Objective function weighted

Finally, Figures 14.9 through 14.11 show a comparison between optimal control and state variable profiles, as function of the objective function weights. In the first case, we have run the problem with co, = 1 x 10 and 0)2=2x 1(F, while in the second... 

Model calibration is done using the numerical Nelder-Mead optimization which serves to find the minimum of one of the objective functions cited in section 2.2. The three objective functions, weighted least square function (13), unweighted least square function and multinomial log-likelihood function (14), have been tested. 

Let w, be the weight we wish to assign to point x, y,. The objective function is now defined... 

Competitive Reactions. The prototypical reactions are A B and A —> C. At least two of the three component concentrations should be measured and the material balance closed. Functional forms for the two reaction rates are assumed, and the parameters contained within these functional forms are estimated by minimizing an objective function of the form waS w wcS where Wa, wb, and wc are positive weights that sum to 1. Weighting the three sums-of-squares equally has given good results when the rates for the two reactions are similar in magnitude. 

Step 3. Learning Step Solve the objective function minimization problem with respect to the weights c (and the coefficients, w, if not defined in the previous step). 

Given N measurements of the output vector, the parameters can be obtained by minimizing the Least Squares (LS) objective function which is given below as the weighted sum ofsquares of the residuals, namely,... 

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

Experimental data are available as measurements of the output vector as a function of time, i.e., [yj, t ], i=l,...,N where withyj we denote the measurement of the output vector at time t,. These are to be matched to the values calculated by the model at the same time, y(t,), in some optimal fashion. Based on the statistical properties of the experimental error involved in the measurement of the output vector, we determine the weighting matrices Qj (i=l,...,N) that should be used in the objective function to be minimized as mentioned earlier in Chapter 2. The objective function is of the form,... 

The functions essentially place an equally weighted penalty for small or large-valued parameters on the overall objective function. If penalty functions for... 

The unknown parameter vector k is obtained by minimizing the corresponding least squares objective function where the weighting matrix Q, is chosen based on the statistical characteristics of the error term e, as already discussed in Chapter 2. 

The parameters are estimated by minimizing the usual LS objective function where the weighting matrix is often chosen as a diagonal matrix that normalizes the data and makes all measurements be of the same order of magnitude. Statistically, this is the correct approach if the error in the measurements is proportional to the magnitude of the measured variable. 

These functions are also multiplied by a user supplied weighting constant, (>0) which should have a large value during the early iterations of the Gauss-Newton method when the parameters are away from their optimal values. In general, should be reduced as the parameters approach the optimum so that the contribution of the penalty function is essentially negligible (so that no bias is introduced in the parameter estimates). If p penalty functions are incorporated then the overall objective function becomes... 

The objective to be minimized is a weighted sum of deviations of the produced amounts from the demanded amounts d at the due dates im. Overproduction and underproduction, i.e., positive differences — d% and d — p respectively, are weighted by the nonnegative factors am and fim. If the value of the objective function is represented by z e R the objective can be stated as follows ... 

As the value-based planning part is handled within SNP, the PP/DS optimizer uses a different objective function than the SNP optimizer. The following goals can be weighted in the objective function, which is subject to minimization ... 

The three preceding assumptions require that the objective function be expressed in dollars since area and weight are no longer directly proportional to cost... 

Historically, treatment of measurement noise has been addressed through two distinct avenues. For steady-state data and processes, Kuehn and Davidson (1961) presented the seminal paper describing the data reconciliation problem based on least squares optimization. For dynamic data and processes, Kalman filtering (Gelb, 1974) has been successfully used to recursively smooth measurement data and estimate parameters. Both techniques were developed for linear systems and weighted least squares objective functions. 

Extended Kalman filtering has been a popular method used in the literature to solve the dynamic data reconciliation problem (Muske and Edgar, 1998). As an alternative, the nonlinear dynamic data reconciliation problem with a weighted least squares objective function can be expressed as a moving horizon problem (Liebman et al., 1992), similar to that used for model predictive control discussed earlier. 

The value objective function is oriented at the company s profit and loss definitions. Guiding principle is to only use value parameters that can be found in the cost controlling of the company signed by controlling. Penalty costs and without currency and weighting factors being applied to steer optimization results but having no actual financial impact - as it can be often found in supply chain optimization models - do not meet this requirement. 

An objective function can be defined with respect to the positive impact by the productivity of LE and the negative contribution by that of BP weighted by a factor a. 

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

Now let us define a more general form of the quadratic objective function, which permits us to assign predetermined weights to the components. Consider the general quadratic objective... 

For most applications, the objective function is simply the weighted least squares (Liebman etal., 1992)... 

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