Convergence solver

A number of improvements that can be made to the branching rules will accelerate the convergence of this method. A comprehensive discussion of all these options can be found in Nemhauser and Wolsey (1988). Also, a number of efficient MILP codes have recently been developed, including CPLEX, OSL, XPRESS, and ZOOM. All these serve as excellent large-scale optimization codes as well. A detailed description and availability of these and other MILP solvere... 

SLP Widely used in practice Rapid convergence when optimum is at a vertex Can handle very large problems Does not attempt to satisfy equalities at each iteration Can benefit from improvements to LP solvers May converge slowly on problems with nonvertex optima Will usually violate nonlinear constraints until convergence to optimum, often by large amounts... 

This material is based on work supported by NSF grant 9811143 and CAMIRO grant 96E01, with additional support from Noranda Exploration. Paul Manhardt provided solver core maintenance and coding support, and Jenn Adams was essential to the convergence of the simulations. 

Within the above scheme, we implemented the generalized minimal residual (GM-RES) method [52], which is a robust linear solver that ensures convergence of the iterative solution. 

The time step Tstep = lOu determines each point in time starting from zero that the transient solver will calculate a solution. A safe estimation of the time step is an order of magnitude less than the period of a switching waveform. For example, the time step for a 100 kHz oscillator (period = 10 /xs) should be approximately 1 /xs. Tmax, the maximum time step, can be left out (at default) or specified to increase (decrease TMAX) or decrease (increase TMAX) simulation accuracy. This allows the simulator to take larger steps when the voltage levels in the circuit experience little change. A transient time domain analysis can prove to be the most difficult to get to converge. 

Using Jacobi s method to compute the inverse of the Laplacian is rather slow. Faster convergence may be achieved using successive over-relaxation (SOR) (Bronstein et al. 2001 Demmel 1996). The iterative solver can also be written in the Gauss-Seidel formulation where already computed results are reused. 

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. 

In sum, the greatest virtues of CG methods are their modest storage and computational requirements (both order n) and their better convergence than the SD method. These properties have made them popular linear solvers and minimization choices in many applications18-20 84-88 and perhaps the only candidates for very large problems. The linear CG is often applied to systems arising from discretizations of partial differential equations,81 89 90 where the matrices are frequently positive-definite, sparse, and structured. 

Carry out the least-squares minimization of the quantity in Eq. (7) according to an appropriate algorithm (presumably normal equations if the observational equations are linear in the parameters to be determined otherwise some other such as Marquardf s ). The linear regression and Solver operations in spreadsheets are especially useful (see Chapter HI). Convergence should not be assumed in the nonlinear case until successive cycles produce no significant change in any of the parameters. 

---