Optimization Toolbox

Problems Covered by Large-Scale Methods

Not all possible problem formulations are covered by the large-scale algorithms. The following table describes what functionality is covered by the large-scale methods. For example, for fmincon, the large-scale algorithm covers the case where there are only bound constraints or only linear equalities. For each problem formulation, the third column describes what additional information is needed for the large-scale algorithms. For fminunc and fmincon, the gradient must be computed along with the objective in the user-supplied function (the gradient is not required for the medium-scale algorithms).

Since these methods can also be used on small- to medium-scale problems that are not necessarily sparse, the last column of the table emphasizes what conditions are needed for large-scale problems to run efficiently without exceeding your computer system's memory capabilities, e.g., the linear constraint matrices should be sparse. For smaller problems the conditions in the last column are unnecessary.

Note    The following table lists the functions in order of increasing problem complexity.

Several examples, which follow this table, clarify the contents of the table.

Table 2-4: Large-Scale Problem Coverage and Requirements 

Function	Problem Formulations	Additional Information Needed	For Large Problems
fminunc		Must provide gradient for f(x) in fun.	Provide sparsity structure of the Hessian, or compute the Hessian in fun. The Hessian should be sparse.
fmincon	such that where such that , and is an m-by-n matrix where	Must provide gradient for f(x) in fun.	Provide sparsity structure of the Hessian or compute the Hessian in fun. The Hessian should be sparse. should be sparse.
lsqnonlin	such that where F(x) must be overdetermined (have at least as many equations as variables).	None	Provide sparsity structure of the Jacobian or compute the Jacobian in fun. The Jacobian should be sparse.
lsqcurvefit	such that where must be overdetermined (have at least as many equations as variables).	None	Provide sparsity structure of the Jacobian or compute the Jacobian in fun. The Jacobian should be sparse.
fsolve	must have the same number of equations as variables.	None	Provide sparsity structure of the Jacobian or compute the Jacobian in fun. The Jacobian should be sparse.
lsqlin	such that where is an m-by-n matrix where i.e., the problem must be overdetermined.	None	should be sparse.
linprog	such that and , where	None	and should be sparse.
quadprog	such that where such that , and is an m-by-n matrix where	None	should be sparse. should be sparse.

In the following examples, many of the M-file functions are available in the Optimization Toolbox optim directory. Most of these do not have a fixed problem size, i.e., the size of your starting point xstart determines the size problem that is computed. If your computer system cannot handle the size suggested in the examples below, use a smaller-dimension start point to run the problems. If the problems have upper or lower bounds or equalities, you must adjust the size of those vectors or matrices as well.

Large-Scale Examples

Nonlinear Equations with Jacobian