Algorithm For Nonlinear Problems Research Articles

On the Extension of the DIRECT Algorithm to Multiple Objectives

Deterministic global optimization algorithms like Piyavskii–Shubert, direct, ego and many more, have a recognized standing, for problems with many local optima. Although many single objective optimization algorithms have been extended to multiple objectives, completely deterministic algorithms for nonlinear problems with guarantees of convergence to global Pareto optimality are still missing. For instance, deterministic algorithms usually make use of some form of scalarization, which may lead to incomplete representations of the Pareto optimal set. Thus, all global Pareto optima may not be obtained, especially in nonconvex cases. On the other hand, algorithms attempting to produce representations of the globally Pareto optimal set are usually based on heuristics. We analyze the concept of global convergence for multiobjective optimization algorithms and propose a convergence criterion based on the Hausdorff distance in the decision space. Under this light, we consider the well-known global optimization algorithm direct, analyze the available algorithms in the literature that extend direct to multiple objectives and discuss possible alternatives. In particular, we propose a novel definition for the notion of potential Pareto optimality extending the notion of potential optimality defined in direct. We also discuss its advantages and disadvantages when compared with algorithms existing in the literature.

Adaptive frame methods for nonlinear elliptic problems

This article is concerned with adaptive numerical wavelet frame methods for nonlinear elliptic operator equations. The starting point of our considerations are the works of Cohen et al. [A. Cohen, W. Dahmen, and R. DeVore, Adaptive wavelet schemes for nonlinear variational problems, SIAM J. Numer. Anal. 41(5) (2003), pp. 1785–1823, A. Cohen, W. Dahmen, and R. DeVore, Sparse evaluation of compositions of functions using multiscale expansions, SIAM J. Math. Anal. 35(2) (2003), pp. 279–303] in which an adaptive, asymptotically optimal evaluation scheme for certain nonlinear expressions was established using wavelet Riesz bases. Due to existing problems in the construction of such bases on domains, we adjust their ideas to the case of wavelet frames. In particular, we show that the discretization of nonlinear problems by wavelet frames, which are constructed by an overlapping partition of the domain, allows us to establish a linear convergent, adaptive algorithm for the iterative solution of the discretized problem. We then focus on main aspects of the numerical implementation of this iterative scheme. More precisely, we point out that specific non-canonical frame expansions enable us to use tree approximation ideas known from the usage of wavelet Riesz bases for the estimation of the index set of significant frame expansion coefficients of nonlinear expressions. We then prove that the ideas from Barinka [A. Barinka, Fast computation tools for adaptive wavelet schemes, Ph.D. thesis, RWTH Aachen, 2005] can be used to approximate the afore selected expansion coefficients by quadrature in linear time. Finally, concerning the complexity of the approximation, we show that all building blocks needed in the algorithm can be realized with asymptotical optimality compared with a best N-term approximation respecting tree structures, which yields an asymptotically optimal algorithm for nonlinear problems discretized by wavelet frames.

A Study on Deriving a Method for Chromosome Similarities Suitable for the Search Space

Evolutionary Computations (ECs) are powerful search algorithms for nonlinear problems, and have been. widely studied. Generally, calculation of EC to acquire expected solutions takes much time because they need repeated calculation for search solutions. Hanaki et al. proposed the fitness inference to reduce evaluation time by simplifying calculation of the fitness of chromosomes. In fitness inference, how chromosome similarities is defined is very important corresponding to that of solutions. We studied chromosome similarities considering the solution similarities, and propose new chromosome similarities with the weight on each locus determined by the change of information on the locus using Sequential Difference Fitness Value Allocation. We used benchmark functions to study the feasibility of the proposed method and found that effective weights on loci suitable for the search space are automatically generated, and that the proposed method enables effective fitness inference.

Simultaneous multigrid techniques for nonlinear eigenvalue problems: Solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions

Algorithms for nonlinear eigenvalue problems (EP's) often require solving self-consistently a large number of EP's. Convergence difficulties may occur if the solution is not sought in an appropriate region, if global constraints have to be satisfied, or if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP's obtained from discretizations of partial differential EP have often been shown to be more efficient than single level algorithms. This paper presents MG techniques and a MG algorithm for nonlinear Schr\odinger Poisson EP's. The algorithm overcomes the above mentioned difficulties combining the following techniques: a MG simultaneous treatment of the eigenvectors and nonlinearity, and with the global constrains; MG stable subspace continuation techniques for the treatment of nonlinearity; and a MG projection coupled with backrotations for separation of solutions. These techniques keep the solutions in an appropriate region, where the algorithm converges fast, and reduce the large number of self-consistent iterations to only a few or one MG simultaneous iteration. The MG projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues. Computational examples for the nonlinear Schr\odinger-Poisson EP in two and three dimensions, presenting special computational difficulties that are due to the nonlinearity and to the equal and closely clustered eigenvalues are demonstrated. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N and for the corresponding eigenvalues. One MG simultaneous cycle per fine level was performed. The total computational cost is equivalent to only a few Gauss-Seidel relaxations per eigenvector. An asymptotic convergence rate of 0.15 per MG cycle is attained.

Adaptive dual control of discrete-time distributed-parameter stochastic systems

Abstract The problem of adaptive dual control of discrete-time distributed-parameter stochastic systems is examined. It is shown that there exists an important difference between feedback and closed-loop policies of control for this type of system as for the lumped parameter case. This difference is based on the adaptivity feature of the control. Namely, when the control policy affects both the state and its uncertainty (dual effect) it possesses the so-called feature of active adaptivity and can only be a characteristic of a closed-loop policy, whereas a feedback policy can only be passively adaptive. These results can be used to develop a control algorithm for non-linear problems for which the realization of optimal control laws involves control strategies with both learning and control features.

Hierarchical optimisation and control – an applications orientated survey

In this paper a survey is given of the “state of the art” of dynamical hierarchical control from a practical applications point of view. The subject is essentially divided into three main parts. In the first part hierarchical optimisation techniques which lead to open loop control are studied. Here the recent work on algorithms for non-linear problems is particularly emphasised. In the second part, hierarchical calculation of feedback control for both linear quadratic as well as non linear systems is described. In the final part, the stochastic control problem is considered and it is shown how a deterministic controller hierarchy can be superposed on that of a new decentralised filter to achieve stochastic optimal control. In each case the algorithms are illustrated on practical examples taken from diverse fields like river pollution control, power systems, etc.

Iterative algorithms for nonlinear problems in remote sensing

First a brief review of the Backus and Gilbert method for the linear problems is given. Then a comprehensive presentation of iterative algorithms for constructing approximate solutions of nonlinear problems from erroneous inadequate data is given. Proofs of convergence and error estimates of these iterative algorithms are obtained. It is found that locally the limit of the iterates does not satisfy the original nonlinear operator equation, but a somewhat different nonlinear equation which depends on the initial iterate. A nonlinear radiative transfer equation in remote sensing of the atmospheric temperature profiles is used as an example to demonstrate the applicability of the iterative algorithms. Finally, a discussion of the present status of research in this domain and some personal views on what should be the useful lines for future research is given.

Acceleration technique applied to conjugate-direction algorithms for nonlinear problems

Acceleration technique applied to conjugate-direction algorithms for nonlinear problems