Nonlinear convergence problem

The SQP strategy applies the equivalent of a Newton step to the KKT conditions of the nonlinear programming problem, and this leads to a fast rate of convergence. By adding slack variables s, the first-order KKT conditions can be rewritten as... 

These nonlinear equations must be solved simultaneously at each point in time. Usually an iterative method is used and sometimes convergence problems occur. The complexity grows as the number of chemical species increases. 

High-level DAE software (e.g., Dassl) makes a time-step selection based on an estimate of the local truncation error, which depends on the difference between a predictor and a corrector step [13,46]. If the difference is too great, the time step is reduced. In the limit of At 0, the predictor is just the initial condition. For the simple linear problem illustrated here, the corrector will always converge to the correct solution y2 = 1, independent of the time step. However, if the initial condition is y2 1, then there is simply no time step for which the predictor and corrector values will be sufficiently close, and the error estimate will always fail. Based on this simple problem, it may seem like a straightforward task to build software that identifies and avoids the problem, and there is current research on the subject [13], The problem is that in highly nonlinear, coupled, problems the inconsistent initial conditions can be extremely difficult to identify and fix in a general way. 

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. 

One handy way to get all the solutions for several pressure drops (and avoid convergence problems) is to use the Parametric Nonlinear option in the Solver Parameters. Call the variable presdropx and set its value to 0 1 7. Then in the Physics/Subdomain Settings, where you put the Ap, replace it with lO presdropx. The program FEMLAB... 

Han, S. P. (1976), Superlinearly Convergent Variable Metric Algorithms for General Nonlinear Programming Problems, Mathematical Programming, Vol. 11, pp. 263—282. 

A rigorous simulation and optimization of reactive distillation processes usually is based on nonlinear fimctions for a realistic description of the reaction kinetics and the vapor-liquid-equilibrium. Within GAMS models, this description leads to very complex models that often face convergence problems. By using the new so-called external functions, the situation can be improved by transferring calculation procedures to an external module. 

As demonstrated by former publications, the GAMS modeling system has been successfully used for the MINLP-optimization of single reactive distillation columns (Poth et al., 2001 Jackson and Grossmann 2001). The strong nonlinear functions required for a realistic description of the reaction kinetics and the vapor-liquid-equilibrium in these cases lead to very complex GAMS models that may face convergence problems. 

Other limitations of ANN method are the convergence problem and the danger in extrapolation. When the data file exhibits strong noise, the training of ANN usually cannot converge. Since the nonlinear nature of ANN, it is generally accepted that extrapolation is not allowable. 

The goal of an optimization problem is to find a vector p in the search space S so that certain quality criterion is satisfied, namely, the error norm g(p) in Eq. 14 is minimized. The vector p is a solution to the minimization problem if g(p ) is a global minimum in S. For the constrained nonlinear optimization problem associated with the identification of differential hysteresis, the error surface defined by the objective function and constraints can exhibit many local minima or can even be multimodal. For this reason, different solution techniques will have dramatically different performance. The primary consideration in evaluating an optimization algorithm may be convergence speed or the minimum error achieved. Secondary consideration may be consistency, robustness, computational efficiency, or tracking capabilities. 

Due to the nrmlinear dependence of the a priori, unknown residual sequence E on the parameter vector 0, the last equation leads to a nonlinear-weighted least squares problem, which has to be tackled by nrmlinear optimization methods. However, nonUnear least squares techniques are sensitive to the initial parameter values and if no acciuate estimates are available, the nuniniization procedure is very likely to converge to a local minimum. In order to avoid potential inaccurate convergence problems associated with arbitrary initial estimates, initial values for the coefficients of projection may be obtained by identifying conventional ARMA models for each of the K data... 

In this section, the experimental order of convergence of the proposed ADER schemes on two-dimensional linear and nonlinear advection problems are determined numerically in order to compare them with the theoretically expected orders. 

Aspen Plus is used for the steady-state designs of the real chemical systems. Convergence problems can occur because of the difficulty of trying to solve the large set of very nonlinear simultaneous algebraic equations. Another problem is that the current version of... 

Nonlinear circuit problems such as those discussed in these two examples are usually solved in Electrical Engineering by use of the SPICE circuit analysis program. For complicated circuits, this should certainly be the means of solving such problems. This program has built-in models for all standard electronic devices and is very advanced in approaches to achieve convergence. However, at the heart of the SPICE program is an approach very similar to that of the much simpler nsolv() program used here. SPICE will automatically set up the equation set to be solved, but uses first-order linearization and iteration to solve the nonlinear equations just as employed in nsolvQ. While SPICE is the preferred tool for its domain of application, a tool such an nsolv() can be readily embedded into other computer code for specialized solutions to problems not appropriate for an electronic simulation. 

---