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Penalty barrier function

Penalty function methods, barrier function methods, and hybrid penalty-barrier function methods are discussed in Chapter 12 they are implemented and adopted... [Pg.518]

Barrier function methods are very similar to penalty function methods except that they start at an interior point of the feasible region and set a barrier against leaving the feasible region. In this case, the feasible region must have an interior, so this method is generally restricted to inequality constraints. Consider the nonlinear problem with inequality constraints. [Pg.2561]

Constrained Minimization Penalty and Barrier Functions The program is... [Pg.422]

It is possible to use a penalty function for the equality constraints and a barrier function for the inequality constraints. The following function represents one of... [Pg.437]

Tethering may be a reversible or an irreversible process. Irreversible grafting is typically accomplished by chemical bonding. The number of grafted chains is controlled by the number of grafting sites and their functionality, and then ultimately by the extent of the chemical reaction. The reaction kinetics may reflect the potential barrier confronting reactive chains which try to penetrate the tethered layer. Reversible grafting is accomplished via the self-assembly of polymeric surfactants and end-functionalized polymers [59]. In this case, the surface density and all other characteristic dimensions of the structure are controlled by thermodynamic equilibrium, albeit with possible kinetic effects. In this instance, the equilibrium condition involves the penalties due to the deformation of tethered chains. [Pg.46]

Murray, W., and Wright, M., Projected Lagrangian methods based on trajectories of barrier and penalty functions, SOL Report 78-23, Stanford University, Stanford, California (1978). [Pg.255]

The chiral discrimination in the self-association of chiral l,3a,4,6a-tetrahydroi-midazo[4,5-d]imidazoles 3 has been studied using density functional theory methods [37], (Scheme 3.20). Clusters from dimers to heptamers have been considered. The heterochiral dimers (RR SS or SS RR) are more stable than the homochiral ones (RR RR or SS SS) with energy differences up to 17.5 kJ mol-1. Besides, in larger clusters, the presence of two adjacent homochiral molecules imposes an energetic penalty when compared to alternated chiral systems (RR SS RR SS...). The differences in interaction energy within the dimers of the different derivatives have been analyzed based on the atomic energy partition carried out within the AIM framework. The mechanism of proton transfer in the homo- and heterochiral dimers shows large transition-state barriers, except in those cases where a third additional molecule is involved in the transfer. The optical rotatory power of several clusters of the parent compound has been calculated and rationalized based on the number of homochiral interactions and the number of monomers of each enantiomer within the complexes. [Pg.63]

Srinivasan B., Biegler L.T. and Bonvin D., Tracking the necessary conditions of optimality with changing set of active constraints using a barrier-penalty function , Comput Chem Eng 32(3y.512-519, 2008. [Pg.16]

Figure 6 An illustration of the problems that can arise when parts of parameter space are nonphysical. Assume that one is interested in sampling the potential for energies less than the maximum of the spline fit function. Classical trajectories coming from the right cannot surmount the barrier and will behave almost tihe same on the spline fit function as on the true potential. However, the GA can sample anywhere and has the possibility of accessing the nonphysical region to the left of the maximum in the spline function. One solution is to add a penalty function whose value is zero except for regions of small distance. The sum of die spline fit function and the penalty function will be at least as large as the true potential in this classically forbidden. region and will thereby push solutions to larger distances. Figure 6 An illustration of the problems that can arise when parts of parameter space are nonphysical. Assume that one is interested in sampling the potential for energies less than the maximum of the spline fit function. Classical trajectories coming from the right cannot surmount the barrier and will behave almost tihe same on the spline fit function as on the true potential. However, the GA can sample anywhere and has the possibility of accessing the nonphysical region to the left of the maximum in the spline function. One solution is to add a penalty function whose value is zero except for regions of small distance. The sum of die spline fit function and the penalty function will be at least as large as the true potential in this classically forbidden. region and will thereby push solutions to larger distances.

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See also in sourсe #XX -- [ Pg.64 , Pg.115 ]




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