Wolf algorithm

Quadratic programming (QP) is a special problem including a product of two decision variables in the objective function e.g. maximization of turnover max p x with p and x both variable requiring a concave objective function and that can be solved if the so-called Kuhn-Tucker-Conditions are fulfilled, e g. by use of the Wolf algorithm (Dom-schke/DrexI 2004, p. 192)... 

In practice, it is necessary to make many such measurements and average the results. In the Wolf algorithm, one trajectory is followed for the entire calculation, whereas the second point is picked anew after multiples of At have elapsed (see Figure 39). If M distances are measured, Lq, Lj, L2,..., Ljy, the average maximum Lyapunov exponent can be calculated from... 

Essentially, MLE is a measure on time-evolution of the distance between orbits in an attractor. When the dynamics are chaotic, a positive MLE occurs which quantifies the rate of separation of neighboring (initial) states and give the period of time where predictions are possible. Due to the uncertain nature of experimental data, positive MLE is not sufficient to conclude the existence of chaotic behavior in experimental systems. However, it can be seen as a good evidence. In [50] an algorithm to compute the MLE form time series was proposed. Many authors have made improvements to the Wolf et al. s algorithm (see for instance [38]). However, in this work we use the original algorithm to compute the MLE values. 

Jameson BA, Wolf H (1988) The antigenic index a novel algorithm for predicting antigenic determinants. Comput Appl Biosci 4 181-186... 

Gautam and Seider (25) implemented the quadratic programming algorithm of Wolfe (30, 31) because compositions satisfy inequality constraints (n. 0) without transformations. Moles of solid... 

An algorithm due to Wolf et al. 2 jg the most widely used for calculating Lyapunov exponents. It can be applied to a reconstructed phase portrait or one found by measuring more than one dynamical variable. Two nearby points, A and B, that are not part of the same orbit around the attractor are located the latter criterion can be ensured by not considering the first m points in the time series after the selection of one of the two points. The evolution of the two points can be followed since the time series is available, and it is known which points on the attractor follow in time. In a chaotic system, the initial distance... 

Other solution procedures for quadratic programming problems include conjugate gradient methods and the Dantzig-Wolfe method (see Dantzig 1963), which uses a modification of the simplex algorithm for linear programming. 

Algorithms for the solution of quadratic programs, such as the Wolfe (1959) algorithm, are very reliable and readily available. Hence, these have been used in preference to the implementation of the Newton-Raphson method. For each iteration, the quadratic objective function is minimized subject to linearized equality and inequality constraints. Clearly, the most computationally expensive step in carrying out an iteration is in the evaluation of the Lapla-cian of the Lagrangian, V xL x , X which is also the Hessian matrix of the La-grangian that is, the matrix of second derivatives with respect to X . 

Zhang and Li (2007) presented an article that analyzes multi-periodic vehicle fleet size and routing problem, and dynamic vehicle fleet size. The authors decompose the model with Dantzig-Wolf decomposition method, and derive an exact algorithm for the model based on simplex method, dynamic programming method, and branch and bound method. 

For catalyst optimization using high-throughput experimentation, evolutionary strategies have been shown to be very effective if vast parameter spaces (many nominal catalyst components) have to be searched for the optimal catalyst composition. The concept of the algorithm has been described in detail by Wolf et al. [21], some application examples are discussed by Rodemerck et al. [22] and Holena [23]. 

T.J. Kowalski, D.G. Geiger, W.H. Wolf, and W. Fichtner, The VLSI Design Automation Assistant From Algorithms to Silicon , IEEE J sign and Test, pages 33-43, August 1985. 

Wayne Wolf. An algorithm for nearly-minimal collapsing of finite-state machine networks. In Proceedings, ICCAD-90, pages 80-83, November 1990. 

Rodemerck, U., Baerns, M., Helena, M. and Wolf, D. (2004). Application of a genetic algorithm and a neural network for the discovery and optimization of new solid catalytic materials, Appl. Surf. Sci., 223,168-174. 

For example, the first genetic algorithm that was developed specifically for the optimisation of solid catalysts relied on the following heuristic parameters (Wolf et al., 2000) ... 

Starting stmctures for all simulations have been prepared by pre-equilibration of a number of simulations systems with an adequate number of water molecules employing periodic, cubic simulation cells for 100 000 MD steps. The velocity-Verlet algorithm has been employed to integrate the equations of motion with a time step of 0.2 fs. The Nose Hoover thermostat [42, 64] was employed to maintain constant temperature of 298.15 K. To account for long-range Coulombic interactions, the Wolf summation technique [108] with a cutoff distance of 10.0 A was applied as defined in the Garofalini model [57, 106]. The proton transfer update criterion p was set to the recommended value of 0.585 [40]. 

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