Optimization Toolbox

Large-Scale Examples

Some of the optimization functions include algorithms for continuous optimization problems especially targeted to large problems with sparsity or structure. The main large-scale algorithms are iterative, i.e., a sequence of approximate solutions is generated. In each iteration a linear system is (approximately) solved. The linear systems are solved using the sparse matrix capabilities of MATLAB and a variety of sparse linear solution techniques, both iterative and direct.

Generally speaking, the large-scale optimization methods preserve structure and sparsity, using exact derivative information wherever possible. To solve the large-scale problems efficiently, some problem formulations are restricted (such as only solving overdetermined linear or nonlinear systems), or require additional information (e.g., the nonlinear minimization algorithm requires that the gradient be computed in the user-supplied function).

This section summarizes the kinds of problems covered by large-scale methods and provides these examples:

• Nonlinear Equations with Jacobian
• Nonlinear Equations with Jacobian Sparsity Pattern
• Nonlinear Least-Squares with Full Jacobian Sparsity Pattern
• Nonlinear Minimization with Gradient and Hessian
• Nonlinear Minimization with Gradient and Hessian Sparsity Pattern
• Nonlinear Minimization with Bound Constraints and Banded Preconditioner
• Nonlinear Minimization with Equality Constraints
• Nonlinear Minimization with a Dense but Structured Hessian and Equality Constraints
• Quadratic Minimization with Bound Constraints
• Quadratic Minimization with a Dense but Structured Hessian
• Linear Least-Squares with Bound Constraints
• Linear Programming with Equalities and Inequalities
• Linear Programming with Dense Columns in the Equalities

Multiobjective Examples

Problems Covered by Large-Scale Methods