Constrained Least Squares Problems

If the condition is not much higher than the condition of M, the described method was successful. 

The condition estimator is usually used in conjunctionjwith a solver for linear equations, which is used here to compute and M K in order to write 

Example 2.2.1 To summarize this procedurej we will apply it to the constrained truck example (Ex. 1.3.2). This results in the following MATLAB statements 

As result for the constrained truck, linearized around its nominal position we obtain 

We will frequently encounter in the sequel constrained least squares problems of the form 

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Solution of the above constrained least squares problem requires the repeated computation of the equilibrium surface at each iteration of the parameter search. This can be avoided by using the equilibrium surface defined by the experimental VLE data points rather than the EoS computed ones in the calculation of the stability function. The above minimization problem can be further simplified by satisfying the constraint only at the given experimental data points (Englezos et al. 1989). In this case, the constraint (Equation 14.25) is replaced by... 

The constrained least squares problem for the overall plant can now be replaced by the equivalent two-problem formulation. 

As shown before, the general data reconciliation procedure for the overall system must solve the following constrained least squares problem ... 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

Hanson, R.J. and Haskell, K.H., Two algorithms for the linearly constrained least-squares problem, ACM Trans. Math. Softw., 8, 323-333, 1982. 

Apart from the original method mentioned above, Morrison and eo-workers [143,144] formulated a new iterative teehnique ealled CAEDMON (Computed Adsorption Energy Distribution in the Monolayer) for the evaluation of the energy distribution from adsorption data without any a priori assumption about the shape of this function. In this case, the local adsorption is calculated numerically from the two-dimensional virial equation. The problem is to find a discrete distribution function that gives the best agreement between the experimental data and calculated isotherms. In this order, the optimization procedure devised for the solution of non-negative constrained least-squares problems is used [145]. The CAEDMON algorithm was applied to evaluate x(fi) for several adsorption systems [137,140,146,147]. Wesson et al. [147] used this procedure to estimate the specific surface area of adsorbents. 

The particular steps in a numerical computation of the solution of a constrained least squares problem will be discussed in Sec. 2.3.2. 

Numerical Computation of the Solution of a Constrained Least Squares Problem ... 

The leading matrix in this equation has the same structure as matrices occuring in constrained least squares problems and solution techniques discussed in Sec. 2.3 can be applied. This leads to an additional gain by a factor two in the number of operations for decomposing the Jacobian. 

These values are then projected orthogonally back onto the manifolds given by the constraints on position and velocity level, i.e. the projected values are defined as the solution of the nonlinear constrained least squares problem... 

The numerical solution of the projection step can be computed iteratively by a Gaufi-Newton method, see also Ch. 7.2.2. There, a nonlinear constrained least squares problem is solved iteratively. The nonlinear functions are linearized about the current iterate and the problem is reduced to a sequence of linear constrained least squares problems. 

Formal insertion of the discontinuous representation of the solution into Problem 7.1.2 yields a constrained least squares problem of the form... 

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