Non-linear optimization

Predictions of ANN with Non-Linear Optimization Training Routine... 

Scales, L.E., Introduction to Non-linear Optimization, Springer-Verlag, New York, NY, 1985. 

Hence, through the LCAO expansion we have translated the non-linear optimization problem, which required a set of difficult to tackle coupled integro-differential equations, into a linear one, which can be expressed in the language of standard linear algebra and can easily be coded into efficient computer programs. 

MENDES, P., KELL, D.B., Non-linear optimization of biochemical pathways applications to metabolic engineering and parameter estimation, Bioinformatics, 1998, 14, 869-883. 

Other authors concentrated their attention on the evaluation of the time constants. It is well known that multiexponential fit is an ill-posed problem, and its solution, obtained with typical non-linear optimization... 

As is the case for standard orthogonal-orbital MCSCF calculations, the optimization of VB wavefimctions can be a complicated task, and a program such as CASVB should therefore not be treated as a black box . This is true, to a greater or lesser extent, for most procedures that involve orbital optimization (and, hence, non-linear optimization problems), but these difficulties are compounded in valence bond theory by the... 

SimSim performs a pressure match of measured and calculated reservoir or compartment pressures with an automatic, non-linear optimization technique, called the Nelder-Mead simplex algorithm3. During pressure matching SimSim s parameters (e.g. hydrocarbons in place, aquifer size and eigentime, etc.) are varied in a systematic manner according to the simplex algorithm to achieve pressure match. In mathematical terms the residuals sum of squares (least squares) between measured and calculated pressures is minimized. The parameters to be optimized can be freely selected by the user. 

Since %abs values close to, or equal to, 0 or 100 would lead to numerical problems in this conversion, we introduced a special non-linear optimization procedure for the fitting of the coefficients, which was able to handle them appropriately. In this way we derived a model for logKia ... 

M.J. Box, D. Davies and W.H. Swann, Non-linear Optimization Techniques, Oliver and Boyd, Edinburgh, 1969. 

For obtaining of the minimum (29) we may use methods from theory of non-linear optimization, for example methods of Gauss-Zaidel or Markuardt [23], We may assume that weights Wj = 1. As a result of such optimization the following values of parameters were obtained medium capacity b/a = 5,075 rate of population chain growth p = 0,25. 

In this formulation the problem is no different from other types of non-linear optimizations, and the same types of techniques, such as steepest descent,"... 

The model predictive control used includes all features of Quadratic Dynamic Matrix Control [19], furthermore it is able to take into account soft output constraints as a non linear optimization. The programs are written in C++ with Fortran libraries. The manipulated inputs (shown in cm Vs) calculated by predictive control are imposed to the full nonlinear model of the SMB. The control simulations were made to study the tracking of both purities and the influence of disturbances of feed flow rate or feed composition. Only partial results are shown. 

The particular iterative technique chosen by Car and Parrinello to iteratively solve the electronic structure problem in concert with nuclear motion was simulated annealing [11]. Specifically, variational parameters for the electronic wave function, in addition to nuclear positions, were treated like dynamical variables in a molecular dynamics simulation. When electronic parameters are kept near absolute zero in temperature, they describe the Bom-Oppenheimer electronic wave function. One advantage of the Car-Parrinello procedure is rather subtle. Taking the parameters as dynamical variables leads to robust prediction of values at a new time step from previous values, and cancellation in errors in the value of the nuclear forces. Another advantage is that the procedure, as is generally true of simulated annealing techniques, is equally suited to both linear and non-linear optimization. If desired, both linear coefficients of basis functions and non-linear functional parameters can be optimized, and arbitrary electronic models employed, so long as derivatives with respect to electronic wave function parameters can be calculated. 

Typically, non-linear optimization and development of solid-state NMR pulse sequences have been closely linked to an analytical derivation of a first generation of pulse sequences through which the formalism with appropriate variables may readily be set up. One simple (and very early) example is the optimization of TOtal Suppression of... 

Indeed, non-linear optimization has been used as an easy tool to find such sequences... 

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