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On second–order Taylor expansion of critical values

Stephan Bütikofer, Diethard Klatte, Bernd Kummer (2010)

Kybernetika

Studying a critical value function ϕ in parametric nonlinear programming, we recall conditions guaranteeing that ϕ is a C 1 , 1 function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of D ϕ . Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization....

On solution to an optimal shape design problem in 3-dimensional linear magnetostatics

Dalibor Lukáš (2004)

Applications of Mathematics

In this paper we present theoretical, computational, and practical aspects concerning 3-dimensional shape optimization governed by linear magnetostatics. The state solution is approximated by the finite element method using Nédélec elements on tetrahedra. Concerning optimization, the shape controls the interface between the air and the ferromagnetic parts while the whole domain is fixed. We prove the existence of an optimal shape. Then we state a finite element approximation to the optimization...

On Solving the Maximum Betweenness Problem Using Genetic Algorithms

Savić, Aleksandar (2009)

Serdica Journal of Computing

In this paper a genetic algorithm (GA) is applied on Maximum Betweennes Problem (MBP). The maximum of the objective function is obtained by finding a permutation which satisfies a maximal number of betweenness constraints. Every permutation considered is genetically coded with an integer representation. Standard operators are used in the GA. Instances in the experimental results are randomly generated. For smaller dimensions, optimal solutions of MBP are obtained by total enumeration. For those...

On stable least squares solution to the system of linear inequalities

Evald Übi (2007)

Open Mathematics

The system of inequalities is transformed to the least squares problem on the positive ortant. This problem is solved using orthogonal transformations which are memorized as products. Author’s previous paper presented a method where at each step all the coefficients of the system were transformed. This paper describes a method applicable also to large matrices. Like in revised simplex method, in this method an auxiliary matrix is used for the computations. The algorithm is suitable for unstable...

On the best choice of a damping sequence in iterative optimization methods.

Leonid N. Vaserstein (1988)

Publicacions Matemàtiques

Some iterative methods of mathematical programming use a damping sequence {αt} such that 0 ≤ αt ≤ 1 for all t, αt → 0 as t → ∞, and Σ αt = ∞. For example, αt = 1/(t+1) in Brown's method for solving matrix games. In this paper, for a model class of iterative methods, the convergence rate for any damping sequence {αt} depending only on time t is computed. The computation is used to find the best damping sequence.

On the convergence of SCF algorithms for the Hartree-Fock equations

Eric Cancès, Claude Le Bris (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The present work is a mathematical analysis of two algorithms, namely the Roothaan and the level-shifting algorithms, commonly used in practice to solve the Hartree-Fock equations. The level-shifting algorithm is proved to be well-posed and to converge provided the shift parameter is large enough. On the contrary, cases when the Roothaan algorithm is not well defined or fails in converging are exhibited. These mathematical results are confronted to numerical experiments performed by chemists.

On the Convergence of the Approximate Free Boundary for the Parabolic Obstacle Problem

Paola Pietra, Claudio Verdi (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si discretizza il problema dell'ostacolo parabolico con differenze all'indietro nel tempo ed elementi finiti lineari nello spazio e si dimostrano stime dell'errore per la frontiera libera discreta.

On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations

Guy Barles, Espen Robstad Jakobsen (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference...

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