Linear optimization

M Baroni, G Costantmo, G Craciam, D Riganelli, R Valigi, S dementi. Generating optimal linear PLS estimations (GOLPE) An advanced chemometnc tool for handling 3D-QSAR problems. Quant Struct-Act Relat 12 9-20, 1993. 

Continuous Optimal Linear Quadratic Regulator (LQR) Design A=[0 1 -1 -2]... 

Easy availability of ultrafast high intensity lasers has fuelled the dream of their use as molecular scissors to cleave selected bonds (1-3). Theoretical approaches to laser assisted control of chemical reactions have kept pace and demonstrated remarkable success (4,5) with experimental results (6-9) buttressing the theoretical claims. The different tablished theoretical approaches to control have been reviewed recently (10). While the focus of these theoretical approaches has been on field design, the photodissociation yield has also been found to be extremely sensitive to the initial vibrational state from which photolysis is induced and results for (11), HI (12,13), HCl (14) and HOD (2,3,15,16) reveal a crucial role for the initial state of the system in product selectivity and enhancement. This critical dependence on initial vibrational state indicates that a suitably optimized linear superposition of the field free vibrational states may be another route to selective control of photodissociation. 

SELECTION OF OPTIMAL LINEAR SOLVENT STRENGTH GRADIENTS IN LIQUID CHROHATOGRAPHy... 

From the discussion in this chapter, it is clear that the difficulties associated with optimizing nonlinear problems are far greater than those for optimizing linear problems. For linear problems, finding the global optimum can, in principle, be guaranteed. 

Figure 18.29 Starting with a simpler superstructure removes many structural options (some of which might be desirable, but makes the optimization linear.

Baroni, M., Constantino, G., Cruciani, G., Riganelli, D., Valigli, R. and Clementi, S. (1993) Generating optimal linear pis estimations (GOLPE) An advanced chemometric tool for handling 3D-QSAR problems. Quantitative Stmcture-Activity Relationships, 12, 9-20. 

Figure 7.14 Conversion curves obtained with the optimized linear feed and with the accumulation controlled feed time (h).

G., Riganelli, D, Valigli, R., Clementi, S. Generating Optimal Linear PLS Estimations (GOLPE) an Advanced Chemo-metric Tool for Handling 3D QSAR Problems. Quant. Struct.-Act. Relat. 

Farhat et al. considered both optimal constant and optimal linear reflux ratio for this problem (Figure 6.15). Final time was fixed and 4 time intervals were considered. The length of each time interval was also optimised. Table 6.12 presents the summary of the optimisation results using both options of reflux ratio profiles. A significant gain of 10.7% in specified products can be observed between the optimal linear reflux policy and the optimal constant reflux policy. 

Newman, M. M., "On Attempts to Reduce the Sensitivity of the Optimal Linear Regulator to a Parameter Change,",... 

However, in a quantum chemical context there is often one overwhelming difficulty that is common to both Newton-like and variable-metric methods, and that is the difficulty of storing the hessian or an approximation to its inverse. This problem is not so acute if one is using such a method in optimizing orbital exponents or internuclear distances, but in optimizing linear coefficients in LCAO type calculations it can soon become impossible. In modern calculations a basis of say fifty AOs to construct ten occupied molecular spin-orbitals would be considered a modest size, and that would, even in a closed-shell case, give one a hessian of side 500. In a Newton-like method the problem of inverting a matrix of such a size is a considerable... 

Optimal linear coefficients can be computed using the compact matrix form of equation (4.4), supposing both sides equal ... 

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