Local minima of the trust region problem

We consider the minimization of a quadratic formzVz+2zq subject to the two-norm constraint z=. The problem received considerable attention in the literature, notably due to its applications to a class of trust region methods in nonlinear optimization. While the previous studies were concerned with just the global minimum of the problem, we investigate the existence of all local minima. The problem is approached via the dual Lagrangian, and the necessary and sufficient conditions for the existence of all local minima are derived. We also examine the suitability of the conventional numerical techniques used to solve the problem to a class of single-instruction multiple-data computers known as processor arrays (in our case, AMT DAP 610). Simultaneously, we introduce certain hardware-oriented multisection algorithms, showing their efficiency in the case of small to medium size problems.This research was partially supported by the National Physical Laboratories of England under Contract RTP2/155/127.     

Trigonometric wavelets for Hermite interpolation

The aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials. The pair of scaling functions which span the sample spaces are fundamental functions for Hermite interpolation on a dyadic partition of nodes on the interval . Two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions. Together with the corresponding dual functions, these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences. Consequently, this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment.

     

Heuristic methods for evolutionary computation techniques

Evolutionary computation techniques, which are based on a powerful principle of evolution—survival of the fittest, constitute an interesting category of heuristic search. In other words, evolutionary techniques are stochastic algorithms whose search methods model some natural phenomena: genetic inheritance and Darwinian strife for survival.Any evolutionary algorithm applied to a particular problem must address the issue of genetic representation of solutions to the problem and genetic operators that would alter the genetic composition of offspring during the reproduction process. However, additional heuristics should be incorporated in the algorithm as well; some of these heuristic rules provide guidelines for evaluating (feasible and infeasible) individuals in the population. This paper surveys such heuristics and discusses their merits and drawbacks.An abridged version of this paper appears in the volume entitled META-HEURISTICS: Theory & Application, edited by Ibrahim H. Osman and James P. Kelly, to be published by Kluwer Academic Publishers in March 1996.     

Nonmonotone bundle-type scheme for convex nonsmooth minimization

In this paper, we present a general scheme for bundle-type algorithms which includes a nonmonotone line search procedure and for which global convergence can be proved. Some numerical examples are reported, showing that the nonmonotonicity can be beneficial from a computational point of view.This work was partially supported by the National Research Program on Metodi di ottimizzazione per le decisioni, Ministero dell' Universitá e della Ricerca Scientifica e Tecnologica and by ASI: Agenzia Spaziale Italiana.     

The application of the genetic algorithm to the minimization of potential energy functions

We adapted the genetic algorithm to minimize the AMBER potential energy function. We describe specific recombination and mutation operators for this task. Next we use our algorithm to locate low energy conformation of three polypeptides (AGAGAGAGA, A9, and [Met]-enkephalin) which are probably the global minimum conformations. Our potential energy minima are –94.71, –98.50, and –48.94 kcal/mol respectively. Next, we applied our algorithm to the 46 amino acid protein crambin and located a non-native conformation which had an AMBER potential energy 150 kcal/mol lower than the native conformation. This is not necessarily the global minimum conformation, but it does illustrate problems with the AMBER potential energy function. We believe this occurred because the AMBER potential energy function does not account for hydration.     

Analysis of grid file algorithms

Grid file algorithms were suggested in [12] to provide multi-key access to records in a dynamically growing file. We specify here two algorithms and derive the average sizes of the corresponding directories. We provide an asymptotic analysis. The growth of the indexes appears to be non-linear for uniform distributions:O(v ^c ) orO(v ), wherec=1+b^–1, =1+(s-1)/(sb+1),s is the number of attributes being used,v the file size, andb the page capacity of the system. Finally we give corresponding results for biased distributions and compare transient phases.     

Lattice models of peptide aggregation: evaluation of conformational search algorithms

We present a series of conformational search calculations on the aggregation of short peptide fragments that form fibrils similar to those seen in many protein mis-folding diseases. The proteins were represented by a face-centered cubic lattice model with the conformational energies calculated using the Miyazawa-Jernigan potential. The searches were performed using algorithms based on the Metropolis Monte Carlo method, including simulated annealing and replica exchange. We also present the results of searches using the tabu search method, an algorithm that has been used for many optimization problems, but has rarely been used in protein conformational searches. The replica exchange algorithm consistently found more stable structures then the other algorithms, and was particularly effective for the octamers and larger systems.     

MM3 parametrization of four‐ and five‐coordinated rhenium complexes by a genetic algorithm—Which factors influence the optimization performance?

Genetic algorithms (GA) were used to solve one of the multidimensional problems in computational chemistry, the optimization of force field parameters. The correlation between the composition of the GA, its parameters (p(c), p(m)) and the quality of the results were investigated. The composition was studied for all combinations of a Simple GA/Steady State GA with a Roulette Wheel/Tournament Selector using different values each for crossover (0.5, 0.7, 0.9) and mutation rates (0.01, 0.02, 0.05, 0.10, 0.20). The results show that the performance is strongly dependent on the GA scheme, where the Simple GA/Tournament Selector yields the best results. Two new MM3 parameters were introduced for rhenium compounds with coordination number four (204) and coordination number five (205), the formal oxidation states of rhenium ranging from +V to +VII. A manifold of parameters (Re-C, N, O, S) was obtained by using a diverse set of CSD structures. The advantage of the GA vs. UFF calculations is shown by comparison of several examples. The GA optimized parameters were able to reproduce the geometrical data of the X-ray structures.     

Selecting variables for k-means cluster analysis by using a genetic algorithm that optimises the silhouettes

The goal of present work is to analyse the effect of having non-informative variables (NIV) in a data set when applying cluster analysis and to propose a method computationally capable of detecting and removing these variables. The method proposed is based on the use of a genetic algorithm to select those variables important to make the presence of groups in data clear. The procedure has been implemented to be used with k-means and using the cluster silhouettes as fitness function for the genetic algorithm.The main problem that can appear when applying the method to real data is the fact that, in general, we do not know a priori what the real cluster structure is (number and composition of the groups).The work explores the evolution of the silhouette values computed from the clusters built by using k-means when non-informative variables are added to the original data set in both a literature data set as well as some simulated data in higher dimension. The procedure has also been applied to real data sets.     

Complete graph conjecture for inner-core electrons: homogeneous index case

The complete graph conjecture that encodes the inner-core electrons of atoms with principal quantum number n >or= 2 with complete graphs, and especially with odd complete graphs, is discussed. This conjecture is used to derive new values for the molecular connectivity and pseudoconnectivity basis indices of hydrogen-suppressed chemical pseudographs. For atoms with n = 2 the new values derived with this conjecture are coincident with the old ones. The modeling ability of the new homogeneous basis indices, and of the higher-order terms, is tested and compared with previous modeling studies, which are centered on basis indices that are either based on quantum concepts or partially based on this new conjecture for the inner-core electrons. Two similar algorithms have been proposed with this conjecture, and they parallel the two "quantum" algorithms put forward by molecular connectivity for atoms with n > 2. Nine properties of five classes of compounds have been tested: the molecular polarizabilities of a class of organic compounds, the dipole moment, molar refraction, boiling points, ionization energies, and parachor of a series of halomethanes, the lattice enthalpy of metal halides, the rates of hydrogen abstraction of chlorofluorocarbons, and the pED(50) of phenylalkylamines. The two tested algorithms based on the odd complete graph conjecture give rise to a highly interesting model of the nine properties, and three of them can even be modeled by the same set of basis indices. Interesting is the role of some basis indices all along the model.