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1

A locally polynomial method for solving a system of linear inequalities

Evtushenko Yuri, Szkatuła Krzysztof, Tretyakov Alexey

Control and Cybernetics

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2021

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Vol. 50, No. 2

301--314

EN

The paper proposes a method for solving systems of linear inequalities. This method determines in a finite number of iterations whether the given system of linear ineqalities has a solution. If it does, the solution for the given system of linear inequalities is provided. The computational complexity of the proposed method is locally polynomial.

2

Symmetrized semi-smooth Newton method for solving 3D contact problems

Kucera R., Haslinger J., Motycková K., Markopoulos A.

Systemy Wspomagania w Inżynierii Produkcji

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2017

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Vol. 6, iss. 4

286--293

EN

The semi-smooth Newton method for solving discretized contact problems with Tresca friction in three space dimensions is analyzed. The slanting function is approximated to get symmetric inner linear systems. The primal-dual algorithm is transformed into the dual one so that the conjugate gradient method can be used. The R-linear convergence rate is proved for an inexact globally convergent variant of the method. Numerical experiments conclude the paper. The contact problems are important in many practical applications, e.g., biological processes, design of machines, transportation systems, metal forming, or medicine (bone replacements).

CS

V práci je analyzována nehladká Newtonova metoda pro rešení diskretizovaných kontaktních úloh s Trescovým trením ve trech prostorových dimenzích. Slanting funkce je aproximována za úcelem získání symetrických vnitrních lineárních úloh. Pro použití metody sdružených gradientu je primárne-duální algoritmus preveden na duální. R-lineární rychlost konvergence je dokázána pro nepresnou globálne konvergentní variantu metody. Záverem jsou uvedeny numerické experimenty. Kontaktní úlohy mají radu významných aplikací, napr. biologické procesy, design stroju a prepravních systému, tvárení kovu nebo medicína (modelování kostních náhrad).

3

Solution of singular optimal control problems using the improved differential evolution algorithm

Lobato F. S., Steffen, Jr V., Silva Neto A. J.

Journal of Artificial Intelligence and Soft Computing Research

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2011

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Vol. 1, No. 3

195--206

EN

The Differential Evolution algorithm, like other evolutionary techniques, presents as main disadvantage the high number of objective function evaluations as compared with classical methods. To overcome this disadvantage, this work proposes a new strategy for the dynamic updating of the population size to reduce the number of objective function evaluations. This strategy is based on the definition of convergence rate to evaluate the homogeneity of the population in the evolutionary process. The methodology is applied to the solution of singular optimal control problems in chemical and mechanical engineering. The results demonstrated that the methodology proposed represents a promising alternative as compared with other competing strategies.

4

Finite difference equations and convergence rates in the central limit theorem

Lindhagen L.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 2

153--166

EN

We apply the theory of finite difference equations to the central limit theorem, using interpolation of Banach spaces and Fourier multipliers. Let S*n be a normalized sum of i.i.d. random vectors, converging weakly to a standard normal vector N. When does ǁEg (x + S*n) -E g (X + N)ǁLp(dx)tend to zero at a specified rate? We show that, under moment conditions, membership of g in various Besov spaces is often sufficient and sometimes necessary. The results extend to signed probability.

5

Convergence rate in CLT for vector-valued random fields with self-normalization

Bulinski A., Kryzhanovskaya N.

Probability and Mathematical Statistics

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2006

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Vol. 26, Fasc. 2

261--281

EN

Statistical version of the central limit theorem (CLT) with random matrix normalization is established for random fields with values in a space Rk (k ≥ 1). Dependence structure of the field under consideration is described in terms of the covariance inequalities for the class of bounded Lipschitz ”test functions” defined on finite disjoint collections of random vectors constituting the field. The main result provides an estimate of the convergence rate, over a family of convex bounded sets, in the CLT with random normalization.