Algorithms optimization

Optimization methods are generally based on the typical algorithm as follows [9] Choose an initial set of coordinates (variables) x° and calculate the energy E(x°) and gradients g(x°), then 

Perform a line search to determine how far along this direction to move to reduce the value of E(x ). In the simplest case one chooses an a value in x + = x + 

Test for convergence. If the system has not converged, increase k to k + 1, and go to Step 4. If convergence is obtained, the optimization is complete. 

Optimization algorithms are often used to find stationary points on a potential energy surface, i.e., local and global minima and saddle points. The only place where they directly enter MD is in the case of Born-Oppenheimer AIMD, in order to converge the SCF wavefunction for each MD step. It is immediately obvious that the choice of optimization algorithm crucially affects the speed of the simulation. 

Note that, in principle, geometry optimization could be a separate chapter of this text. In its essence, geometry optimization is a problem in applied mathematics. How does one find a minimum in an arbitrary function of many variables [Indeed, we have already discussed that 

Because this text is designed primarily to illuminate the conceptual aspects of computational chemistry, and not to provide detailed descriptions of algorithms, we will examine only the most basic procedures. Much more detailed treatises of more sophisticated algorithms are available (see, for instance, Jensen 1999). 

In the multi-dimensional case, the simplest generalization of this procedure is to carry out the process iteratively. Thus, for LiOH, for example, we might first find a parabolic minimum for the OH bond, then for the LiO bond, then for the LiOH bond angle (in each case holding 

What we really want to do at any given point in the multi-dimensional case is move not in the direction of a single coordinate, but rather in the direction of the greatest downward slope in the energy with respect to all coordinates. This direction is the opposite of the gradient vector, g, which is defined as 

The bond length rAi was defined in Eq. 2.15, and its partial derivative with respect to xA is 

we may quickly assemble the bond stretching contributions to this particular component of the gradient. Contributions from the other terms in the force field can be somewhat more tedious to derive, but are nevertheless available analytically. This makes force fields highly efficient for the optimization of geometries of very large systems. 

A more robust method is tire Newton-Raphson procedure. In Eq. (2.26), we expressed the full force-field energy as a multidimensional Taylor expansion in arbitrary coordinates. If we rewrite this expression in matrix notation, and truncate at second order, we have 

A few studies have found potential surfaces with a stable minimum at the transition point, with two very small barriers then going toward the reactants and products. This phenomenon is referred to as Lake Eyring Henry Eyring, one of the inventors of transition state theory, suggested that such a situation, analogous to a lake in a mountain cleft, could occur. In a study by Schlegel and coworkers, it was determined that this energy minimum can occur as an artifact of the MP2 wave function. This was found to be a mathematical quirk of the MP2 wave function, and to a lesser extent MP3, that does not correspond to reality. The same effect was not observed for MP4 or any other levels of theory. 

The best way to predict how well a given level of theory will describe a transition structure is to look up results for similar classes of reactions. Tables of such data are provided by Hehre in the book referenced at the end of this chapter. 

As mentioned above, a structure with a higher symmetry than is obtained for the ground state may satisfy the mathematical criteria defining a reaction structure. In a few rare (but happy) cases, the transition structure can be rigorously defined by the fact that it should have a higher symmetry. An example of this would be the symmetric Sn2 reaction  

In this case, the transition structure must have symmetry, with the two F atoms arranged axially and the H atoms being equatorial. In fact, the transition structure is the lowest energy compound that satisfies this symmetry criteria. 

For systems where the transition structure is not defined by symmetry, it may be necessary to ensure that the starting geometry does not have any symmetry. This helps avoid converging to a solution that is an energy maximum of some other type. 

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Note The segmentation operation yields a near-optimal estimate x that may be used as initialization point for an optimization algoritlim that has to find out the global minimum of the criterion /(.). Because of its nonlinear nature, we prefer to minimize it by using a stochastic optimization algorithm (a version of the Simulated Annealing algorithm [3]). 

Polak E 1997 Optimization Algorithms and Consistent Approximations (New York Springer)... 

We may conclude that the matter of optimal algorithms for integrating Newton s equations of motion is now nearly settled however, their optimal and prudent use [28] has not been fully exploited yet by most programs and may still give us an improvement by a factor 3 to 5. 

There are many different algorithms for finding the set of coordinates corresponding to the minimum energy. These are called optimization algorithms because they can be used equally well for finding the minimum or maximum of a function. 

Artieles addressing the merits of varions optimization algorithms are... 

Davidsou-Fletcher-Powell (DFP) a geometry optimization algorithm De Novo algorithms algorithms that apply artificial intelligence or rational techniques to solving chemical problems density functional theory (DFT) a computational method based on the total electron density... 

Dirac equation one-electron relativistic quantum mechanics formulation direct integral evaluation algorithm that recomputes integrals when needed distance geometry an optimization algorithm in which some distances are held fixed... 

There are two basic types of unconstrained optimization algorithms (I) those reqmring function derivatives and (2) those that do not. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an ac tual process measurement (such as yield) can be the objec tive function, and no mathematical model for the process is required. Methods that do not reqmre derivatives are called direc t methods and include sequential simplex (Nelder-Meade) and Powell s method. The sequential simplex method is quite satisfac tory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell s method is more efficient than the simplex method and is based on the concept of conjugate search directions. 

Most optimization algorithms also estimate or compute the value of the second derivative of the energy with respect to the molecular coordinates, updating the matrix of force constants (known as the Hessian). These force constants specify the curvature of the surface at that point, which provides additional information useful for determining the next step. 

This paper presents the results of ab initio calculation investigating the pressure dependence of properties of rutile, anatase and brookite, as well as of columbite and hypothetical fluorite phases. The main emphasis is on lattice properties since it was possible to locate transitions and investigate transformation precursors by using constant-pressure optimization algorithm. 

It has been shown that ab initio total energy DFT approach is a suitable tool for studies of phase equilibria at low temperatures and high pressures even when small energy differences of the order of 0.01 eV/mol are involved. The constant pressure optimization algorithm that has been developed here allows for the calculation of the equation of state for complex structures and for the study of precursor effects related to phase transitions. 

Owing to the constraints, no direct solution exists and we must use iterative methods to obtain the solution. It is possible to use bound constrained version of optimization algorithms such as conjugate gradients or limited memory variable metric methods (Schwartz and Polak, 1997 Thiebaut, 2002) but multiplicative methods have also been derived to enforce non-negativity and deserve particular mention because they are widely used RLA (Richardson, 1972 Lucy, 1974) for Poissonian noise and ISRA (Daube-Witherspoon and Muehllehner, 1986) for Gaussian noise. 

Three paths can be advanced (1) expansion, e.g., Taylor series (2) trial and error, e.g., generating curves on the plotter and (3) simplex optimization algorithm. (See Section 3.1.)... 

This work describes one approach for optimizing recovering systems using a simulation package in conjunction with standard statistical techniques such as designed experiments, multiple correlation analyses and optimization algorithms. The approach is illustrated with an actual industrial process. 

Very often empirical equations can be developed from plant data using multiple regression techniques. The main advantage of this approach is that the correlations are often linear, can be easily coupled to optimization algorithms, do not cause convergence problems and are easily transferred from one computer to another. However, there are disadvantages, namely,... 

The statement of the problem of finding an optimal algorithm depends on how it is to be applied (for individual variants or a great number of variants). 

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