Mesh-based Nelder–Mead algorithm for inequality constrained optimization

Despite the lack of theoretical and practical convergence support, the Nelder–Mead (NM) algorithm is widely used to solve unconstrained optimization problems. It is a derivative-free algorithm, that attempts iteratively to replace the worst point of a simplex by a better one. The present paper proposes a way to extend the NM algorithm to inequality constrained optimization. This is done through a search step of the mesh adaptive direct search (Mads) algorithm, inspired by the NM algorithm. The proposed algorithm does not suffer from the NM lack of convergence, but instead inherits from the totality of the Mads convergence analysis. Numerical experiments show an important improvement in the quality of the solutions produced using this search step.     

Stochastic Nelder–Mead simplex method – A new globally convergent direct search method for simulation optimization

Nelder–Mead simplex method (NM), originally developed in deterministic optimization, is an efficient direct search method that optimizes the response function merely by comparing function values. While successful in deterministic settings, the application of NM to simulation optimization suffers from two problems: (1) It lacks an effective sample size scheme for controlling noise; consequently the algorithm can be misled to the wrong direction because of noise, and (2) it is a heuristic algorithm; the quality of estimated optimal solution cannot be quantified. We propose a new variant, called Stochastic Nelder–Mead simplex method (SNM), that employs an effective sample size scheme and a specially-designed global and local search framework to address these two problems. Without the use of gradient information, SNM can handle problems where the response functions are nonsmooth or gradient does not exist. This is complementary to the existing gradient-based approaches. We prove that SNM can converge to the true global optima with probability one. An extensive numerical study also shows that the performance SNM is promising and is worthy of further investigation.     

The Farrell–Hsiang method revisited

We present a sufficient condition for groups to satisfy the Farrell–Jones Conjecture in algebraic K-theory and L-theory. The condition is formulated in terms of finite quotients of the group in question and is motivated by work of Farrell–Hsiang.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

The preconditioned Gauss–Seidel method faster than the SOR method

In recent years, a number of preconditioners have been applied to linear systems [A.D. Gunawardena, S.K. Jain, L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154–156 (1991) 123–143; T. Kohno, H. Kotakemori, H. Niki, M. Usui, Improving modified Gauss–Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997) 113–123; H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+_Smax)$(I+S_{\mathit{max}})$, J. Comput. Appl. Math. 145 (2002) 373–378; H. Kotakemori, H. Niki, N. Okamoto, Accelerated iteration method for Z$Z$-matrices, J. Comput. Appl. Math. 75 (1996) 87–97; M. Usui, H. Niki, T.Kohno, Adaptive Gauss-Seidel method for linear systems, Internat. J. Comput. Math. 51(1994)119–125 [10]]. Since these preconditioners are constructed from the elements of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the n$n$th row of matrix A. In the present paper, in order to deal with this drawback, we propose a new preconditioner. In addition, the convergence and comparison theorems of the proposed method are established. Simple numerical examples are also given, and we show that the convergence rate of the proposed method is better than that of the optimum SOR.     

On an Aitken–Newton type method

We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is $\sqrt[5]{6},$ which is higher than the efficiency index of the Aitken or Newton methods. Monotone sequences are obtained for initial approximations farther from the solution, if they satisfy the Fourier condition and the nonlinear mapping satisfies monotony and convexity assumptions on the domain.     

Issues with the Smith–Wilson method

We analyse various features of the Smith–Wilson method used for discounting under the EU regulation Solvency II, with special attention to hedging. In particular, we show that all key rate duration hedges of liabilities beyond the Last Liquid Point will be peculiar. Moreover, we show that there is a connection between the occurrence of negative discount factors and singularities in the convergence criterion used to calibrate the model. The main tool used for analysing hedges is a novel stochastic representation of the Smith–Wilson method.     

On the complex Falk–Langemeyer method

Numerical Algorithms - A new algorithm for the simultaneous diagonalization of two complex Hermitian matrices is derived. It is a proper generalization of the known Falk–Langemeyer algorithm...     

Energy–vector method in mechanical oscillations

The new conception of the space, and its construction are introduced. The space allows for a geometrical view on energy changes in mechanical vibrating systems. It posses all the possibilities of the system dynamics analysis that are used in the traditional phase space but it also shows an amount of the energy accumulated in the system, an energy changes, flow, dissipation, synchronization, an energy attractors. Thus it increases our knowledge and intuition on energy changes in vibrating systems. Dynamics of the chaotic system is studied using the energy space. New types of maps are introduced.     

A stabilized structured Dantzig–Wolfe decomposition method

We discuss an algorithmic scheme, which we call the stabilized structured Dantzig–Wolfe decomposition method, for solving large-scale structured linear programs. It can be applied when the subproblem of the standard Dantzig–Wolfe approach admits an alternative master model amenable to column generation, other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron. Stabilization is achieved by the same techniques developed for the standard Dantzig–Wolfe approach and it is equally useful to improve the performance, as shown by computational results obtained on an application to the multicommodity capacitated network design problem.     

The Uzawa–HSS method for saddle-point problems

Based on the Hermitian and skew-Hermitian splitting iteration scheme, we propose a Uzawa-type iteration method for solving a class of saddle-point problems whose coefficient matrix has non-Hermitian positive definite (1, 1)-block. The convergence properties of this novel method are analyzed, which show that the Uzawa-type iteration method is convergent if the iteration parameters satisfy suitable restrictions.     

Projection–difference method for controlled Fourier filtering

A projection–difference method is developed for approximating controlled Fourier filtering for quasilinear parabolic functional-differential equations. The method relies on a projection–difference scheme (PDS) for the approximation of the differential problem and derives a O(τ^1/2 + h) bound on the rate of convergence of PDS in the weighted energy norm without prior assumptions of additional smoothness of the generalized solutions. The PDS leads to a natural approximation of the objective functional in the optimal Fourier filtering problem. A bound of the same order is obtained for the rate of convergence in the functional of the problems approximating the Fourier filter control problem.     

Generalizations of the Goulden–Jackson cluster method

We give several modifications of the Goulden–Jackson cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various ‘weights’ on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at the website for this paper.     

Feasible smooth method based on Barzilai–Borwein method for stochastic linear complementarity problem

In this paper, we propose a feasible smooth method based on Barzilai–Borwein (BB) for stochastic linear complementarity problem. It is based on the expected residual minimization (ERM) formulation for the stochastic linear complementarity problem. Numerical experiments show that the method is efficient.     

Numerical method for the KdV–Kawahara and Benney–Lin equations

We are concerned with the initial-boundary-value problem associated to the Korteweg – de Vries – Kawahara (KdVK) equation and Benney – Lin (BL) equation, which are transport equations perturbed by dispersive terms of 3rd and 5th order and a term of 4th order in the case of (BL) equation. These equations appear in several fluid dynamics problems. We obtain local smoothing effects that are uniform with respect to the size of the interval. We also propose a simple finite-difference scheme for the problem and prove its stability. Finally, we give some numerical examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)