A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

A method for solving an inverse biharmonic problem

This paper deals with the problem of determining of an unknown coefficient in an inverse boundary value problem. Using a nonconstant overspecified data, it has been shown that the solution to this inverse problem exists and is unique.     

A convergent Jacobi method for solving the eigenproblem of arbitrary real matrices

Summary A global convergence proof of a Jacobi-type norm-reducing diagonalisation process for arbitrary real matrices is given. The convergence to a fixed matrix in Murnaghan form is obtained with the well-known exception of complex conjugate pairs of eigenvalues whose real parts are more than double.     

Modified Newton-MDPMHSS method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matrices

Numerical Algorithms - In this study, an efficient iterative method is given to solve large sparse nonlinear systems with block two-by-two complex symmetric Jacobian matrices. Based on the...     

Matrix method for solving the Sturm-Liouville problem

We consider the application of the matrix method to constructing an approximate solution of the regular Sturm-Liouville problem with conditions of first, second, third, and mixed types. All these cases are reduced to a homogeneous algebraic system and its characteristic equation. A numerical example is given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 43–49, 1985     

A visual interactive method for solving the multiple criteria problem

This paper proposes a new method for solving multiple criteria mathematical programming problems. The method does not rely on explicit knowledge of the properties of the utility function. However, if the utility function is assumed to be pseudoconcave and differentiable when the procedure terminates, sufficient conditions for optimality can be established.The method has some features in common with the well-known GDF method of Geoffrion, Dyer and Feinberg (1972). However, our method operates solely in terms of efficient solutions contrary to the GDF method. Moreover, the original procedure of the GDF method for solving the direction-finding problem has been abandoned, since it has turned out that it is not very workable in practice. Instead, the idea of using reference goals or aspiration levels, originated by Wierzbicki, is utilized for determining a direction of improvement. The step-size problem is dealt with in a manner analogous to the GDF method. Computer graphics are used as an aid in solving the step-size problem.The method is easy to implement and convenient to use. Any standard LP package with a parametric optimization routine can be used. In addition, a graphical display device is required.     

A method for solving the general parametric linear complementarity problem

This paper presents a solution method for the general (mixed integer) parametric linear complementarity problem pLCP(q(θ),M), where the matrix M has a general structure and integrality restriction can be enforced on the solution. Based on the equivalence between the linear complementarity problem and mixed integer feasibility problem, we propose a mixed integer programming formulation with an objective of finding the minimum 1-norm solution for the original linear complementarity problem. The parametric linear complementarity problem is then formulated as multiparametric mixed integer programming problem, which is solved using a multiparametric programming algorithm. The proposed method is illustrated through a number of examples.     

A problem expanding parametric programming method for solving the job shop scheduling problem

A new zero-one integer programming model for the job shop scheduling problem with minimum makespan criterion is presented. The algorithm consists of two parts: (a) a branch and bound parametric linear programming code for solving the job shop problem with fixed completion time; (b) a problem expanding algorithm for finding the optimal completion time. Computational experience for problems having up to thirty-six operations is presented. The largest problem solved was limited by memory space, not computation time. Efforts are under way to improve the efficiency of the algorithm and to reduce its memory requirements.This report was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract No. N00014-82-K-0329 NR 047-048 with the U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

A method for solving the problem of isometric realization of developments

We suggest a new algorithmic solution for the problem of isometric realization of developments. For any development, a system of polynomial equations is composed such that its solutions are, in some sense, in bijective correspondence with all possible isometric realizations of the development. An important advantage of this method is the fact that it can be used in practical computations. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 167–203, 2006.     

A method for solving a general multi-valued complementarity problem

We propose an extended version of Chandrasekaran’s method for general complementarity problems with multi-valued weakly off-diagonally antitone costmappings. It allows one either to construct a sequence converging to a solution or to recognize that the problem has no solutions. We also propose versions of Jacobi’s methods for multi-valued inclusions subject to one- and two-side constraints.     

A certain spline method for solving a boundary-value problem

Translated from Ukrainskii Matematichskii Zhurnal, Vol. 41, No. 5, pp. 703–707, May, 1989.     

A method for solving a quadratic optimal control problem

Hestenes' method of multipliers is used to approximate a quadratic optimal control problem. The global existence of a family of unconstrained problems is established. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the original differentially constrained problem.The preparation of this paper was sponsored in part by the U.S. Army Research Office under Grant No. DA-31-124-ARO(D)-355.     

A new iterative method for solving split feasibility problem

In this paper, we construct a new iterative algorithm and show that the newly introduced iterative algorithm converges faster than a number of existing iterative algorithms for contractive-like mappings. We present a numerical example followed by graphs to validate our claim. We prove strong and weak convergence results for approximating fixed points of generalized $\alpha$-nonexpansive mappings. Again we reconfirm our results by an example and table. Further, we utilize our proposed algorithm to solve split feasibility problem.     

A new hybrid method for solving global optimization problem

In this paper we present a new hybrid method, called the SASP method. The purpose of this method is the hybridization of the simulated annealing (SA) with the descent method, where we estimate the gradient using simultaneous perturbation. Firstly, the new hybrid method finds a local minimum using the descent method, then SA is executed in order to escape from the currently discovered local minimum to a better one, from which the descent method restarts a new local search, and so on until convergence.The new hybrid method can be widely applied to a class of global optimization problems for continuous functions with constraints. Experiments on 30 benchmark functions, including high dimensional functions, show that the new method is able to find near optimal solutions efficiently. In addition, its performance as a viable optimization method is demonstrated by comparing it with other existing algorithms. Numerical results improve the robustness and efficiency of the method presented.     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.