error of approximation is bounded from above by a majorant error estimate. The sharpness of the majorant error estimate is discussed.
We verify functional a posteriori error estimates proposed by S. Repin for a class of obstacle problems in two space dimensions. New benchmarks with known analytical solution are constructed based on one dimensional benchmark introduced by P. Harasim and J. Valdman. Numerical approximation of the solution of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. Error of the approximation is measured by a functional majorant. The majorant value...
Unilateral deflection problem of a clamped plate above a rigid inner obstacle is considered. The variable thickness of the plate is to be optimized to reach minimal weight under some constraints for maximal stresses. Since the constraints are expressed in terms of the bending moments only, Herrmann-Hellan finite element scheme is employed. The existence of an optimal thickness is proved and some convergence analysis for approximate penalized optimal design problem is presented.
Extending the results of the previous paper [1], the authors consider elastic bodies with two design variables, i.e. "curved trapezoids" with two curved variable sides. The left side is loaded by a hydrostatic pressure. Approximations of the boundary are defined by cubic Hermite splines and piecewise linear finite elements are used for the displacements. Both existence and some convergence analysis is presented for approximate penalized optimal design problems.
Shape optimization of a two-dimensional elastic body is considered, provided the material is weakly supporting tension. The problem generalizes that of a masonry dam subjected to its own weight and to the hydrostatic presure. Existence of an optimal shape is proved. Using a penalty method and finite element technique, approximate solutions are proposed and their convergence is analyzed.
In der vorliegenden Arbeit werden Voraussetzungen für die Konvergenz eines Verfahrens zur Lösung nichtlinearer Optimierungsprobleme ohne Restriktionen mitgeteilt. Das betrachtete Verfahren gehört zur Klasse der direkten oder ableitungsfreien Verfahren, für die in der Regel Konvergenzbedingungen bisher nicht angegeben wurden. Bei diesen Bedingungen spielen Eigenschaften der Zielfunktion eine Rolle, die Verallgemeinerungen der Unimodalität darstellen, aber auch mit verallgemeinerten Konvexitätsbegriffen...
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