Infeasible Path

Fig. 1. Schematic of a typical planning problem (solid line signifies a feasible path and dotted line an infeasible path).

Infeasible path algorithms. The equality constraints and active inequality constraints are satisfied only at the stage on which the optimal solution is reached. 

Commercial process simulators mainly use a form of SQP. To use LP, you must balance the nonlinearity of the plant model (constraints) and the objective function with the error in approximation of the plant by linear models. Infeasible path, sequential modular SQP has proven particularly effective. 

By using open-equation formats and infeasible path optimization algorithms, the type of difficulty described above can be avoided. All the equations in the NLP problem can be solved simultaneously, driving the residuals to zero. The open-equation format for the heat exchanger is... 

With feasible path strategies, as the name implies, on each iteration you satisfy the equality and inequality constraints. The results of each iteration, therefore, provide a candidate design or feasible set of operating conditions for the plant, that is, sub-optimal. Infeasible path strategies, on the other hand, do not require exact solution of the constraints on each iteration. Thus, if an infeasible path method fails, the solution at termination may be of little value. Only at the optimal solution will you satisfy the constraints. 

Biegler, L. T. Improved Infeasible Path Optimization for Sequential Modular Simulators—I The Interface. Comput Chem Eng 9 245-256 (1985). 

Biegler. L. T., and Cuthrell, 1. E., Improved infeasible path optimization for sequential modular simulators — II The optimization algorithm, Comp, and Chem. Eng. 9(3), 257-267 (1985). 

Cuthrell and Biegler (1987) and Renfro et al. (1987) developed dynamic optimisation methods based on the infeasible path approach. The main advantage of this approach is that it avoids repetitive simulations during iteration of the... 

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

NLP Based Dynamic Optimisation Problem- Infeasible Path Approach... 

Our judgment is that feasible path methods in which the solution of the model equations over time is carried out by conventional integration software, which has been extensively developed and refined, are at present more reliable than infeasible path methods. Feasible path optimization methods are also easier to implement as the size of the optimization problem is much smaller. For these reasons, we have pursued feasible path methods despite evidence that infeasible path methods are more efficient on some problems. 

Infeasible path tear streams and constraints are converged simultaneously. 

In the infeasible path approach, as illustrated in Figure 18.1 lb, both d and w are adjusted simultaneously by the optimizer (with w x for the next iteration), usually using the SQP algorithm. This algorithm involves just one pass through the flowsheet per iteration, so the tear equations are normally not satisfied until the optimum is located. As will be seen in the ex-... 

Before implementing an infeasible path optimization, it is very helpfiil to carry out preliminary searches by varying the key decision variables, somewhat randomly, to gain insights into the key trade-offs. For these searches, it is probably best not to use optimization algorithms that require derivatives, or approximations to them, such as SQP. A common approach is to use the sensitivity analysis facilities of the process simulators referred to earlier. 

Understand the advantages of performing optimization and converging recycle calculations and design specifications simultaneously, as implemented using an infeasible path optimization algorithm. 

For an efficient numerical solution, problem (1.2) is first reformulated as a nonlinear programming (NLP) problem by parameterization of the controls and discretization of the model equations. Then, an infeasible-path optimization method is applied to solve the discretized problem. By this approach, simulation and optimization proceed simulianeously i.e., the model equations are satisfied only at the final solution. The necessary gradients can be calculated efficiently and reliably by internal numerical differentiation. In addition, reduced-space strategies allow to considerably reduce the number of gradient evaluations. 

A much better alternative is to explicitly discretize the DAE model as well as any additional path constraints at a finite number of points, and to use multiple shooting togeihei with an infeasible-path optimization method [4, 5, 8]. To this end, choose a multiple shooting mesh... 

The diffusion matrix can be estimated from a single experiment with good precision using the incremental approach. It should be noted that the four diffusion coefficients are not identifiable from Eq. (5). But the insertion of the constitutive law for the diffusion coefficient (Eq. (3)) into the flux expression allows to overcome this situation. Purthermore, it should be stressed that this estimation problem is very difficult to solve by the simultaneous approach. The Pick matrix is positive definite which is enforced by three inequality constraints (Taylor and Krishna, 1993). In parameter estimation, a sequential approach with an infeasible path optimization routine is often used. This may not be possible since the model cannot be integrated if the matrix is not positive definite. This limitation does not apply to the incremental approach since no solution of the direct problem is required. It could therefore also be used to initialize the simultaneous procedure. 

Other techniques employ a discretization approach whereby the optimal control problem is converted to an NLP through the discretization of all variables. This can be done using the finite difference and orthogonal collocations methods [22, 23]. The characteristic of the discretization method is that the optimization is carried out in the full space of the descretized variables and the discretized constraints are satisfied at the solution of the optimization problem only. This is therefore called the infeasible path approach. Another... 

The other possibility would be to solve fc s, Sj,0) = 0 for Sj as function of sj and thus Sj is no longer a degree of freedom for the Gaufi Newton method. The former type of procedure is often called infeasible path method as for the iterates the constraints are not enforced. This is in contrast to the latter type which belongs to feasible path methods. Feasible path methods are known to converge in general slower than infeasible path methods. 

Additionally, the infeasible path method avoids the elimination of the algebraic variables A in favor of y,0. The information available from measurements for both types of variables can be brought in. Although fcis Sj O) 0 during the solution process the solution of the DAE remains a well-posed problem because the consistent extension of the IVP always leads to consistent initial values for the DAE (7.3.1). 

Value analysis aims at statically determining the contents of the registers and memory cells at each program point and for each execution context. The results of the value analysis are used to predict the addresses of data accesses, computed calls and branches, and to find infeasible paths caused by conditions that always evaluate to true, or always... 

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