Feasible Path Approach

Using this nonlinear programming approach (also termed the embedded model or feasible path approach), we denote as x the vector of parameters representing l/(t) as well as the parameters x. For example, if U t) is assumed piecewise constant over a variable distance, we include w, and t in x. Problem... 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

Morison (1984) and Vassialidis (1993) developed sequential model solution and optimisation strategy which is commonly known as Feasible Path Approach. 

Nonlinear Programming (NLP) Based Dynamic Optimisation Problem-Feasible Path Approach... 

Figure 18.11 Optimization of a process with recycle (a) repeated simulation (feasible path approach) ...

With the feasible path approach the optimization algorithm automatically performs case studies by variing input data. There are several drawbacks the process equations (32c) have to be solved every time the performance index is evaluated. Efficient gradient-based optimization techniques can only be used with great difficulties because derivatives can only be evaluated by perturbing the entire flowsheet with respect to the decision variables. This is very time consuming. Second, process units are often described by discrete and discontinuous relations or by functions that may be nondifferentiable at certain points. To overcome these problems quadratic module models can be... 

Successive Quadratic Programming (SQP) The above approach to finding the optimum is called a feasible path method, as it attempts at all times to remain feasible with respect to the equahty and inequahty constraints as it moves to the optimum. A quite different method exists called the Successive Quadratic Programming (SQP) method, which only requires one be feasible at the final solution. Tests that compare the GRG and SQP methods generaUy favor the SQP method so it has the reputation of being one of the best methods known for nonlinear optimization for the type of problems considered here. 

A feasible path optimization approach can be very expensive because an iterative calculation is required to solve the undetermined model. A more efficient way is to use an unfeasible path approach to solve the NLP problem however, many of these large-scale NLP methods are only efficient in solving problems with few degrees of freedom. A decoupled SQP method was proposed by Tjoa and Biegler (1991) that is based on a globally convergent SQP method. 

These early approaches suffered from two drawbacks. First, simultaneous approaches lead to much larger nonlinear programs than embedded model approaches. Consequently, nonlinear programming methods available at that time were too slow to compete with smaller feasible path formulations. Second, care must be taken in the formulation in order to yield an accurate algebraic representation of the differential equations. 

Except for very small n, this procedure is obviously not feasible. The approach presented below is a variation of this procedure, and limits the number of representations that must be generated by establishing a partial order of atoms, restricting the numbering permitted, and saving the results of the path tracing. 

Chen (1988) provided detailed accounts on feasible and infeasible path approaches in optimisation. 

The second purity constraint over the whole prediction horizon acts as a terminal (stability) constraint, forcing the process to converge towards the optimal cyclic steady state. The goal of feedback control in a standard control approach (i.e. to fulfill the extract purity) is introduced as a constraint here. A feasible path SQP algorithm is used for the optimization (Zhou et al., 1997), which generates a feasible point before it starts to minimize the objective function. 

The second approach is to expand Eq. (47) for small values of the experimental argument. This is discussed briefly by Jansson in Section I.E.2 of Chapter 2. For very-high-resolution data where very short paths and low pressures are used (i.e., where X is small) this approach is potentially feasible though signal-to-noise ratio requirements would be very demanding for weak apparent absorptions. 

Classical Path. Another approach to scattering calculations uses a quantum-mechanical description of the internal states, but classical mechanics for the translational motion. This "classical path" method has been popular in line-shape calculations (37,38). It is almost always feasible to carry out such calculations in the perturbation approximation for the internal states (37). Only recently have practical methods been developed to perform non-perturbative calculations in this approach (39). 

Density functional methods are competitive with the above traditional wave function methods for numerous applications such as the computation of ground-state PES. A few applications of transition metal photochemistry have been proposed on the basis of the A-SCF approach implying several approximations on the excited-state reaction-path definition by symmetry constraints not always appropriate in a coordinate driving scheme. Excited-state gradients have been recently implemented in DFT for various functionals, the feasibility of the approach having been tested for small molecules... 

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