## Advanced Methods

The Large-Scale GRG Solver can use Premium Solver Platform's For some smooth nonlinear problems, multistart methods will |

#### Sparse Matrix Storage

Sparse matrix storage methods take advantage of **sparsity in large models**, where subsets of the constraints typically depend on only a small subset of the variables.

For example, the **Jacobian matrix** (the matrix of partial derivatives of the objective and constraints with respect to the variables) for a problem with 2,000 variables and 2,000 constraints would take about 32 megabytes for matrix storage using dense storage methods, but if this problem has the sparsity typical of larger models, it would take as little as 1 to 1.5 megabytes using the sparse storage methods in the Large-Scale GRG Solver.

#### Improved Methods for Numerical Stability

Large nonlinear models require hundreds of thousands to millions of floating-point arithmetic calculations. Because of the finite precision inherent in computer arithmetic, **small numerical errors** occur in these calculations. Nonlinear models are particularly susceptible to the cumulative effect of these errors, which can lead to a numerically unstable or ill-conditioned matrix representation of the problem.The Large-Scale GRG Solver uses several advanced methods to deal with numerical stability, including methods for estimating the **conditioning** of the Jacobian and Hessian matrix, special methods for dealing with **degeneracy**, and advanced methods for selecting **changes in the basis** (which represents the currently binding constraints in the problem).

Thanks to these methods, the Large-Scale GRG Solver can find very good or optimal solutions to problems that could not be solved at all with more primitive nonlinear optimizers.

#### Options for Mixed-Integer Nonlinear Problems

The Large-Scale GRG Solver uses the Premium Platform's You can control the number of Branch and Bound subproblems and the number of integer feasible solutions found before the Solver stops. And you can speed up the solution of problems with integer constraints by supplying an |