Large Number Of Local Minima Research Articles

An orthogonal genetic algorithm with quantization for global numerical optimization

We design a genetic algorithm called the orthogonal genetic algorithm with quantization for global numerical optimization with continuous variables. Our objective is to apply methods of experimental design to enhance the genetic algorithm, so that the resulting algorithm can be more robust and statistically sound. A quantization technique is proposed to complement an experimental design method called orthogonal design. We apply the resulting methodology to generate an initial population of points that are scattered uniformly over the feasible solution space, so that the algorithm can evenly scan the feasible solution space once to locate good points for further exploration in subsequent iterations. In addition, we apply the quantization technique and orthogonal design to tailor a new crossover operator, such that this crossover operator can generate a small, but representative sample of points as the potential offspring. We execute the proposed algorithm to solve 15 benchmark problems with 30 or 100 dimensions and very large numbers of local minima. The results show that the proposed algorithm can find optimal or close-to-optimal solutions.

Mesoscopic magnetization fluctuations for metallic grains close to the Stoner instability

This paper is devoted to the magnetic properties of isolated mesoscopic grains. We demonstrate that under very general conditions the electron-electron interactions in such grains can be taken into account by a simple interaction Hamiltonian. This Hamiltonian involves only three coupling constants, which correspond to charging, exchange interaction, and superconducting correlations. The most important condition for such a description is that Thouless conductance of each grain is large. Under this condition sample-to-sample fluctuations of the coupling constants can be neglected. However, the thermodynamic properties can still remain sample specific due to the one-electron part of the Hamiltonian. If the grain is made from a material that is close to the threshold of ferromagnetic instability, the mesoscopic fluctuations of the magnetization are especially strong. Moreover, the situation becomes multistable: free energy of each grain as a function of the magnetization is characterized by a large number of local minima. We analyze the statistics of these minima and show that it possesses simple scaling properties. Numerical simulations confirm this scaling.

Sparse Representations for Image Decompositions

We are given an image I and a library of templates {\cal L} , such that {\cal L} is an overcomplete basis for I. The templates can represent objects, faces, features, analytical functions, or be single pixel templates (canonical templates). There are infinitely many ways to decompose I as a linear combination of the library templates. Each decomposition defines a representation for the image I, given {\cal L} . What is an optimal representation for I given {\cal L} and how to select it? We are motivated to select a sparse/compact representation for I, and to account for occlusions and noise in the image. We present a concave cost function criterion on the linear decomposition coefficients that satisfies our requirements. More specifically, we study a “weighted L norm” with 0 < p < 1. We prove a result that allows us to generate all local minima for the L norm, and the global minimum is obtained by searching through the local ones. Due to the computational complexity, i.e., the large number of local minima, we also study a greedy and iterative “weighted L Matching Pursuit” strategy.

On Computing the Least Quantile of Squares Estimate

In linear regression, an important role is played by the least quantile of squares (LQS) estimate, which involves the minimization of the qth smallest squared residual for a given set of data. This function is nondifferentiable and nonconvex and may have a large number of local minima. This paper is mainly concerned with the efficient calculation of the global solution, and some different approaches are considered.

Exploring the conformational energy landscape of proteins

Proteins possess a complex energy landscape with a large number of local minima called conformational substates that are arranged in a hierarchical fashion. Here we discuss experiments aimed at the elucidation of the energy landscape in carbonmonoxy myoglobin (MbCO). In the highest tier of the hierarchy, a few taxonomic substates exist. Because of their small number, these substates are accessible to detailed structural investigations. Spectroscopic experiments are discussed that elucidate the role of protonations of amino acid side chains in creating the substates. The lower tiers of the hierarchy contain a large number of statistical substates. Substate interconversions are observed in the entire temperature range from below 1 K up to the denaturation temperature, indicating a wide spectrum of energy barriers that separate the substates.

Single-Coordinate-Driving Method Coupled with Simulated Annealing. An Efficient Tool To Search Conformational Space

A single-strand AUG RNA trimer has been studied by computational tools using a single internal coordinate driving method incorporated in the CICADA program interfaced with AMBER molecular mechanics. CICADA, previously appearing a good tool for the global description of the small and middle-sized flexible organic molecules conformational space, failed in this particular case, giving the best minimum of 10 kcal/mol higher than molecular dynamics. We then combined the single-coordinate-driving (SCD) method with simulated annealing (SA), obtaining a global minimum substantially better than that from both CICADA and molecular dynamics. In addition, an interconversion path between both minima has been revealed.The new global minimum exhibits violation of stacking at the G terminal as reported in the experimental data. In comparison with the data obtained by other methods and reported in the literature, the new method yields better results. Additionally, the new method is able to find pathways of conformational interconversions connecting distant parts of the conformational space. The method is now implemented in the updated version of the CICADA program. The results have confirmed that even the PES of such a relatively small molecule exhibits a large number of local minima and that a clustering of the minima into conformational families appears necessary.

