Parallelization of the global extremum searching process

The parallel algorithm for searching the global extremum of the function of several variables is designed. The algorithm is based on the method of nonuniform coverings proposed by Yu.G. Evtushenko for functions that comply with the Lipschitz condition. The algorithm is realized in the language C and message passing interface (MPI) system. To speed up computations, auxiliary procedures for founding the local extremum are used. The operation of the algorithm is illustrated by the example of atomic cluster structure calculations.     

On global extremum seeking in the presence of local extrema

We propose a global extremum seeking scheme which can seek the global optimal value in the presence of local extrema. It is shown that the proposed global extremum seeking scheme can converge to an arbitrarily small neighborhood of the global extremum from an arbitrarily large set of initial conditions if sufficient conditions are satisfied. A simple example illustrates the effectiveness of the proposed scheme.     

基于互信息图像配准中的局部极值问题研究

基于互信息的图像配准方法具有自动化程度高、配准精度高等优点,近年来在医学图像配准中得到广泛应用。但是当变换后像素坐标位于非采样网格点时,插值算法有时会使目标函数产生局部极值,使得最优化搜索终止于局部极值,得到错误的配准结果。分析了两种常见的插值算法和产生局部极值的原因,在此基础上提出一种新的插值算法。实验结果表明,该算法有效地抑制了基于互信息的目标函数的局部极值问题,使目标函数更加平滑。     

The stability of a proportional rate extremum regulator

When the signal component of the input to a linear unity feedback sampled-data system is additively contaminated by a relatively wide-band noise process, it has been shown previously that the mean-square values of the true and apparent error quantities are minimized by essentially the same value of loop gain. This paper presents an investigation into the stability of a proportional-rate extremum regulator, which utilizes this fact to establish the optimum gain setting from finite estimates of the measurable mean-square apparent error. Assuming a parabolic operating characteristic, the stability and dynamic behavior of the adaptive loop is studied using path tangent curves in the incremental phase plane. The ability of the regulator to establish the optimum gain setting for a given system is verified experimentally for a number of different operating conditions. In addition, the experimental results show that the transient disturbances generated by the corrections to the gain parameter may adversely affect the regulator stability if they exist over a sufficient fraction of the measurement period.     

Interactive search for the global maximum of a function of many variables

This paper discusses a new algorithm to locate the global maximum of a function defined in a multidimensional rectangular domain. The number of dimensions is as large as, or even more than, 5 or 10. There are two important elements in this algorithm. One is the transformation of the object function in such a way that its global maximum corresponds to infinity while other secondary maxima are reduced to zero. Actually there is some departure from the ideal transformation because of possible overflow on the computer. This portion of the algorithm precedes the interactive (or conversational) use of a graphic display system. This interactive part makes the other element of the algorithm. A multidimensional point is represented as a curve on the display screen. By projecting numerous points in the multidimensional space to similarly numerous curves on the screen of the graphic display device, the human eye can make overall recognition much more efficiently than computers. This fact is exploited to reduce the problem to that of a set of unimodal peaks. Once the supporting domain for each of these peaks is separated by visual aid, one may leave the computer to handle the rest of the problem for itself. A number of numerical experiments are done and discussed to provide evidence regarding the feasibility of the proposed algorithm.     

Solution of the problem concerning a conditional extremum under fuzzy constraints

The solution of the conditional extremum problem by Lagrange’s method of undetermined coefficients under fuzzy constraints on the expenditure function was proposed. The problem is solved by using a special example of the availability maximization of a double-link communication system. A disadvantage of the solution is pointed out, and problems for further study are stated.     

Single-machine multi-agent scheduling problems with a global objective function

In this paper, we consider the problem of scheduling independent jobs when several agents compete to perform their jobs on a common single processing machine. Each agent wants to minimise its cost function, which depends exclusively on its jobs and we assume that a global cost function concerning the whole set of jobs has to be minimised. This cost function may correspond to the global performance of the workshop or to the global objective of the company, independent of the objectives of the agents. Classical regular objective functions are considered and both the ε-constraint and a linear combination of criteria are used for finding compromise solutions. This new multi-agent scheduling problem is introduced into the literature and simple reductions with multicriteria scheduling and multi-agent scheduling problems are established. In addition, the complexity results of several problems are proposed and a dynamic programming algorithm is given.     

A simple reliable solver for all the roots of a nonlinear function in a given domain

A simple and reliable solver based on an exclusion method is formulated to find all the zeros of a nonlinear function in a given bounded domainD. The algorithm automatically searchesD and returns small domains which contain all the zeros inD up to some prespecified accuracy ε. The Running Time isO(log1/ε)when two simple exclusion functions are implemented in the solver. Global information in the form of bounds on derivatives,M, is employed in these exclusion functions. This guarantees a reliable result without the risk of missing any zeros. The concept of a Dominating Function is introduced and an algorithm is formulated which computesM on every subdomaind inD automatically and efficiently. Pre-processing can be used for a class of functions to find a bounded subdomain ofD (which may itself be unbounded) which contains all the zeros inD. Three computational examples are given.     

Navigation of a unicycle-like mobile robot for environmental extremum seeking

We consider a single Dubins-like mobile robot traveling with a constant longitudinal speed in a planar region supporting an unknown field distribution. A single sensor provides the distribution value at the current robot location. We present a new sliding mode navigation strategy that drives the robot to the location where the field distribution attains its maximum. The proposed control algorithm does not employ gradient estimation and is non-demanding with respect to both computation and motion. Its mathematically rigorous analysis and justification are provided. Simulation results confirm the applicability and performance of the proposed guidance approach.     

Necessary and sufficient condition of extremum for the Lebesgue-Stieltjes integral on a class of distributions

We consider the optimization problem for the Lebesgue-Stieltjes integral defined on the set of distribution functions with two fixed power moments. The Krein-Nudel'man theorem on the necessary and sufficient condition of extremum of this functional is generalized to the case of a piecewise-continuous integrand.Translated from Kibernetika, No. 1, pp. 72–75, January–February, 1990.     

On the equivalence between global recurrence and the existence of a smooth Lyapunov function for hybrid systems

We study a weak stability property called recurrence for a class of hybrid systems. An open set is recurrent if there are no finite escape times and every complete trajectory eventually reaches the set. Under sufficient regularity properties for the hybrid system we establish that the existence of a smooth, radially unbounded Lyapunov function that decreases along solutions outside an open, bounded set is a necessary and sufficient condition for recurrence of that set. Recurrence of open, bounded sets is robust to sufficiently small state dependent perturbations and this robustness property is crucial for establishing the existence of a Lyapunov function that is smooth. We also highlight some connections between recurrence and other well studied properties like asymptotic stability and ultimate boundedness.     

Determination of a transfer function matrix in multivariablesystems

A conceptually simple method is presented for determining the system transfer function matrix of linear multivariable systems described by their state-space equations. The polynomial coefficients of the transfer function matrix are computed directly by using the controllability matrix method and the transformation to canonical (phase-variable) form matrices, even if the transformation matrices are singular. The method does not require matrix inversion and calculation of eigenvalues, and requires only O(n³) operations