Reconsidering the conformational flexibility of β-cyclodextrin

Abstract Conformational analysis of β-cyclodextrin in vacuo has been carried out using two complementary searching techniques to answer the question: what is the relationship between the conformational changes in the Φ,Ψ torsional angles around the glycosidic bonds and the fluctuations of the hydroxyl pendant groups? Because of the large number of local minima on the conformation and potential energy surface of cyclodextrin, a standard systematic search involving molecular mechanics minimization at points on a regular, fixed torsional angle space grid would generate so many points as to be impractical for conformational sampling. Instead the RAMM (RAndom Molecular Mechanics) procedure, a molecular mechanics calculation based on a random walk within torsional angle space, is used here and is compared to the results of nanosecond molecular dynamics simulation. The RAMM procedure is a semi-automatic calculation of the n-dimensional potential energy surface of a molecule which combines a grid-based conformation and search for one pair of bonds with random generation of a conformational ensemble of rotatable bonds and optimization of molecular geometry. Results are presented for six conformers of low symmetry and three conformers with higher symmetry. For all cases, random sampling of the 287-dimensional hydroxy and hydroxymethyl pendant group torsional angle conformational space improved the molecular energy. Torsional angles involving the primary hydroxyl groups exhibited larger conformational freedom than those involving secondary hydroxyls. The secondary hydroxyls of the symmetric forms are involved in a homodromic O2…O3 hydrogen bonding network. The results of the RAMM modeling agree with results from molecular dynamics simulations at 300 K (1 ns), 400 K (2 ns), and at 1000 K (1 ns) with dielectric constant 1. At the two lower temperatures, the molecule fluctuates within the Φ,Ψ space at values around 0 °,0 °. The occupancy profile, drawn in two-dimensional Φ,Ψ plots, is similar for each of the seven combinations of Φi, Ψi and has a characteristic half-moon shape. A stabilizing hydrogen bond network between O2(i)…O3(i − 1) is present during the entire simulation with a consequent decrease in the mobility of HO2 and HO3 (oscillating around χ i2 ≅ −60 °, χ i3 t-60 ° ). No conformational transitions of these groups were observed at 300 K and the first and only reorientation (χi2 ≅ 180 °, χi3 ≅ 180 °) occurred at approximately 1.7 ns at 400 K. At 1000 K, the molecule explores regions beyond Φ,Ψ equal to 0 °,0 ° and the chair conformer of the pyranose rings is not preserved. An additional 2 ns molecular dynamics simulation at 400 K with dielectric constant 4 revealed the “flip-flop” character of O2…O3 hydrogen bonding between adjacent glucose residues.

An Infeasible Point Method for Minimizing the Lennard-JonesPotential

Minimizing the Lennard-Jones potential, the most-studied model problem for molecular conformation, is an unconstrained global optimization problem with a large number of local minima. In this paper, the problem is reformulated as an equality constrained nonlinear programming problem with only linear constraints. This formulation allows the solution to approached through infeasible configurations, increasing the basin of attraction of the global solution. In this way the likelihood of finding a global minimizer is increased. An algorithm for solving this nonlinear program is discussed, and results of numerical tests are presented.

Free energy barriers to melting in atomic clusters

We employ an order parameter approach to investigate melting in clusters bound by the Lennard-Jones potential containing 13, 55, and 147 atoms. We find well-defined Landau free energy barriers between solidlike and liquidlike states for the two larger clusters. A barrier is also revealed in an approximate analytical calculation using only information derived from the potential energy surface. For the two smaller clusters the order parameters are calculated for a large number of local minima. This helps us to interpret the Landau free energy calculations and to comment upon the suitability of the various order parameters for the cluster melting process. Systematic quenching offers us further insight into melting events for the 55-atom cluster. Finally, we elaborate further upon the relationships between S-bends and probability distributions in different ensembles.

The number of metastable states of a simple perceptron with gradient descent learning algorithm

The number of metastable states of a simple perceptron with gradient descent learning algorithm has been calculated as a function of the storage capacity alpha and the gain parameter beta . For alpha not=0 and beta >0 exponential large number of local minima, similar to the spin glasses and analogue attractor networks, has been found.

Correlated walk model of the melting transition in small clusters

A novel criterion to locate the melting temperature, Tm, in clusters is proposed. Based on the characteristics of the configuration space of the clusters we identify a class of clusters that: (1) have the global minimum of the potential-energy surface well detached by an energy gap from all other local minima; (2) present a large number of local minima above the gap that can be accessed by the system before evaporation takes place and (3) have a mean energy spacing between the local minima that is very small compared to the gap. For this class of clusters, one trajectory in phase space split into short time intervals can be mapped onto one state of a one-dimensional walker that steps on the various minima in configuration space. The average number of accessed minima above the gap, f, is obtained in a closed form. This quantity has a sigmoid shape as a function of temperature, i.e. it changes fairly rapid from zero at low temperatures to one at high temperatures. Thus f(T) is identified as the signature of the melting transition, and we define Tm as the temperature at which f(T) reaches the value of ½. This phenomenological model is supported by a comparison with a molecular-dynamics simulation of 12-, 13- and 14-atom Lennard-Jones clusters. Values of the parameters pertaining to the theory are extracted from the simulation and a comparison to Lindemann's criterion for melting is provided.